Mathematics & Statistics Theseshttp://hdl.handle.net/10023/61292020-08-08T03:04:21Z2020-08-08T03:04:21ZNonlinear partial differential equations on fractalsHu, Jiaxinhttp://hdl.handle.net/10023/151802020-07-08T10:10:43Z2001-01-01T00:00:00ZThe study of nonlinear partial differential equations on fractals is a burgeoning inter-disciplinary topic, allowing dynamic properties on fractals to be investigated. In this thesis we will investigate nonlinear PDEs of three basic types on bounded and unbounded fractals. We first review the definition of post-critically finite (p.c.f.) self-similar fractals with regular harmonic structure. A Dirichlet form exists on such a fractal; thus we may define a weak version of the Laplacian. The Sobolev-type inequality, established on p.c.f. self-similar fractals satisfying the separation condition, plays a crucial role in the analysis of PDEs on p.c.f. self-similar fractals. We use the classical approach to study the linear eigenvalue problem on p.c.f. self-similar fractals, which depends on the Sobolev-type inequality. Fundamental solutions such as Green's function, wave propagator and heat kernel are then explicitly expressed in terms of eigenvalues and eigenfunctions. The main aim of the thesis is to study nonlinear PDEs on fractals. We begin with nonlinear elliptic equations on p.c.f. self-similar fractals. We prove the existence of non-trivial solutions to elliptic equations with zero Dirichlet boundary conditions using the mountain pass theorem and the saddle point theorem. For nonlinear wave equations on p.c.f. self-similar fractals, we show the existence of global solutions for appropriate initial and boundary data. We also examine blow up at finite time which may occur for certain initial data. Finally, we consider nonlinear diffusion equations on p.c.f. self-similar fractals and unbounded fractals. Using the upper-lower solution technique, we prove the global existence of solutions of the nonlinear diffusion equation with initial value and boundary conditions on p.c.f. self-similar fractals. For unbounded fractals, starting with a heat kernel satisfying certain assumptions, we prove that the diffusion equation with a nonlinear term of the form uᵖ possesses a global solution if the initial data is small and p > 1 + ds/2, while solutions blow up if p ≤ 1 + ds/2 even for small initial data, where dg is the spectral dimension of the fractal. We investigate smoothness and Holder continuity of solutions when they exist.
2001-01-01T00:00:00ZHu, JiaxinThe study of nonlinear partial differential equations on fractals is a burgeoning inter-disciplinary topic, allowing dynamic properties on fractals to be investigated. In this thesis we will investigate nonlinear PDEs of three basic types on bounded and unbounded fractals. We first review the definition of post-critically finite (p.c.f.) self-similar fractals with regular harmonic structure. A Dirichlet form exists on such a fractal; thus we may define a weak version of the Laplacian. The Sobolev-type inequality, established on p.c.f. self-similar fractals satisfying the separation condition, plays a crucial role in the analysis of PDEs on p.c.f. self-similar fractals. We use the classical approach to study the linear eigenvalue problem on p.c.f. self-similar fractals, which depends on the Sobolev-type inequality. Fundamental solutions such as Green's function, wave propagator and heat kernel are then explicitly expressed in terms of eigenvalues and eigenfunctions. The main aim of the thesis is to study nonlinear PDEs on fractals. We begin with nonlinear elliptic equations on p.c.f. self-similar fractals. We prove the existence of non-trivial solutions to elliptic equations with zero Dirichlet boundary conditions using the mountain pass theorem and the saddle point theorem. For nonlinear wave equations on p.c.f. self-similar fractals, we show the existence of global solutions for appropriate initial and boundary data. We also examine blow up at finite time which may occur for certain initial data. Finally, we consider nonlinear diffusion equations on p.c.f. self-similar fractals and unbounded fractals. Using the upper-lower solution technique, we prove the global existence of solutions of the nonlinear diffusion equation with initial value and boundary conditions on p.c.f. self-similar fractals. For unbounded fractals, starting with a heat kernel satisfying certain assumptions, we prove that the diffusion equation with a nonlinear term of the form uᵖ possesses a global solution if the initial data is small and p > 1 + ds/2, while solutions blow up if p ≤ 1 + ds/2 even for small initial data, where dg is the spectral dimension of the fractal. We investigate smoothness and Holder continuity of solutions when they exist.Adaptive distance samplingPollard, Johnhttp://hdl.handle.net/10023/151762019-04-01T08:37:19Z2002-01-01T00:00:00ZWe investigate mechanisms to improve efficiency for line and point transect surveys of clustered populations by combining the distance methods with adaptive sampling. In adaptive sampling, survey effort is increased when areas of high animal density are located, thereby increasing the number of observations. We begin by building on existing adaptive sampling techniques, to create both point and line transect adaptive estimators, these are then extended to allow the inclusion of covariates in the detection function estimator. However, the methods are limited, as the total effort required cannot be forecast at the start of a survey, and so a new fixed total effort adaptive approach is developed. A key difference in the new method is that it does not require the calculation of the inclusion probabilities typically used by existing adaptive estimators. The fixed effort method is primarily aimed at line transect sampling, but point transect derivations are also provided. We evaluate the new methodology by computer simulation, and report on surveys of harbour porpoise in the Gulf of Maine, in which the approach was compared with conventional line transect sampling. Line transect simulation results for a clustered population showed up to a 6% improvement in the adaptive density variance estimate over the conventional, whilst when there was no clustering the adaptive estimate was 1% less efficient than the conventional. For the harbour porpoise survey, the adaptive density estimate cvs showed improvements of 8% for individual porpoise density and 14% for school density over the conventional estimates. The primary benefit of the fixed effort method is the potential to improve survey coverage, allowing a survey to complete within a fixed time and effort; an important feature if expensive survey resources are involved, such as an aircraft, crew and observers.
2002-01-01T00:00:00ZPollard, JohnWe investigate mechanisms to improve efficiency for line and point transect surveys of clustered populations by combining the distance methods with adaptive sampling. In adaptive sampling, survey effort is increased when areas of high animal density are located, thereby increasing the number of observations. We begin by building on existing adaptive sampling techniques, to create both point and line transect adaptive estimators, these are then extended to allow the inclusion of covariates in the detection function estimator. However, the methods are limited, as the total effort required cannot be forecast at the start of a survey, and so a new fixed total effort adaptive approach is developed. A key difference in the new method is that it does not require the calculation of the inclusion probabilities typically used by existing adaptive estimators. The fixed effort method is primarily aimed at line transect sampling, but point transect derivations are also provided. We evaluate the new methodology by computer simulation, and report on surveys of harbour porpoise in the Gulf of Maine, in which the approach was compared with conventional line transect sampling. Line transect simulation results for a clustered population showed up to a 6% improvement in the adaptive density variance estimate over the conventional, whilst when there was no clustering the adaptive estimate was 1% less efficient than the conventional. For the harbour porpoise survey, the adaptive density estimate cvs showed improvements of 8% for individual porpoise density and 14% for school density over the conventional estimates. The primary benefit of the fixed effort method is the potential to improve survey coverage, allowing a survey to complete within a fixed time and effort; an important feature if expensive survey resources are involved, such as an aircraft, crew and observers.Contractive Markov systemsWerner, Ivanhttp://hdl.handle.net/10023/151732019-04-01T08:38:03Z2004-01-01T00:00:00ZWe introduce a theory of contractive Markov systems (CMS) which provides a unifying framework in so-called "fractal" geometry. It extends the known theory of iterated function systems (IFS) with place dependent probabilities [1][8] in a way that it also covers graph directed constructions of "fractal" sets [18]. Such systems naturally extend finite Markov chains and inherit some of their properties. In Chapter 1, we consider iterations of a Markov system and show that they preserve the essential structure of it. In Chapter 2, we show that the Markov operator defined by such a system has a unique invariant probability measure in the irreducible case and an attractive probability measure in the aperiodic case if the restrictions of the probability functions on their vertex sets are Dini-continuous and bounded away from zero, and the system satisfies a condition of a contractiveness on average. This generalizes a result from [1]. Furthermore, we show that the rate of convergence to the stationary state is exponential in the aperiodic case with constant probabilities and a compact state space. In Chapter 3, we construct a coding map for a contractive Markov system. In Chapter 4, we calculate Kolmogorov-Sinai entropy of the generalized Markov shift. In Chapter 5, we prove an ergodic theorem for Markov chains associated with the contractive Markov systems. It generalizes the ergodic theorem of Elton [8].
2004-01-01T00:00:00ZWerner, IvanWe introduce a theory of contractive Markov systems (CMS) which provides a unifying framework in so-called "fractal" geometry. It extends the known theory of iterated function systems (IFS) with place dependent probabilities [1][8] in a way that it also covers graph directed constructions of "fractal" sets [18]. Such systems naturally extend finite Markov chains and inherit some of their properties. In Chapter 1, we consider iterations of a Markov system and show that they preserve the essential structure of it. In Chapter 2, we show that the Markov operator defined by such a system has a unique invariant probability measure in the irreducible case and an attractive probability measure in the aperiodic case if the restrictions of the probability functions on their vertex sets are Dini-continuous and bounded away from zero, and the system satisfies a condition of a contractiveness on average. This generalizes a result from [1]. Furthermore, we show that the rate of convergence to the stationary state is exponential in the aperiodic case with constant probabilities and a compact state space. In Chapter 3, we construct a coding map for a contractive Markov system. In Chapter 4, we calculate Kolmogorov-Sinai entropy of the generalized Markov shift. In Chapter 5, we prove an ergodic theorem for Markov chains associated with the contractive Markov systems. It generalizes the ergodic theorem of Elton [8].Finding "small' matrices P,Q such that PDQ = SWainwright, Robert J.http://hdl.handle.net/10023/151712019-04-01T08:38:52Z2002-01-01T00:00:00ZGiven an integer matrix A, there is a unique matrix S of a particular form, called the Smith Normal Form, and non-unique unimodular matrices P and Q such that PAQ = S. It is often the case that these matrices P and Q will be used for further calculation, and as such it is desirable to find P and Q with small entries. In this thesis we address the problem of finding such P and Q with small entries, in particular in the case where A is a diagonal matrix, which arises as a final step in many published algorithms. Heuristic algorithms are developed which appear to do well in practice and some theory is developed to explain this behaviour. We also give an account of the implementation of an alternative algorithm which bypasses this intermediary diagonal form. The basic theoretical development of this is work by Storjohan.
2002-01-01T00:00:00ZWainwright, Robert J.Given an integer matrix A, there is a unique matrix S of a particular form, called the Smith Normal Form, and non-unique unimodular matrices P and Q such that PAQ = S. It is often the case that these matrices P and Q will be used for further calculation, and as such it is desirable to find P and Q with small entries. In this thesis we address the problem of finding such P and Q with small entries, in particular in the case where A is a diagonal matrix, which arises as a final step in many published algorithms. Heuristic algorithms are developed which appear to do well in practice and some theory is developed to explain this behaviour. We also give an account of the implementation of an alternative algorithm which bypasses this intermediary diagonal form. The basic theoretical development of this is work by Storjohan.Automatic S-acts and inverse semigroup presentationsDombi, Erzsebet Ritahttp://hdl.handle.net/10023/151232019-04-01T08:38:02Z2005-01-01T00:00:00ZTo provide a general framework for the theory of automatic groups and semigroups, we introduce the notion of an automatic semigroup act. This notion gives rise to a variety of definitions for automaticity depending on the set chosen as a semigroup act. Namely, we obtain the notions of automaticity, Schutzenberger automaticity, R- and L-class automaticity, etc. We discuss the basic properties of automatic semigroup acts. We show that if S is a semigroup with local right identities, then automaticty of a semigroup act is independent of the choice of both the generators of S and the generators of the semigroup act. We also discuss the equality problem of automatic semigroup acts. To give a geometric approach, we associate a directed labelled graph to each S-act and introduce the notion of the fellow traveller property in the associated graph. We verify that if S is a regular semigroup with finitely many idempotents, then Schutzenberger automaticity is characterized by the fellow traveller property of the Schutzenberger graph. We also verify that a Schutzenberger automatic regular semigroup with finitely many idempotents is finitely presented. We end Chapter 3 by proving that an inverse free product of Schutzenberger automatic inverse semigroups is Schutzenberger automatic. In Chapter 4, we first introduce the notion of finite generation and finite presentability with respect to a semigroup action. With the help of these concepts we give a necessary and sufficient condition for a semidirect product of a semilattice by a group to be finitely generated and finitely presented as an inverse semigroup. We end Chapter 4 by giving a necessary and sufficient condition for the semidirect product of a semilattice by a group to be Schutzenberger automatic. Chapter 5 is devoted to the study of HNN extensions of inverse semigroups from finite generation and finite presentability point of view. Namely, we give necessary and sufficient conditions for finite presentability of Gilbert's and Yamamura's HNN extension of inverse semigroups. The majority of the results contained in Chapter 5 are the result of a joint work with N.D. Gilbert and N. Ruskuc.
2005-01-01T00:00:00ZDombi, Erzsebet RitaTo provide a general framework for the theory of automatic groups and semigroups, we introduce the notion of an automatic semigroup act. This notion gives rise to a variety of definitions for automaticity depending on the set chosen as a semigroup act. Namely, we obtain the notions of automaticity, Schutzenberger automaticity, R- and L-class automaticity, etc. We discuss the basic properties of automatic semigroup acts. We show that if S is a semigroup with local right identities, then automaticty of a semigroup act is independent of the choice of both the generators of S and the generators of the semigroup act. We also discuss the equality problem of automatic semigroup acts. To give a geometric approach, we associate a directed labelled graph to each S-act and introduce the notion of the fellow traveller property in the associated graph. We verify that if S is a regular semigroup with finitely many idempotents, then Schutzenberger automaticity is characterized by the fellow traveller property of the Schutzenberger graph. We also verify that a Schutzenberger automatic regular semigroup with finitely many idempotents is finitely presented. We end Chapter 3 by proving that an inverse free product of Schutzenberger automatic inverse semigroups is Schutzenberger automatic. In Chapter 4, we first introduce the notion of finite generation and finite presentability with respect to a semigroup action. With the help of these concepts we give a necessary and sufficient condition for a semidirect product of a semilattice by a group to be finitely generated and finitely presented as an inverse semigroup. We end Chapter 4 by giving a necessary and sufficient condition for the semidirect product of a semilattice by a group to be Schutzenberger automatic. Chapter 5 is devoted to the study of HNN extensions of inverse semigroups from finite generation and finite presentability point of view. Namely, we give necessary and sufficient conditions for finite presentability of Gilbert's and Yamamura's HNN extension of inverse semigroups. The majority of the results contained in Chapter 5 are the result of a joint work with N.D. Gilbert and N. Ruskuc.Automatic semigroups : constructions and subsemigroupsDescalco, L.http://hdl.handle.net/10023/151222019-04-01T08:38:10Z2002-01-01T00:00:00ZIn this thesis we start by considering conditions under which some standard semigroup constructions preserve automaticity. We first consider Rees matrix semigroups over a semigroup, which we call the base, and work on the following questions: (i) If the base is automatic is the Rees matrix semigroup automatic? (ii) If the Rees matrix semigroup is automatic must the base be automatic as well? We also consider similar questions for Bruck-Reilly extensions of monoids and wreath products of semigroups. Then we consider subsemigroups of free products of semigroups and we study conditions that guarantee them to be automatic. Finally we obtain a description of the subsemigroups of the bicyclic monoid that allow us to study some of their properties, which include finite generation, automaticity and finite presentability.
2002-01-01T00:00:00ZDescalco, L.In this thesis we start by considering conditions under which some standard semigroup constructions preserve automaticity. We first consider Rees matrix semigroups over a semigroup, which we call the base, and work on the following questions: (i) If the base is automatic is the Rees matrix semigroup automatic? (ii) If the Rees matrix semigroup is automatic must the base be automatic as well? We also consider similar questions for Bruck-Reilly extensions of monoids and wreath products of semigroups. Then we consider subsemigroups of free products of semigroups and we study conditions that guarantee them to be automatic. Finally we obtain a description of the subsemigroups of the bicyclic monoid that allow us to study some of their properties, which include finite generation, automaticity and finite presentability.Presentations for subsemigroups of groupsCain, Alan Jameshttp://hdl.handle.net/10023/151192019-04-01T08:37:43Z2005-01-01T00:00:00ZThis thesis studies subsemigroups of groups from three perspectives: automatic structures, ordinary semigroup presentations, and Malcev presentaions. [A Malcev presentation is a presentation of a special type for a semigroup that can be embedded into a group. A group-embeddable semigroup is Malcev coherent if all of its finitely generated subsemigroups admit finite Malcev presentations.] The theory of synchronous and asynchronous automatic structures for semigroups is expounded, particularly for group-embeddable semigroups. In particular, automatic semigroups embeddable into groups are shown to inherit many of the pleasant geometric properties of automatic groups. It is proved that group- embeddable automatic semigroups admit finite Malcev presentations, and such presentations can be found effectively. An algorithm is exhibited to test whether an automatic semigroup is a free semigroup. Cancellativity of automatic semigroups is proved to be undecidable. Study is made of several classes of groups: virtually free groups; groups that satisfy semigroup laws (in particular [virtually] nilpotent and [virtually] abelian groups); polycyclic groups; free and direct products of certain groups; and one-relator groups. For each of these classes, the question of Malcev coherence is considered, together with the problems of whether finitely generated subsemigroups are finitely presented or automatic. This study yields closure and containment results regarding the class of Malcev coherent groups. The property of having a finite Malcev presentation is shown to be preserved under finite Rees index extensions and subsemigroups. Other concepts of index are also studied.
2005-01-01T00:00:00ZCain, Alan JamesThis thesis studies subsemigroups of groups from three perspectives: automatic structures, ordinary semigroup presentations, and Malcev presentaions. [A Malcev presentation is a presentation of a special type for a semigroup that can be embedded into a group. A group-embeddable semigroup is Malcev coherent if all of its finitely generated subsemigroups admit finite Malcev presentations.] The theory of synchronous and asynchronous automatic structures for semigroups is expounded, particularly for group-embeddable semigroups. In particular, automatic semigroups embeddable into groups are shown to inherit many of the pleasant geometric properties of automatic groups. It is proved that group- embeddable automatic semigroups admit finite Malcev presentations, and such presentations can be found effectively. An algorithm is exhibited to test whether an automatic semigroup is a free semigroup. Cancellativity of automatic semigroups is proved to be undecidable. Study is made of several classes of groups: virtually free groups; groups that satisfy semigroup laws (in particular [virtually] nilpotent and [virtually] abelian groups); polycyclic groups; free and direct products of certain groups; and one-relator groups. For each of these classes, the question of Malcev coherence is considered, together with the problems of whether finitely generated subsemigroups are finitely presented or automatic. This study yields closure and containment results regarding the class of Malcev coherent groups. The property of having a finite Malcev presentation is shown to be preserved under finite Rees index extensions and subsemigroups. Other concepts of index are also studied.Finiteness conditions of wreath products of semigroups and related properties of diagonal actsThomson, Michael R.http://hdl.handle.net/10023/151172019-04-01T08:39:24Z2001-01-01T00:00:00ZThe purpose of this thesis is to consider finite generation, finite presentability and related properties of restricted wreath products of semigroups. We show that the wreath product Awr B of two monoids is finitely generated if and only if A and B are finitely generated and the action by right multiplication on B of the group of units of B has only finitely many orbits. Also we show that the wreath product AwrB of two non-trivial monoids is finitely presented if and only if A is finitely presented and B is finite. The situation is more complicated in the case of the wreath product SₑwrT of two semigroups with respect to an idempotent e ϵ S. We give a complete characterization for finite generation in the case where T is finite. This result depends on the properties of the diagonal action of S on S x S. We also prove that if this action is not finitely generated, then SₑwrT (with S infinite and T finite) is finitely presented if and only if S x S is finitely presented and T is the direct product of a monoid and a left zero semigroup. In the case where T is infinite, we prove that S must be a monoid in order for SwrT to be finitely generated. We show that the finiteness properties of periodicity and local finiteness are preserved under the wreath product construction. We conclude the thesis with a systematic investigation into the properties of diagonal acts of semigroups, and make some interesting connections between diagonal acts and power semigroups.
2001-01-01T00:00:00ZThomson, Michael R.The purpose of this thesis is to consider finite generation, finite presentability and related properties of restricted wreath products of semigroups. We show that the wreath product Awr B of two monoids is finitely generated if and only if A and B are finitely generated and the action by right multiplication on B of the group of units of B has only finitely many orbits. Also we show that the wreath product AwrB of two non-trivial monoids is finitely presented if and only if A is finitely presented and B is finite. The situation is more complicated in the case of the wreath product SₑwrT of two semigroups with respect to an idempotent e ϵ S. We give a complete characterization for finite generation in the case where T is finite. This result depends on the properties of the diagonal action of S on S x S. We also prove that if this action is not finitely generated, then SₑwrT (with S infinite and T finite) is finitely presented if and only if S x S is finitely presented and T is the direct product of a monoid and a left zero semigroup. In the case where T is infinite, we prove that S must be a monoid in order for SwrT to be finitely generated. We show that the finiteness properties of periodicity and local finiteness are preserved under the wreath product construction. We conclude the thesis with a systematic investigation into the properties of diagonal acts of semigroups, and make some interesting connections between diagonal acts and power semigroups.Todd-Coxeter methods for inverse monoidsCutting, Andrewhttp://hdl.handle.net/10023/150522019-04-01T08:37:42Z2001-01-01T00:00:00ZLet P be the inverse monoid presentation (X|U) for the inverse monoid M, let π be the set of generators for a right congruence on M and let u Є M. Using the work of J. Stephen [15], the current work demonstrates a coset enumeration technique for the R-class Rᵤ similar to the coset enumeration algorithm developed by J. A. Todd and H. S. M. Coxeter for groups. Furthermore it is demonstrated how to test whether Rᵤ = Rᵥ, for u, v Є M and so a technique for enumerating inverse monoids is described. This technique is generalised to enumerate the H-classes of M. The algorithms have been implemented in GAP 3.4.4 [25], and have been used to analyse some examples given in Chapter 6. The thesis concludes by a related discussion of normal forms and automaticity of free inverse semigroups.
2001-01-01T00:00:00ZCutting, AndrewLet P be the inverse monoid presentation (X|U) for the inverse monoid M, let π be the set of generators for a right congruence on M and let u Є M. Using the work of J. Stephen [15], the current work demonstrates a coset enumeration technique for the R-class Rᵤ similar to the coset enumeration algorithm developed by J. A. Todd and H. S. M. Coxeter for groups. Furthermore it is demonstrated how to test whether Rᵤ = Rᵥ, for u, v Є M and so a technique for enumerating inverse monoids is described. This technique is generalised to enumerate the H-classes of M. The algorithms have been implemented in GAP 3.4.4 [25], and have been used to analyse some examples given in Chapter 6. The thesis concludes by a related discussion of normal forms and automaticity of free inverse semigroups.Fibonacci length and efficiency in group presentationsCampbell, Peter P.http://hdl.handle.net/10023/150482019-04-01T08:39:25Z2003-01-01T00:00:00ZIn this thesis we shall consider two topics that are contained in combinatorial group theory and concern properties of finitely presented groups. The first problem we examine is that of calculating the Fibonacci length of certain families of finitely presented groups. In pursuing this we come across ideas and unsolved problems from number theory. We mainly concentrate on finding the Fibonacci length of powers of dihedral groups, certain Fibonacci groups and a family of metacyclic groups. The second problem we investigate in this thesis is finding if the group PGL(2, p), for p a prime, is efficient on a minimal generating set. We find various presentations that define PGL(2,p) or C₂ x PSL(2,p) and direct products of these groups. As in the previous sections we come across number theoretic problems. We also have occasion to use results from tensor theory and homological algebra in order to obtain our results.
2003-01-01T00:00:00ZCampbell, Peter P.In this thesis we shall consider two topics that are contained in combinatorial group theory and concern properties of finitely presented groups. The first problem we examine is that of calculating the Fibonacci length of certain families of finitely presented groups. In pursuing this we come across ideas and unsolved problems from number theory. We mainly concentrate on finding the Fibonacci length of powers of dihedral groups, certain Fibonacci groups and a family of metacyclic groups. The second problem we investigate in this thesis is finding if the group PGL(2, p), for p a prime, is efficient on a minimal generating set. We find various presentations that define PGL(2,p) or C₂ x PSL(2,p) and direct products of these groups. As in the previous sections we come across number theoretic problems. We also have occasion to use results from tensor theory and homological algebra in order to obtain our results.Automated theorem proving for mathematics : real analysis in PVSGottliebsen, Hannehttp://hdl.handle.net/10023/150462019-04-01T08:37:27Z2002-01-01T00:00:00ZComputer Algebra Systems (CASs), such as Maple and Mathematica, are now widely used in both industry and education. In many areas of mathematics they perform well. However, many well-established methods in mathematics, such as definite integration via the fundamental theorem of calculus, rely on analytic side conditions which CASs in general do not support. This thesis presents our work with automatic, formal mathematics using the theorem prover PVS. Based on an existing real analysis library for PVS, we have implemented transcendental functions such as exp, cos, sin, tan and their inverses, and we have provided strategies to prove that a function is continuous at a given point. In general, this is undecidable, but using certain restrictions we can still provide proofs for a large collection of functions. Similarly, we can prove that a function has a limit at a point. We illustrate how the extended library may be used with Maple to provide correct results where Maple's are incorrect. We present a case study of definite integration in the CASs axiom. Maple, Mathematica and Matlab. The case study clearly shows that apart from axiom the systems do not fully check the necessary conditions for the definite integral to exist, thus giving results varying from plain incorrect to correct, even if the latter is difficult to detect without manipulating the result. The extension and correction of the PVS library consists of around 1000 theorems proven by around 18000 PVS proof commands. We also have a test suite of 88 lemmas for the automatic checks for continuity and existence of limits. Thus we have devised and tested automatic computational logic support for the use of formal mathematics in applications, particularly computer algebra.
2002-01-01T00:00:00ZGottliebsen, HanneComputer Algebra Systems (CASs), such as Maple and Mathematica, are now widely used in both industry and education. In many areas of mathematics they perform well. However, many well-established methods in mathematics, such as definite integration via the fundamental theorem of calculus, rely on analytic side conditions which CASs in general do not support. This thesis presents our work with automatic, formal mathematics using the theorem prover PVS. Based on an existing real analysis library for PVS, we have implemented transcendental functions such as exp, cos, sin, tan and their inverses, and we have provided strategies to prove that a function is continuous at a given point. In general, this is undecidable, but using certain restrictions we can still provide proofs for a large collection of functions. Similarly, we can prove that a function has a limit at a point. We illustrate how the extended library may be used with Maple to provide correct results where Maple's are incorrect. We present a case study of definite integration in the CASs axiom. Maple, Mathematica and Matlab. The case study clearly shows that apart from axiom the systems do not fully check the necessary conditions for the definite integral to exist, thus giving results varying from plain incorrect to correct, even if the latter is difficult to detect without manipulating the result. The extension and correction of the PVS library consists of around 1000 theorems proven by around 18000 PVS proof commands. We also have a test suite of 88 lemmas for the automatic checks for continuity and existence of limits. Thus we have devised and tested automatic computational logic support for the use of formal mathematics in applications, particularly computer algebra.Photometry of star clustersEvans, Thomas Harry Hope Lloydhttp://hdl.handle.net/10023/142832019-04-01T08:37:25Z1968-01-01T00:00:00ZThe suitability of the Cassegrain Schmidt telescopes at St. Andrews University Observatory for the measurement of stellar magnitudes and colours by in-focus multicolour photography has been examined. A major requirement is that the photographic plate should coincide with the focal surface. Thermal effects in the Scott Lang Telescope and optical and mechanical problems in the James Gregory Telescope cause difficulty in, attaining this. These difficulties have been overcome in the case of the Scott Lang Telescope but no certain method for focussing the James Gregory Telescope was found. The photometric field limited by field error, is approximately one degree in diameter in each case. Colour equations between the instrumental and standard B, V systems depend on magnitude and, in the case of the Scott Lang Telescope, on exposure time as well. The methods used to measure UBV magnitudes and colours with the Radcliffe 74-inch reflector are described and the accuracy of the results discussed. Magnitudes and colours of stars brighter than V - 15.5 in the open cluster IC 2581 have been measured, together with MK spectral types for a few of the brighter stars. The interstellar absorption provides a criterion for the recognition of cluster members. A discrepancy between the shape of the cluster main sequence and that of the zero age main sequence is attributed to an error in the derivation of the standard zero age main sequence. The cluster is found to be at a distance of 2500 parsecs and may form part of the Carinae complex. The positions of the brightest stars in the colour magnitude diagram are discussed in the light of modern theories of stellar evolution and an age of approximately 10 million years is deduced. The colour magnitude diagram of the open cluster NGC 6383 has been obtained for stars brighter than V - 18.1; the limiting magnitudes in B and U are 19.7 and 17.9, respectively. MK spectral types have permitted the cluster membership of several bright B stars to be established; some stars of later type are non-members. The observation of this cluster are more complete than for most young clusters studied to date, but the poorness of the cluster and the unfavourable distribution of interstellar absorption with distance make it impossible to be certain of the membership of stars fainter than V = 13. The lack of stars fainter than V = 12.8 on the zero age main sequence indicates a contraction age of 5 million years. The distance is 1300 parsecs, like those of other young groups in the vicinity. The dense dust clouds which divide the Milky Way in Scorpius are immediately beyond this. Several faint variable stars may be of the T Tauri type.
1968-01-01T00:00:00ZEvans, Thomas Harry Hope LloydThe suitability of the Cassegrain Schmidt telescopes at St. Andrews University Observatory for the measurement of stellar magnitudes and colours by in-focus multicolour photography has been examined. A major requirement is that the photographic plate should coincide with the focal surface. Thermal effects in the Scott Lang Telescope and optical and mechanical problems in the James Gregory Telescope cause difficulty in, attaining this. These difficulties have been overcome in the case of the Scott Lang Telescope but no certain method for focussing the James Gregory Telescope was found. The photometric field limited by field error, is approximately one degree in diameter in each case. Colour equations between the instrumental and standard B, V systems depend on magnitude and, in the case of the Scott Lang Telescope, on exposure time as well. The methods used to measure UBV magnitudes and colours with the Radcliffe 74-inch reflector are described and the accuracy of the results discussed. Magnitudes and colours of stars brighter than V - 15.5 in the open cluster IC 2581 have been measured, together with MK spectral types for a few of the brighter stars. The interstellar absorption provides a criterion for the recognition of cluster members. A discrepancy between the shape of the cluster main sequence and that of the zero age main sequence is attributed to an error in the derivation of the standard zero age main sequence. The cluster is found to be at a distance of 2500 parsecs and may form part of the Carinae complex. The positions of the brightest stars in the colour magnitude diagram are discussed in the light of modern theories of stellar evolution and an age of approximately 10 million years is deduced. The colour magnitude diagram of the open cluster NGC 6383 has been obtained for stars brighter than V - 18.1; the limiting magnitudes in B and U are 19.7 and 17.9, respectively. MK spectral types have permitted the cluster membership of several bright B stars to be established; some stars of later type are non-members. The observation of this cluster are more complete than for most young clusters studied to date, but the poorness of the cluster and the unfavourable distribution of interstellar absorption with distance make it impossible to be certain of the membership of stars fainter than V = 13. The lack of stars fainter than V = 12.8 on the zero age main sequence indicates a contraction age of 5 million years. The distance is 1300 parsecs, like those of other young groups in the vicinity. The dense dust clouds which divide the Milky Way in Scorpius are immediately beyond this. Several faint variable stars may be of the T Tauri type.Solar intense magnetic fieldsWebb, Andrew Roberthttp://hdl.handle.net/10023/142772019-04-01T08:37:55Z1980-01-01T00:00:00ZThe nature of motions in intense magnetic fields is investigated. For a flux tube in a uniform atmosphere a dispersion relation is derived for the modes of vibration and analytic approximations are obtained for a slender tube. In a stratified atmosphere an expansion procedure is used to derive an equation for the vertical velocity perturbation. The behaviour of motions within the flux tube is shown to depend upon a transition frequency 𝜔[sub]v such that vertically propagating waves are possible only for frequencies greater than 𝜔[sub]v. Also, the nature of convective instability in a slender magnetic flux tube is explored. A sufficient condition for stability is derived for the case of an arbitrary temperature profile in the external medium. For a tube of infinite depth, with a uniform-temperature gradient inside the tube equal to that in the exterior, a necessary and sufficient condition for convective stability to occur inside the tube is derived. Under the assumptions of the model, intense flux tubes are convectively stable if sufficiently shallow (with depths 1 - 2 x 10³ km or less). Tubes that extend deeper into the convection zone are potentially (convectively) unstable, but may be stabilised for sufficiently strong magnetic fields. Radiative damping of waves is important in the upper photosphere and the effect of radiative relaxation on the propagation of waves in an intense flux tube is examined both for a uniform and stratified atmosphere. The cut-off frequency is generalized to include the effects of radiative relaxation. The phase-shift between velocity oscillations at two different levels and the phase difference between temperature and velocity perturbations are derived and compared with the available observations. Finally, the consequences of the observed steady downflow are discussed.
1980-01-01T00:00:00ZWebb, Andrew RobertThe nature of motions in intense magnetic fields is investigated. For a flux tube in a uniform atmosphere a dispersion relation is derived for the modes of vibration and analytic approximations are obtained for a slender tube. In a stratified atmosphere an expansion procedure is used to derive an equation for the vertical velocity perturbation. The behaviour of motions within the flux tube is shown to depend upon a transition frequency 𝜔[sub]v such that vertically propagating waves are possible only for frequencies greater than 𝜔[sub]v. Also, the nature of convective instability in a slender magnetic flux tube is explored. A sufficient condition for stability is derived for the case of an arbitrary temperature profile in the external medium. For a tube of infinite depth, with a uniform-temperature gradient inside the tube equal to that in the exterior, a necessary and sufficient condition for convective stability to occur inside the tube is derived. Under the assumptions of the model, intense flux tubes are convectively stable if sufficiently shallow (with depths 1 - 2 x 10³ km or less). Tubes that extend deeper into the convection zone are potentially (convectively) unstable, but may be stabilised for sufficiently strong magnetic fields. Radiative damping of waves is important in the upper photosphere and the effect of radiative relaxation on the propagation of waves in an intense flux tube is examined both for a uniform and stratified atmosphere. The cut-off frequency is generalized to include the effects of radiative relaxation. The phase-shift between velocity oscillations at two different levels and the phase difference between temperature and velocity perturbations are derived and compared with the available observations. Finally, the consequences of the observed steady downflow are discussed.Nonlinear stability of flows over rigid and flexible boundariesThomas, Michael Dhttp://hdl.handle.net/10023/142732019-04-01T08:39:26Z1990-01-01T00:00:00ZThis work assesses the importance of nonlinearity in the stability of flows over compliant and rigid walls, and comprises three main parts. The first part considers inviscid flow with a free surface over a flexible boundary. The dispersion relation is obtained, and the conditions for linear instability investigated. The linear dispersion relation is then used to show that the conditions for nonlinear three-wave resonance are often met. In some circumstances, the resonance may be of 'explosive' sort, involving waves of opposite energy sign; but non-explosive resonant configurations are most common. Next, the wave- amplitude evolution equations for three-wave resonance are derived, firstly by a 'direct' approach, and then via a variational (averaged Lagrangian) method. Results agree with those of Case & Chiu (1977) for capillary-gravity waves, and Craik & Adam (1979), for three-layer fluid flow, on taking the appropriate limits. We also consider a nonlinear model for the flexible boundary. In the second part, stability of Blasius flow over a compliant surface is studied. This extension of rigid-wall work of Craik (1971) and Hendriks (appendix to Usher & Craik 1975) determines the quadratic interaction coefficients of three-wave resonance, and complements the linear analysis of Carpenter & Garrad (1985, 1986) and others. First, the linear eigenvalue spectrum is investigated for various values of the wall parameters. Then, resonant triads are located and the quadratic interaction coefficients determined numerically. By way of introduction some rigid-wall results are also presented, extending those of Hendriks.
1990-01-01T00:00:00ZThomas, Michael DThis work assesses the importance of nonlinearity in the stability of flows over compliant and rigid walls, and comprises three main parts. The first part considers inviscid flow with a free surface over a flexible boundary. The dispersion relation is obtained, and the conditions for linear instability investigated. The linear dispersion relation is then used to show that the conditions for nonlinear three-wave resonance are often met. In some circumstances, the resonance may be of 'explosive' sort, involving waves of opposite energy sign; but non-explosive resonant configurations are most common. Next, the wave- amplitude evolution equations for three-wave resonance are derived, firstly by a 'direct' approach, and then via a variational (averaged Lagrangian) method. Results agree with those of Case & Chiu (1977) for capillary-gravity waves, and Craik & Adam (1979), for three-layer fluid flow, on taking the appropriate limits. We also consider a nonlinear model for the flexible boundary. In the second part, stability of Blasius flow over a compliant surface is studied. This extension of rigid-wall work of Craik (1971) and Hendriks (appendix to Usher & Craik 1975) determines the quadratic interaction coefficients of three-wave resonance, and complements the linear analysis of Carpenter & Garrad (1985, 1986) and others. First, the linear eigenvalue spectrum is investigated for various values of the wall parameters. Then, resonant triads are located and the quadratic interaction coefficients determined numerically. By way of introduction some rigid-wall results are also presented, extending those of Hendriks.Time dependent heating of the solar coronaWalsh, Robert Williamhttp://hdl.handle.net/10023/142672019-04-01T08:38:57Z1996-01-01T00:00:00ZThe problem of how the Sun's corona is heated is of central importance in Solar Physics research. In this thesis, a model is constructed of a typical coronal magnetic loop in order to investigate the response of coronal plasma to a time-dependent heating source. It is not the aim of the research to study in detail a particular heating mechanism but rather to understand the important features arising from time-dependent heating in general. A time-varying energy input into the coronal loop is required because it is likely that none of the suggested theoretical heating methods can provide a constant supply of heat to the corona. The magnetic field is taken to be strong enough that the loop dynamics reduce to a one-dimensional problem along the field. In addition, it is assumed that the radiative timescale in the corona is much longer than the sound travel time and thus, the plasma evolves isobarically. The thermal equilibria profiles along the coronal loop are then investigated for a simplified form of the optically thin radiation. Initially, a heating function that displays a regular, sinusoidal variation in time is introduced and it is found that there is a critical heating frequency above which a hot coronal loop solution can be maintained and below which the plasma temperature cools to chromospheric values. Pulse heating and the deposition of random-sized energy quanta in a loop are also investigated. An evaluation of the isobaric assumption to the corona is presented by allowing sound waves to propagate back and forth along the loop. It is found that the system can exhibit isobaric-like behaviour provided the acoustic timescale is short enough. Possible extensions of the developed loop model are discussed as well as the implications of time-dependent heating upon observations from the SOHO satellite.
1996-01-01T00:00:00ZWalsh, Robert WilliamThe problem of how the Sun's corona is heated is of central importance in Solar Physics research. In this thesis, a model is constructed of a typical coronal magnetic loop in order to investigate the response of coronal plasma to a time-dependent heating source. It is not the aim of the research to study in detail a particular heating mechanism but rather to understand the important features arising from time-dependent heating in general. A time-varying energy input into the coronal loop is required because it is likely that none of the suggested theoretical heating methods can provide a constant supply of heat to the corona. The magnetic field is taken to be strong enough that the loop dynamics reduce to a one-dimensional problem along the field. In addition, it is assumed that the radiative timescale in the corona is much longer than the sound travel time and thus, the plasma evolves isobarically. The thermal equilibria profiles along the coronal loop are then investigated for a simplified form of the optically thin radiation. Initially, a heating function that displays a regular, sinusoidal variation in time is introduced and it is found that there is a critical heating frequency above which a hot coronal loop solution can be maintained and below which the plasma temperature cools to chromospheric values. Pulse heating and the deposition of random-sized energy quanta in a loop are also investigated. An evaluation of the isobaric assumption to the corona is presented by allowing sound waves to propagate back and forth along the loop. It is found that the system can exhibit isobaric-like behaviour provided the acoustic timescale is short enough. Possible extensions of the developed loop model are discussed as well as the implications of time-dependent heating upon observations from the SOHO satellite.Time-dependent MHD wave coupling in non-uniform mediaMann, Ian R.http://hdl.handle.net/10023/142642019-04-01T08:37:06Z1996-01-01T00:00:00ZThis thesis studies the time dependent evolution of MHD waves in cold, fully compressible non-uniform plasmas. We used a 1-D box model (e.g., Southwood (1974)) to study wave mode coupling, and concentrate upon developing an understanding of the underlying physics that governs waves in the Earth's magnetosphere. We begin by discussing the form of the (often singular) governing eigenmodes of the system, and subsequently use these eigenmodes as a basis with which to construct the solution to a variety of initial value problems. We consider a detailed analysis of both the widths and the internal length scales developed by cavity mode driven held line resonances (FLRs), and compare our results to observations presented in the literature. We find that (especially asymptotically in time) the coupled waves derive their dominant characteristics from the form of undriven decoupled toroidal Alfvén eigenmodes. Ideal numerical solutions show that fine spatial scales are developed across the background magnetic field, and we demonstrate that this is accurately estimated as the decoupled phase mixing length
L[sub]p[sub]h = 2π/𝜔ⁱ[sub]A = d 𝜔[sub]A/dx
We also discuss the likely ionospheric and kinetic modifications to our theory. Later, we consider the evolution of poloidal Alfvén waves having large azimuthal wavenumber (𝜆). We find that the 𝜆 → ∞ decoupled poloidal Alfvén wave evaluation (Dungey, 1967) is modified for finite 𝜆 lambda, approaching decoupled toroidal field line oscillations for large t. We define a poloidal lifetime 𝛵, when toroidal and poloidal displacements become equal, and demonstrate that this is when the phase mixing length is equal to 2pi/lambda. We examine numerically the poloidal Alfvén wave evolution for 𝜆 ≫ k[sub]z, and k[sub]≳ lambda, when k[sub]x(x,t = 0) ≪ lambda or k[sub]z. We interpret the lambda ≪ kz results (applicable to the Earth's magnetosphere) in the context of poloidal Alfvén wave observations, and compare our study to the numerical analysis of Ding et al. (1995). We conclude the thesis by undertaking an asymptotic derivation of the large 𝜆 solutions by using the method of multiple time scales. We find our analytic solutions are in excellent agreement with those determined numerically. A central result of the thesis is the importance and dominance of the phase mixing length for time dependent solutions, irrespective of the value of 𝜆.
1996-01-01T00:00:00ZMann, Ian R.This thesis studies the time dependent evolution of MHD waves in cold, fully compressible non-uniform plasmas. We used a 1-D box model (e.g., Southwood (1974)) to study wave mode coupling, and concentrate upon developing an understanding of the underlying physics that governs waves in the Earth's magnetosphere. We begin by discussing the form of the (often singular) governing eigenmodes of the system, and subsequently use these eigenmodes as a basis with which to construct the solution to a variety of initial value problems. We consider a detailed analysis of both the widths and the internal length scales developed by cavity mode driven held line resonances (FLRs), and compare our results to observations presented in the literature. We find that (especially asymptotically in time) the coupled waves derive their dominant characteristics from the form of undriven decoupled toroidal Alfvén eigenmodes. Ideal numerical solutions show that fine spatial scales are developed across the background magnetic field, and we demonstrate that this is accurately estimated as the decoupled phase mixing length
L[sub]p[sub]h = 2π/𝜔ⁱ[sub]A = d 𝜔[sub]A/dx
We also discuss the likely ionospheric and kinetic modifications to our theory. Later, we consider the evolution of poloidal Alfvén waves having large azimuthal wavenumber (𝜆). We find that the 𝜆 → ∞ decoupled poloidal Alfvén wave evaluation (Dungey, 1967) is modified for finite 𝜆 lambda, approaching decoupled toroidal field line oscillations for large t. We define a poloidal lifetime 𝛵, when toroidal and poloidal displacements become equal, and demonstrate that this is when the phase mixing length is equal to 2pi/lambda. We examine numerically the poloidal Alfvén wave evolution for 𝜆 ≫ k[sub]z, and k[sub]≳ lambda, when k[sub]x(x,t = 0) ≪ lambda or k[sub]z. We interpret the lambda ≪ kz results (applicable to the Earth's magnetosphere) in the context of poloidal Alfvén wave observations, and compare our study to the numerical analysis of Ding et al. (1995). We conclude the thesis by undertaking an asymptotic derivation of the large 𝜆 solutions by using the method of multiple time scales. We find our analytic solutions are in excellent agreement with those determined numerically. A central result of the thesis is the importance and dominance of the phase mixing length for time dependent solutions, irrespective of the value of 𝜆.Magnetic neutral points and nonuniform reconnectionStrachan, N. R.http://hdl.handle.net/10023/142502019-04-01T08:39:39Z1994-01-01T00:00:00ZEver since the first recorded observation of a solar flare in September 1859, it has been a key question - for physics as a whole and for astrophsics in particular - to ask what mechanism lies behind the sudden, violent release of energy from the sun. It has become increasingly apparent that the complex structure of the solar magnetic field lies at the heart of the answer. The process of magnetic reconnection has, over the years, become the accepted explanation by which magnetic energy can be released on both large and small scales in astrophysical and laboratory plasmas. The results of reconnection can be seen, for instance, in star formation, solar flares and the earth's aurorae; indeed the 1859 flare was followed by exceptional auroral activity. The mechanism of magnetic reconnection was first postulated by Giovanelli (1947) as a way of releasing the magnetic energy stored in the Sun. He, and later Dungey (1953), realised that the behaviour of the plasma in the vicinity of a magnetic neutral or null point, where the field disappears, is quite different from other regions of space. In this thesis the nature of magnetic neutral points and their role in the process of reconnection is investigated. Firstly, a general classification of magnetic neutral points is presented. The chapter includes equilibrium and steady-state solutions for two-dimensional magnetic neutral points. The differences in the field behaviour close to each type of neutral point are explained and criteria for the existence of steady-state solutions and equilibria involving pressure balance are presented. In the last section, a self-similar solution for a collapsed X-point is explored. The X-point necessarily becomes cusp-like in nature if shearing is applied in the ignorable direction. Two reconnection models are considered. The first is an extension of the Priest-Lee model (1990). It incorporates large pressure gradients in the inflow corresponding to the Forbes-Priest Almost-Uniform Model. The investigation includes both analytical and numerical solutions and a study of the separatrix jet. In the numerical study, current spikes are found at the end of the current sheets and a much increased reconnection rate is found analytically in the extreme flux file-up limit. The second reconnection model presented is also based on the Priest-Lee configuration. A uniform field is imposed on the basic structure producing a cusp-point with a non-zero field strength as the neutral point is approached from above. This results in the removal of the singularity in the flow above the separatrix. A non-singular solution is found analytically for a double-cusp. A much larger reconnection rate is found and a numerical solution is presented.
1994-01-01T00:00:00ZStrachan, N. R.Ever since the first recorded observation of a solar flare in September 1859, it has been a key question - for physics as a whole and for astrophsics in particular - to ask what mechanism lies behind the sudden, violent release of energy from the sun. It has become increasingly apparent that the complex structure of the solar magnetic field lies at the heart of the answer. The process of magnetic reconnection has, over the years, become the accepted explanation by which magnetic energy can be released on both large and small scales in astrophysical and laboratory plasmas. The results of reconnection can be seen, for instance, in star formation, solar flares and the earth's aurorae; indeed the 1859 flare was followed by exceptional auroral activity. The mechanism of magnetic reconnection was first postulated by Giovanelli (1947) as a way of releasing the magnetic energy stored in the Sun. He, and later Dungey (1953), realised that the behaviour of the plasma in the vicinity of a magnetic neutral or null point, where the field disappears, is quite different from other regions of space. In this thesis the nature of magnetic neutral points and their role in the process of reconnection is investigated. Firstly, a general classification of magnetic neutral points is presented. The chapter includes equilibrium and steady-state solutions for two-dimensional magnetic neutral points. The differences in the field behaviour close to each type of neutral point are explained and criteria for the existence of steady-state solutions and equilibria involving pressure balance are presented. In the last section, a self-similar solution for a collapsed X-point is explored. The X-point necessarily becomes cusp-like in nature if shearing is applied in the ignorable direction. Two reconnection models are considered. The first is an extension of the Priest-Lee model (1990). It incorporates large pressure gradients in the inflow corresponding to the Forbes-Priest Almost-Uniform Model. The investigation includes both analytical and numerical solutions and a study of the separatrix jet. In the numerical study, current spikes are found at the end of the current sheets and a much increased reconnection rate is found analytically in the extreme flux file-up limit. The second reconnection model presented is also based on the Priest-Lee configuration. A uniform field is imposed on the basic structure producing a cusp-point with a non-zero field strength as the neutral point is approached from above. This results in the removal of the singularity in the flow above the separatrix. A non-singular solution is found analytically for a double-cusp. A much larger reconnection rate is found and a numerical solution is presented.Aspects of the MHD stability of coronal and laboratory plasmasClifford, Leo J.http://hdl.handle.net/10023/142482019-04-01T08:39:18Z1993-01-01T00:00:00ZThe magnetohydrodynamic (MHD) model is a simple mathematical model that treats a plasma as a perfectly conducting fluid acted upon by magnetic and pressure-driven forces. Many instabilities in plasmas can be predicted using this model. In this Thesis, aspects of the linear stability of solar and laboratory plasmas are studied using the MHD model. Firstly, we investigate the thermal instability of coronal plasmas with line-tied magnetic fields and with anisotropical heat conduction, using an analytical analysis which concentrates on isobaric perturbations, and a time-dependent numerical code. We find that including perpendicular thermal conduction means that condensations are restricted to a narrow layer around the region where the local isobaric growth rate is largest and that, while the growth rate of the thermal mode is largely unaffected by perpendicular thermal conduction, this may be an important factor in determining the lengthscale for the width of condensations. Secondly, the effect of a finitely conducting wall on the linear stability of Spheromak and Reversed Field Finch equilibria is investigated. We find growth rates for the modes that are present because of the finite resistivity of the wall, which grow proportionally to the "long" time constant of the wall. Finally, we apply a tractable method, derived by De Bruyne (1990), for investigating the stability of 2-D line-tied magnetic fields, to cylindrically symmetric spheromak equilibria. The method involves the solution of two sets of ordinary differential equations, integrated along the field lines, which give necessary and sufficient conditions for stability. The role of plasma pressure and of the width of the entrance region are investigated.
1993-01-01T00:00:00ZClifford, Leo J.The magnetohydrodynamic (MHD) model is a simple mathematical model that treats a plasma as a perfectly conducting fluid acted upon by magnetic and pressure-driven forces. Many instabilities in plasmas can be predicted using this model. In this Thesis, aspects of the linear stability of solar and laboratory plasmas are studied using the MHD model. Firstly, we investigate the thermal instability of coronal plasmas with line-tied magnetic fields and with anisotropical heat conduction, using an analytical analysis which concentrates on isobaric perturbations, and a time-dependent numerical code. We find that including perpendicular thermal conduction means that condensations are restricted to a narrow layer around the region where the local isobaric growth rate is largest and that, while the growth rate of the thermal mode is largely unaffected by perpendicular thermal conduction, this may be an important factor in determining the lengthscale for the width of condensations. Secondly, the effect of a finitely conducting wall on the linear stability of Spheromak and Reversed Field Finch equilibria is investigated. We find growth rates for the modes that are present because of the finite resistivity of the wall, which grow proportionally to the "long" time constant of the wall. Finally, we apply a tractable method, derived by De Bruyne (1990), for investigating the stability of 2-D line-tied magnetic fields, to cylindrically symmetric spheromak equilibria. The method involves the solution of two sets of ordinary differential equations, integrated along the field lines, which give necessary and sufficient conditions for stability. The role of plasma pressure and of the width of the entrance region are investigated.Ducted magnetoacoustic waves in the solar coronaSmith, Jason M.http://hdl.handle.net/10023/142462019-04-01T08:38:54Z1997-01-01T00:00:00ZThis thesis investigates the ducting of magnetoacoustic waves in coronal structures. The propagation of waves in current sheets and coronal loops has been examined in order to understand wave ducting in structured plasmas, and to provide an explanation of the observed oscillatory behaviour in the solar corona. Firstly a comprehensive review of the observations of loops and oscillations in the corona is given. An investigation into how the curvature of the loop alters the ducting of magnetoacoustic waves is then presented by studying the effect of the length, width and the density enhancement of the loop and also the frequency of oscillation. The effect of the curvature is to generate wave leakage from the loop. The guiding of magnetoacoustic waves by a current sheet is also considered. An investigation into the type of modes which may propagate and the time scales of oscillation is performed. Impulsively generated waves exhibit similar temporal signatures to observations of X-ray and radio emission. Periods of oscillation for all the ducted wave models are in good agreement with reported observations. The effect of a random boundary motion on a magnetospheric cavity is examined through numerical simulations. A broadband driving spectrum excites the quasi-monochromatic fast modes whose frequencies lie within the driving spectrum. These fast modes couple to an Alfvén mode if the frequency lies within the Alfvén continuum. The position of the resonant field lines and the Alfvén mode eigenfunction may be accurately calculated by assuming a periodic boundary motion. To conclude the work in this thesis the three-dimensional magnetic topologies surrounding neutral points are studied. The local linear magnetic structure about the null is found to depend only on a 3 X 3 matrix containing four parameters. The type of topology is dependent upon the nature of the eigenvalues and eigenvectors of this matrix.
1997-01-01T00:00:00ZSmith, Jason M.This thesis investigates the ducting of magnetoacoustic waves in coronal structures. The propagation of waves in current sheets and coronal loops has been examined in order to understand wave ducting in structured plasmas, and to provide an explanation of the observed oscillatory behaviour in the solar corona. Firstly a comprehensive review of the observations of loops and oscillations in the corona is given. An investigation into how the curvature of the loop alters the ducting of magnetoacoustic waves is then presented by studying the effect of the length, width and the density enhancement of the loop and also the frequency of oscillation. The effect of the curvature is to generate wave leakage from the loop. The guiding of magnetoacoustic waves by a current sheet is also considered. An investigation into the type of modes which may propagate and the time scales of oscillation is performed. Impulsively generated waves exhibit similar temporal signatures to observations of X-ray and radio emission. Periods of oscillation for all the ducted wave models are in good agreement with reported observations. The effect of a random boundary motion on a magnetospheric cavity is examined through numerical simulations. A broadband driving spectrum excites the quasi-monochromatic fast modes whose frequencies lie within the driving spectrum. These fast modes couple to an Alfvén mode if the frequency lies within the Alfvén continuum. The position of the resonant field lines and the Alfvén mode eigenfunction may be accurately calculated by assuming a periodic boundary motion. To conclude the work in this thesis the three-dimensional magnetic topologies surrounding neutral points are studied. The local linear magnetic structure about the null is found to depend only on a 3 X 3 matrix containing four parameters. The type of topology is dependent upon the nature of the eigenvalues and eigenvectors of this matrix.Inverse polarity prominence equilibriaSchönfelder, Apollonia Maria Oktaviahttp://hdl.handle.net/10023/142432019-04-01T08:38:36Z1995-01-01T00:00:00ZIt has been supposed since the middle of this century that it is the global magnetic field surrounding a quiescent prominence that provides the force to prevent its collapse due to the sun’s gravitational field. Many theoretical models, assuming that the prominence plasma is supported in a dip in the magnetic field lines associated by the magnetic tension force, have since been put forward. The aim of this thesis is to propose further models of quiescent prominences to widen our understanding and knowledge of these remarkable features.
A short overview over the magnetohydrodynamic equations used to describe solar prominences, or most of the solar phenomena for that matter, are discussed in chapter 2, and a short summary of prominence observations and attempts to model them is given in chapter 3.
A brief description of the numerical code used in chapters 5 and 7 is given in chapter 4.
Observations of Kim (1990) and Leroy (1985) have found that most large quiescent prominences are of inverse polarity type for which the magnetic field passes through the prominence in the opposite direction to that expected from the photospheric magnetic field. Many theoretical models have been proposed, but failed. Hence, in chapter 5 we investigate first – without the inclusion of a prominence sheet – when an inverse polarity magnetic field must have the correct topology for an inverse polarity configuration before the formation of the prominence itself. Only very recently, the first basic successful model of an I-type polarity prominence was proposed by Low (1993). In chapter 6 we examine this model and investigate current sheets more complicated and realistic than the one used by Low. These analytical models deal with the force-free solution, which is matched onto an external, unsheared, potential coronal magnetic field. These solutions are mathematically interesting and allow an investigation of different profiles of the current intensity of the magnetic field vector and of the mass density in the sheet. The prominence properties predicted by these models have been examined and have been found to match the observational values. The mathematics of current sheets in general is also briefly discussed.
Chapter 7 deals with numerical solutions of inverse polarity prominences embedded in a force-free magnetic flux tube, matched onto an unsheared potential coronal field. Unfortunately the solutions gained are quite sensitive to the boundary conditions imposed on them through the numerical box, showing a loss of convergence and a tendency for the solution to blow up.
Finally, a short summary as well as possible future work is given in chapter 8.
1995-01-01T00:00:00ZSchönfelder, Apollonia Maria OktaviaIt has been supposed since the middle of this century that it is the global magnetic field surrounding a quiescent prominence that provides the force to prevent its collapse due to the sun’s gravitational field. Many theoretical models, assuming that the prominence plasma is supported in a dip in the magnetic field lines associated by the magnetic tension force, have since been put forward. The aim of this thesis is to propose further models of quiescent prominences to widen our understanding and knowledge of these remarkable features.
A short overview over the magnetohydrodynamic equations used to describe solar prominences, or most of the solar phenomena for that matter, are discussed in chapter 2, and a short summary of prominence observations and attempts to model them is given in chapter 3.
A brief description of the numerical code used in chapters 5 and 7 is given in chapter 4.
Observations of Kim (1990) and Leroy (1985) have found that most large quiescent prominences are of inverse polarity type for which the magnetic field passes through the prominence in the opposite direction to that expected from the photospheric magnetic field. Many theoretical models have been proposed, but failed. Hence, in chapter 5 we investigate first – without the inclusion of a prominence sheet – when an inverse polarity magnetic field must have the correct topology for an inverse polarity configuration before the formation of the prominence itself. Only very recently, the first basic successful model of an I-type polarity prominence was proposed by Low (1993). In chapter 6 we examine this model and investigate current sheets more complicated and realistic than the one used by Low. These analytical models deal with the force-free solution, which is matched onto an external, unsheared, potential coronal magnetic field. These solutions are mathematically interesting and allow an investigation of different profiles of the current intensity of the magnetic field vector and of the mass density in the sheet. The prominence properties predicted by these models have been examined and have been found to match the observational values. The mathematics of current sheets in general is also briefly discussed.
Chapter 7 deals with numerical solutions of inverse polarity prominences embedded in a force-free magnetic flux tube, matched onto an unsheared potential coronal field. Unfortunately the solutions gained are quite sensitive to the boundary conditions imposed on them through the numerical box, showing a loss of convergence and a tendency for the solution to blow up.
Finally, a short summary as well as possible future work is given in chapter 8.The magnetohydrostatic equilibrium of quiescent solar prominencesRidgway, Christopherhttp://hdl.handle.net/10023/142392019-04-01T08:37:44Z1992-01-01T00:00:00ZSince the mid 1900's it has been supposed that the global magnetic field surrounding a quiescent prominence provides the force required to prevent its collapse under the influence of the Sun's gravitational field. Many theoretical models of this magnetic field have been produced in which it is assumed that the prominence plasma is supported in a dip in the field lines by the associated magnetic tension force. It is the aim of this thesis to propose further models of the magnetic field in order to extend our knowledge and understanding of prominences. In doing so we present three distinct models. The first is an extension of the twisted flux tube model for prominences proposed by Priest et al. (1989). Here we present analytical solutions to the magnetohydrostatic equilibrium equation within the tube using the so- called generating function method in which we select two distinct functional forms of the longitudinal field component. Unlike the solutions found by Priest et al., we allow for large deviations of the field from cylindrical symmetry. The prominence is represented by a finite vertical sheet of mass and current and we show that it is possible for such a sheet to be in static equilibrium everywhere along its vertical extent. Next we consider the model of van Ballegooijen and Martens in which photospheric motions drive a reconnection process leading to the formation of a helical magnetic structure capable of supporting dense prominence plasma in the low points of the helical windings. Under the assumption of cylindrical symmetry we analyse two methods of solving the magnetohydrostatic equilibrium equation in which the positions of the field line footpoints at the photosphere are imposed. Using a combination of analytical and numerical techniques, we study the quasi-static evolution of the model as the height of the helical axis increases. Unlike the numerical analysis of van Ballegooijen and Martens we are able to produce inverse polarity configurations without the problem of singular field components at the helical axis. Lastly we present an analysis of the interaction of a finite, vertical sheet of mass and current (representing a prominence) with an external constant-current force-free field. We formalise two distinct boundary-value problems in which the distribution of the normal magnetic field component along the photosphere is imposed along with the distribution of either the normal magnetic field component across the prominence or the prominence surface current. In both cases we demonstrate for particular boundary conditions that it is possible for equilibrium solutions to exist of both normal and inverse polarity in which dense material is supported everywhere along the prominence sheet. In particular we are, for the first time, able to produce an inverse polarity equilibrium configuration in which the field components are locally bounded and closed field lines exist above the prominence sheet while an X-type neutral point lies below it.
1992-01-01T00:00:00ZRidgway, ChristopherSince the mid 1900's it has been supposed that the global magnetic field surrounding a quiescent prominence provides the force required to prevent its collapse under the influence of the Sun's gravitational field. Many theoretical models of this magnetic field have been produced in which it is assumed that the prominence plasma is supported in a dip in the field lines by the associated magnetic tension force. It is the aim of this thesis to propose further models of the magnetic field in order to extend our knowledge and understanding of prominences. In doing so we present three distinct models. The first is an extension of the twisted flux tube model for prominences proposed by Priest et al. (1989). Here we present analytical solutions to the magnetohydrostatic equilibrium equation within the tube using the so- called generating function method in which we select two distinct functional forms of the longitudinal field component. Unlike the solutions found by Priest et al., we allow for large deviations of the field from cylindrical symmetry. The prominence is represented by a finite vertical sheet of mass and current and we show that it is possible for such a sheet to be in static equilibrium everywhere along its vertical extent. Next we consider the model of van Ballegooijen and Martens in which photospheric motions drive a reconnection process leading to the formation of a helical magnetic structure capable of supporting dense prominence plasma in the low points of the helical windings. Under the assumption of cylindrical symmetry we analyse two methods of solving the magnetohydrostatic equilibrium equation in which the positions of the field line footpoints at the photosphere are imposed. Using a combination of analytical and numerical techniques, we study the quasi-static evolution of the model as the height of the helical axis increases. Unlike the numerical analysis of van Ballegooijen and Martens we are able to produce inverse polarity configurations without the problem of singular field components at the helical axis. Lastly we present an analysis of the interaction of a finite, vertical sheet of mass and current (representing a prominence) with an external constant-current force-free field. We formalise two distinct boundary-value problems in which the distribution of the normal magnetic field component along the photosphere is imposed along with the distribution of either the normal magnetic field component across the prominence or the prominence surface current. In both cases we demonstrate for particular boundary conditions that it is possible for equilibrium solutions to exist of both normal and inverse polarity in which dense material is supported everywhere along the prominence sheet. In particular we are, for the first time, able to produce an inverse polarity equilibrium configuration in which the field components are locally bounded and closed field lines exist above the prominence sheet while an X-type neutral point lies below it.Wave-particle dynamics in a hot inhomogenous fusion plasmaTaylor, Michael Anthonyhttp://hdl.handle.net/10023/142352019-04-01T08:38:09Z1996-01-01T00:00:00ZAn outstanding problem in the field of nuclear fusion research is the precise mechanism by which a hot, magnetically inhomogeneous plasma is heated when illuminated by a constant beam of small amplitude radio waves matched in frequency to harmonics of the ion Larmor frequency. An accurate model must include microscopic dynamics and inevitably a kinetic theory is required. Highly energetic ions (> 1MeV) born from fusion reactions or powered by gyroresonance have large Larmor radii (> 10cm) which are comparable in size to the wavelength of the incident radiation. In particular we will focus on fast magnetosonic waves. Exact full wave equations describing a thermal plasma in a weakly inhomogeneous field are presently at least fourth order integro-differential equations (Sauter, 1992). These are computationally taxing. Recently a method was proposed to reduce the problem to a second order integro-differential equation at the expense of information related to the propagation of mode-converted waves (Holt, 1992). We present here a generalisation of the theory to allow for arbitrary velocity-dependent equilibria while at the same time retaining a general functional form for the field profile. We consider the specific case of a bi-Maxwellian plasma immersed in a linearly inhomogenous magnetic field. We find that thermal anisotropy produces resonance localisation when the perpendicular ion temperature is greater than that parallel to the ambient field. A study of the symmetry properties of the conductivity tensor reveals that the Onsager reciprocal relations are obeyed only for an isotropic plasma in an inhomogeneous field. This is a generalisation of the result obtained by Nambu (1995). We present a generalisation of the reduction method to include effects due to changes in wave amplitude. We find that we are able to include the odd-order field derivatives responsible for energy conservation. Our numerical study of fundamental Helium-3 gyroresonance in a majority Deuterium plasma reveals that we have > 99.9% energy conservation in all cases. We show that locally-uniform theory can be very inaccurate (≃ 70% in one case presented in our recent paper, Cairns et al., 1995) particularly for higher energy ions whose non-locality is more extreme. We present a representative sample of results for minority heating and mode conversion heating schemes. We report the appearance of an unexpected cut-off on the low field side of the minority gyroresonance which may have important consequences for antennae presently placed on the outside of Tokamaks.
1996-01-01T00:00:00ZTaylor, Michael AnthonyAn outstanding problem in the field of nuclear fusion research is the precise mechanism by which a hot, magnetically inhomogeneous plasma is heated when illuminated by a constant beam of small amplitude radio waves matched in frequency to harmonics of the ion Larmor frequency. An accurate model must include microscopic dynamics and inevitably a kinetic theory is required. Highly energetic ions (> 1MeV) born from fusion reactions or powered by gyroresonance have large Larmor radii (> 10cm) which are comparable in size to the wavelength of the incident radiation. In particular we will focus on fast magnetosonic waves. Exact full wave equations describing a thermal plasma in a weakly inhomogeneous field are presently at least fourth order integro-differential equations (Sauter, 1992). These are computationally taxing. Recently a method was proposed to reduce the problem to a second order integro-differential equation at the expense of information related to the propagation of mode-converted waves (Holt, 1992). We present here a generalisation of the theory to allow for arbitrary velocity-dependent equilibria while at the same time retaining a general functional form for the field profile. We consider the specific case of a bi-Maxwellian plasma immersed in a linearly inhomogenous magnetic field. We find that thermal anisotropy produces resonance localisation when the perpendicular ion temperature is greater than that parallel to the ambient field. A study of the symmetry properties of the conductivity tensor reveals that the Onsager reciprocal relations are obeyed only for an isotropic plasma in an inhomogeneous field. This is a generalisation of the result obtained by Nambu (1995). We present a generalisation of the reduction method to include effects due to changes in wave amplitude. We find that we are able to include the odd-order field derivatives responsible for energy conservation. Our numerical study of fundamental Helium-3 gyroresonance in a majority Deuterium plasma reveals that we have > 99.9% energy conservation in all cases. We show that locally-uniform theory can be very inaccurate (≃ 70% in one case presented in our recent paper, Cairns et al., 1995) particularly for higher energy ions whose non-locality is more extreme. We present a representative sample of results for minority heating and mode conversion heating schemes. We report the appearance of an unexpected cut-off on the low field side of the minority gyroresonance which may have important consequences for antennae presently placed on the outside of Tokamaks.Models of X-ray bright points and concelling magnetic featuresParnell, Clare Elizabethhttp://hdl.handle.net/10023/142322019-04-01T08:38:00Z1995-01-01T00:00:00ZSmall brightenings called x-ray bright points (Golub et al, 1974) occur in the solar corona. They are observed with the soft x-ray telescope on Skylab to be approximately 22 Mm in diameter with a brighter inner core of width 4-7 Mm although with the Normal Incidence X-ray Telescope their dimensions are observed to be typically 6 Mm x 9 Mm. By comparison with magnetograms of the photosphere it has been noticed recently that there is a high correlation between the occurrence of x-ray bright points and the mutual reduction of flux between two opposite polarity magnetic fragments. These fragments are originally unconnected magnetically, but move towards each other and simultaneously lose equal amounts of flux (cancel): they are called cancelling magnetic features (Martin et al, 1984). The observations relating to these features were reviewed by Priest et al. (1994) who suggested that they naturally evolve through three phases: the pre-interaction, interaction and cancellation phases. From this evidence qualitative pictures of the magnetic field structure for an x-ray bright point and associated cancelling magnetic feature were established. The aim of this thesis has been to build on the ideas of Priest et al. (1994) to produce a detailed theoretical model of an x-ray bright point and a cancelling magnetic feature. The magnetic field structures are estimated, and the position and lifetime of the bright point are calculated, as is the total amount of energy released during the bright point. This work is also extended to study more complex cancelling configurations representing the main basic types of cancelling magnetic feature. The results of these models determine the factors that affect the lifetime and position of a bright point and indicate which types of cancelling magnetic features are most likely to produce bright points that are long-lived, lie directly above the cancellation site and occur simultaneously with the cancellation phase. The complex structure of a bright point cannot be explained from the above two-dimensional models: thus two recently observed bright points were studied to see if the above model could be extended into three dimensions to explain the structure seen in soft x-ray images. The available observational data was used and leads to reasonable explanations for the complex shapes of both bright points. Finally, a more realistic model for the overlying field was set up involving a model of the field above a supergranule cell field with fragments of finite width. The interaction of an ephemeral region within this field was then studied and led to five different scenarios. The results obtained reaffirmed those found in the previous simpler models and suggest where bright points may appear in a cell relative to the cancelling magnetic feature and for how long the bright points might last. Predictions for the lifetimes of cancelling magnetic features are also made, indicating when the cancelling magnetic feature occurs relative to the bright point.
1995-01-01T00:00:00ZParnell, Clare ElizabethSmall brightenings called x-ray bright points (Golub et al, 1974) occur in the solar corona. They are observed with the soft x-ray telescope on Skylab to be approximately 22 Mm in diameter with a brighter inner core of width 4-7 Mm although with the Normal Incidence X-ray Telescope their dimensions are observed to be typically 6 Mm x 9 Mm. By comparison with magnetograms of the photosphere it has been noticed recently that there is a high correlation between the occurrence of x-ray bright points and the mutual reduction of flux between two opposite polarity magnetic fragments. These fragments are originally unconnected magnetically, but move towards each other and simultaneously lose equal amounts of flux (cancel): they are called cancelling magnetic features (Martin et al, 1984). The observations relating to these features were reviewed by Priest et al. (1994) who suggested that they naturally evolve through three phases: the pre-interaction, interaction and cancellation phases. From this evidence qualitative pictures of the magnetic field structure for an x-ray bright point and associated cancelling magnetic feature were established. The aim of this thesis has been to build on the ideas of Priest et al. (1994) to produce a detailed theoretical model of an x-ray bright point and a cancelling magnetic feature. The magnetic field structures are estimated, and the position and lifetime of the bright point are calculated, as is the total amount of energy released during the bright point. This work is also extended to study more complex cancelling configurations representing the main basic types of cancelling magnetic feature. The results of these models determine the factors that affect the lifetime and position of a bright point and indicate which types of cancelling magnetic features are most likely to produce bright points that are long-lived, lie directly above the cancellation site and occur simultaneously with the cancellation phase. The complex structure of a bright point cannot be explained from the above two-dimensional models: thus two recently observed bright points were studied to see if the above model could be extended into three dimensions to explain the structure seen in soft x-ray images. The available observational data was used and leads to reasonable explanations for the complex shapes of both bright points. Finally, a more realistic model for the overlying field was set up involving a model of the field above a supergranule cell field with fragments of finite width. The interaction of an ephemeral region within this field was then studied and led to five different scenarios. The results obtained reaffirmed those found in the previous simpler models and suggest where bright points may appear in a cell relative to the cancelling magnetic feature and for how long the bright points might last. Predictions for the lifetimes of cancelling magnetic features are also made, indicating when the cancelling magnetic feature occurs relative to the bright point.Aspects of MHD wave propagation in solar atmospheric studiesMundie, Cheryl Annhttp://hdl.handle.net/10023/142272019-04-01T08:39:19Z1998-01-01T00:00:00ZThe theme of this thesis is ideal linear MHD wave propagation in structured media, using models relevant to structures in the solar atmosphere. We derive dispersion relations governing the ideal linear MHD modes for stationary states which are discretely structured in velocity and other plasma properties, in a direction transverse to the magnetic field, with field-aligned steady flow; the discrete structures considered are the single interface, uniform slab and uniform cylinder. This represents an extension of earlier models for the static case (Edwin 1984), by the inclusion of structured flows. The basic effects of flow are described, drawing on a discussion of the dispersion relations. The dispersion relations for the case of incompressible surface modes are examined in detail. We identify the qualitative effects of flow, including the onset of instability, by tracing the evolution of stable solutions and their propagation windows, as the relative flow is increased. Our analysis is presented in terms of a general formulation applicable to all three geometries (interface, slab and cylinder), revealing the combined role of dispersion and the ratio of densities in the two media. We go on to consider the relevance of the incompressible approximation to compressible surface modes, with particular reference to the static case of a single interface one side of which is field-free. We present and investigate analytical solutions for several special cases. The properties of the solutions obtained are compared with those for the equivalent incompressible case. Finally, we turn to the topic of global oscillations of quiescent prominences. A uniform slab model (Joarder 1993) yields, under conditions appropriate to the prominence-coronal inhomogeneity with the magnetic field threading the prominence being line-tied in the photosphere, modes which are analogous to the oscillations of a uniform string loaded with a point mass, and a formula approximating the period is given. We investigate the robustness of this formula for various plasma density profiles, assessing the applicability of the results from the uniform slab calculation to more realistic density profiles of the prominence-coronal inhomogeneity.
1998-01-01T00:00:00ZMundie, Cheryl AnnThe theme of this thesis is ideal linear MHD wave propagation in structured media, using models relevant to structures in the solar atmosphere. We derive dispersion relations governing the ideal linear MHD modes for stationary states which are discretely structured in velocity and other plasma properties, in a direction transverse to the magnetic field, with field-aligned steady flow; the discrete structures considered are the single interface, uniform slab and uniform cylinder. This represents an extension of earlier models for the static case (Edwin 1984), by the inclusion of structured flows. The basic effects of flow are described, drawing on a discussion of the dispersion relations. The dispersion relations for the case of incompressible surface modes are examined in detail. We identify the qualitative effects of flow, including the onset of instability, by tracing the evolution of stable solutions and their propagation windows, as the relative flow is increased. Our analysis is presented in terms of a general formulation applicable to all three geometries (interface, slab and cylinder), revealing the combined role of dispersion and the ratio of densities in the two media. We go on to consider the relevance of the incompressible approximation to compressible surface modes, with particular reference to the static case of a single interface one side of which is field-free. We present and investigate analytical solutions for several special cases. The properties of the solutions obtained are compared with those for the equivalent incompressible case. Finally, we turn to the topic of global oscillations of quiescent prominences. A uniform slab model (Joarder 1993) yields, under conditions appropriate to the prominence-coronal inhomogeneity with the magnetic field threading the prominence being line-tied in the photosphere, modes which are analogous to the oscillations of a uniform string loaded with a point mass, and a formula approximating the period is given. We investigate the robustness of this formula for various plasma density profiles, assessing the applicability of the results from the uniform slab calculation to more realistic density profiles of the prominence-coronal inhomogeneity.Magnetohydrodynamic surface waves in the solar atmosphereMiles, Alan J.http://hdl.handle.net/10023/142252019-04-01T08:37:59Z1991-01-01T00:00:00ZIn this thesis the nature of magnetoacoustic surface waves at a single magnetic interface is examined for the case of parallel propagation. Above the interface is an isothermal medium permeated by a horizontal magnetic field. The lower region is a field-free medium of different density to the magnetic atmosphere. We consider both the incompressible and compressible situations and the effect of including gravity. In each case a transcendental dispersion relation is solved numerically for a range of parameters and the resulting dispersion curves plotted. In the first chapter we provide a general introduction to the work, reviewing previous work in this area and considering applications of surface waves. In the second chapter we consider the existence of surface waves for the case when the media are incompressible either side of the interface. We consider the cases of both uniform and non-uniform distributions of densities and the effect of including gravity. We show that the f-mode exists in a restricted band of horizontal wavenumber. In the subsequent chapters we consider the effect of compressibility on surface waves. The media either side of the interface are taken to be isothermal. In the absence of gravity the interface may support one or two surface modes determined by the relative temperatures and magnetism of the two media. This case is studied in Chapter 3 where phase-speeds and penetration depths of the waves and the associated pressure perturbations are investigated for a variety of field strengths and sound speeds. In Chapters 4 and 5 we consider the effect of gravity on the compressible modes described in Chapter 3. In Chapter 4 an exact dispersion relation is obtained for the case of a constant Alfven speed, whilst in Chapter 5 the case of a uniform magnetic field is considered. In the absence of the magnetic field the transcendental dispersion relation may be reduced to a polynomial. This polynomial possesses two acceptable solutions, only one of which may exist at any given circumstance depending on the densities either side of the interface. If the gas density within the field exceeds that in the field-free medium, then the f-mode may propagate; otherwise, a magnetic surface gravity mode propagates. As in the incompressible case, the f-mode exists in a restricted band of horizontal wavenumber. An analytical form for the wave speed of the f-mode is obtained for small values of the Alfven speed. It is shown that the f-mode is related to the fast magnetoacoustic surface wave, merging into that mode at short wavelengths.
1991-01-01T00:00:00ZMiles, Alan J.In this thesis the nature of magnetoacoustic surface waves at a single magnetic interface is examined for the case of parallel propagation. Above the interface is an isothermal medium permeated by a horizontal magnetic field. The lower region is a field-free medium of different density to the magnetic atmosphere. We consider both the incompressible and compressible situations and the effect of including gravity. In each case a transcendental dispersion relation is solved numerically for a range of parameters and the resulting dispersion curves plotted. In the first chapter we provide a general introduction to the work, reviewing previous work in this area and considering applications of surface waves. In the second chapter we consider the existence of surface waves for the case when the media are incompressible either side of the interface. We consider the cases of both uniform and non-uniform distributions of densities and the effect of including gravity. We show that the f-mode exists in a restricted band of horizontal wavenumber. In the subsequent chapters we consider the effect of compressibility on surface waves. The media either side of the interface are taken to be isothermal. In the absence of gravity the interface may support one or two surface modes determined by the relative temperatures and magnetism of the two media. This case is studied in Chapter 3 where phase-speeds and penetration depths of the waves and the associated pressure perturbations are investigated for a variety of field strengths and sound speeds. In Chapters 4 and 5 we consider the effect of gravity on the compressible modes described in Chapter 3. In Chapter 4 an exact dispersion relation is obtained for the case of a constant Alfven speed, whilst in Chapter 5 the case of a uniform magnetic field is considered. In the absence of the magnetic field the transcendental dispersion relation may be reduced to a polynomial. This polynomial possesses two acceptable solutions, only one of which may exist at any given circumstance depending on the densities either side of the interface. If the gas density within the field exceeds that in the field-free medium, then the f-mode may propagate; otherwise, a magnetic surface gravity mode propagates. As in the incompressible case, the f-mode exists in a restricted band of horizontal wavenumber. An analytical form for the wave speed of the f-mode is obtained for small values of the Alfven speed. It is shown that the f-mode is related to the fast magnetoacoustic surface wave, merging into that mode at short wavelengths.Some aspects of solar flare and prominence theoryMilne, Alexander Mitchellhttp://hdl.handle.net/10023/142222019-04-01T08:37:16Z1980-01-01T00:00:00ZSolar flares and solar prominences are amongst the best known features of solar activity. Despite this familiarity, however, there are still significant gaps in our knowledge of these phenomena. In this thesis some theoretical aspects of these events are considered. We first consider solar prominences. We propose a model for the static equilibrium of quiescent prominences which will simultaneously explain the support mechanism for the dense prominence material and take account roughly of the required energy balance. This model contains two parameters, namely the coronal plasma beta and the horizontal shear angle 𝜙, that the magnetic fieldlines make with the prominence normal. We obtain limits on both these parameters which, when exceeded, imply that no equilibrium state is possible. The results obtained provide a possible explanation for several prominence features. For the remainder of the thesis we consider one aspect of the solar flare problem, namely the possibility of a trigger mechanism for the rapid release of energy in a flare. One candidate for this mechanism is the sudden release of energy stored in excess of potential by a force-free magnetic field which becomes unstable as a result of photospheric motions. For this reason we seek simple analytic solutions to the force-free field equations which may exhibit such an instability. An alternative trigger mechanism, which requires the presence of a current sheet, is given by the emerging flux model for solar flares. We thus develop a one-dimensional model for current sheets in general, where the conditions within the current sheet are given in terms of several non-dimensional parameters which describe the external conditions. These results are then applied to the emerging flux model.
1980-01-01T00:00:00ZMilne, Alexander MitchellSolar flares and solar prominences are amongst the best known features of solar activity. Despite this familiarity, however, there are still significant gaps in our knowledge of these phenomena. In this thesis some theoretical aspects of these events are considered. We first consider solar prominences. We propose a model for the static equilibrium of quiescent prominences which will simultaneously explain the support mechanism for the dense prominence material and take account roughly of the required energy balance. This model contains two parameters, namely the coronal plasma beta and the horizontal shear angle 𝜙, that the magnetic fieldlines make with the prominence normal. We obtain limits on both these parameters which, when exceeded, imply that no equilibrium state is possible. The results obtained provide a possible explanation for several prominence features. For the remainder of the thesis we consider one aspect of the solar flare problem, namely the possibility of a trigger mechanism for the rapid release of energy in a flare. One candidate for this mechanism is the sudden release of energy stored in excess of potential by a force-free magnetic field which becomes unstable as a result of photospheric motions. For this reason we seek simple analytic solutions to the force-free field equations which may exhibit such an instability. An alternative trigger mechanism, which requires the presence of a current sheet, is given by the emerging flux model for solar flares. We thus develop a one-dimensional model for current sheets in general, where the conditions within the current sheet are given in terms of several non-dimensional parameters which describe the external conditions. These results are then applied to the emerging flux model.Polynomial generated polygonsSoares, Benedict J.http://hdl.handle.net/10023/141982019-04-01T08:38:31Z1999-01-01T00:00:00ZA turtle geometric construction on the plane, called a polynomial generated polygon (PGP) and represented by 𝒫[sub]f,[sub]pᵐ, is generated from the sequence obtained from evaluating f(𝓍) ∈ ℤ [𝓍] over ℤ modulo pᵐ where p is a prime and m ∈ ℕ. Computational methods are developed to pre-calculate the symmetries exhibited by [sub]f,[sub]pᵐ for a given f and pᵐ.
These include procedures to find whether [sub]f,[sub]pᵐ is bounded or unbounded, the degree of rotational symmetry present, whether lines of reflectional symmetry can be observed, and in the case of 𝒫[sub]f,[sub]pᵐ unbounded, whether the PGP has a glide reflection.
Methods are also sought to find a suitable f and pᵐ to produce a desired 'feasible' shape in a PGP construction, and how the same shape might be generated modulo pᵐ⁺ᵏ if it cannot be produced modulo pᵐ.
1999-01-01T00:00:00ZSoares, Benedict J.A turtle geometric construction on the plane, called a polynomial generated polygon (PGP) and represented by 𝒫[sub]f,[sub]pᵐ, is generated from the sequence obtained from evaluating f(𝓍) ∈ ℤ [𝓍] over ℤ modulo pᵐ where p is a prime and m ∈ ℕ. Computational methods are developed to pre-calculate the symmetries exhibited by [sub]f,[sub]pᵐ for a given f and pᵐ.
These include procedures to find whether [sub]f,[sub]pᵐ is bounded or unbounded, the degree of rotational symmetry present, whether lines of reflectional symmetry can be observed, and in the case of 𝒫[sub]f,[sub]pᵐ unbounded, whether the PGP has a glide reflection.
Methods are also sought to find a suitable f and pᵐ to produce a desired 'feasible' shape in a PGP construction, and how the same shape might be generated modulo pᵐ⁺ᵏ if it cannot be produced modulo pᵐ.Nonlinear magnetic reconnectionColin, A. M.http://hdl.handle.net/10023/141952019-04-01T08:38:30Z1987-01-01T00:00:00ZIn many astrophysical problems magnetic reconnection plays a major role. Despite this reconnection theory remains incompletely understood, partly due to the strong non-linearity of the governing equations and the resulting difficulties in demonstrating analytical solutions. This thesis examines some fundamental aspects of reconnection theory: namely, the dynamics of driven and spontaneously reconnecting systems. We first consider the dynamics of a driven reconnecting system by numerically modelling a configuration consisting of two oppositely oriented flux systems with a variety of different boundary conditions and internal parameters. The results indicate that the rate of reconnection is chiefly dependent on the magnetic Reynolds number, but that the plasma flow is weakly dependent on this parameter, being more affected by the curvature of Incoming magnetic field. Scaling laws for the dimensions of the diffusion region are derived, and the existence of several reconnection regimes is shown. Using the same computer code we also simulate tearing modes in Cartesian geometry under different boundary conditions. By imposing a suitable perturbation a magnetic island is generated. The plasma flows show marked differences for the different boundary conditions implemented. Lastly, we examine some aspects of the coalescence instability. The usual flux function taken to represent a tearing node Island in the linear growth phase is shown to be erroneous, and we derive a correct expression. We show that under certain conditions there exists a threshold to coalescence that depends on the island wavenumbers and the associated perturbation.
1987-01-01T00:00:00ZColin, A. M.In many astrophysical problems magnetic reconnection plays a major role. Despite this reconnection theory remains incompletely understood, partly due to the strong non-linearity of the governing equations and the resulting difficulties in demonstrating analytical solutions. This thesis examines some fundamental aspects of reconnection theory: namely, the dynamics of driven and spontaneously reconnecting systems. We first consider the dynamics of a driven reconnecting system by numerically modelling a configuration consisting of two oppositely oriented flux systems with a variety of different boundary conditions and internal parameters. The results indicate that the rate of reconnection is chiefly dependent on the magnetic Reynolds number, but that the plasma flow is weakly dependent on this parameter, being more affected by the curvature of Incoming magnetic field. Scaling laws for the dimensions of the diffusion region are derived, and the existence of several reconnection regimes is shown. Using the same computer code we also simulate tearing modes in Cartesian geometry under different boundary conditions. By imposing a suitable perturbation a magnetic island is generated. The plasma flows show marked differences for the different boundary conditions implemented. Lastly, we examine some aspects of the coalescence instability. The usual flux function taken to represent a tearing node Island in the linear growth phase is shown to be erroneous, and we derive a correct expression. We show that under certain conditions there exists a threshold to coalescence that depends on the island wavenumbers and the associated perturbation.The nonlinear thermal evolution of coronal structuresMendoza Briceño, César Augustohttp://hdl.handle.net/10023/141932019-04-01T08:39:23Z1996-01-01T00:00:00ZThe thermal equilibrium and evolution of coronal structure is studied in this thesis. A symmetric and constant cross-sectional coronal loop is considered and, because of the strong magnetic field, the plasma is confined to move along the field lines, so that a one-dimensional problem can be assumed. We begin by giving a brief description of the Sun and corresponding phenomena. Then a discussion of the basic MHD equations is given. Here, it is assumed that the heating function is spatially dependent and the cooling function is due to an optically thin plasma. The thermal equilibrium of uniform-pressure coronal loop is investigated. The effects due to varying the values of the parameters involved in the governing equations are studied. It is found that there is a critical decay length of the heating below which a hot coronal loop does not exist. It is suggested that thermal non-equilibrium occurs, allowing the existence of catastrophic cooling. A study of the stability of the equilibrium up to the second order approximation is presented, and it is found that the response of the structure not only depends on the amplitude of the disturbance, but also on whether the disturbance increases or decreases the static temperature. The thermal evolution of the above structure is also investigated by assuming that the inertial terms are small. The previous results related to the critical heating decay length are verified. The numerical simulation shows that an initial hot plasma evolves to a new equilibrium which has a cool summit. This equilibrium is identified as a prominence-like solution. Further investigations are made in order to show how the structure can either evolve to a hot or a cool summit temperature depending on whether the initial conditions are above or below threshold values. The evolution of a disturbance increasing or decreasing an initial equilibrium temperature is followed numerically verifying the prediction made in the stability analysis. Furthermore, the effect of gravity is considered in the thermal equilibrium of loop. Similar results were found as studied for a constant-pressure loop. However, it was found that the critical values in which thermal non-equilibrium can occur is increased. A magnetic dip is also included in this model and the thermal equilibrium is studied. Finally, extensions of the present work is presented and some preliminary results are discussed.
1996-01-01T00:00:00ZMendoza Briceño, César AugustoThe thermal equilibrium and evolution of coronal structure is studied in this thesis. A symmetric and constant cross-sectional coronal loop is considered and, because of the strong magnetic field, the plasma is confined to move along the field lines, so that a one-dimensional problem can be assumed. We begin by giving a brief description of the Sun and corresponding phenomena. Then a discussion of the basic MHD equations is given. Here, it is assumed that the heating function is spatially dependent and the cooling function is due to an optically thin plasma. The thermal equilibrium of uniform-pressure coronal loop is investigated. The effects due to varying the values of the parameters involved in the governing equations are studied. It is found that there is a critical decay length of the heating below which a hot coronal loop does not exist. It is suggested that thermal non-equilibrium occurs, allowing the existence of catastrophic cooling. A study of the stability of the equilibrium up to the second order approximation is presented, and it is found that the response of the structure not only depends on the amplitude of the disturbance, but also on whether the disturbance increases or decreases the static temperature. The thermal evolution of the above structure is also investigated by assuming that the inertial terms are small. The previous results related to the critical heating decay length are verified. The numerical simulation shows that an initial hot plasma evolves to a new equilibrium which has a cool summit. This equilibrium is identified as a prominence-like solution. Further investigations are made in order to show how the structure can either evolve to a hot or a cool summit temperature depending on whether the initial conditions are above or below threshold values. The evolution of a disturbance increasing or decreasing an initial equilibrium temperature is followed numerically verifying the prediction made in the stability analysis. Furthermore, the effect of gravity is considered in the thermal equilibrium of loop. Similar results were found as studied for a constant-pressure loop. However, it was found that the critical values in which thermal non-equilibrium can occur is increased. A magnetic dip is also included in this model and the thermal equilibrium is studied. Finally, extensions of the present work is presented and some preliminary results are discussed.Alfvén waves in low-mass star-forming regionsMartin, Clare E.http://hdl.handle.net/10023/141902019-04-01T08:38:36Z1999-01-01T00:00:00ZLow-mass star-forming regions have a lifetime which is greater than their dynamical time and must therefore be, in an average sense, in mechanical equilibrium. The work presented here proposes that an equilibrium exists between the self-gravity, gas pressure, and the magnetic field and the waves it supports. Specifically the equilibrium in the direction perpendicular to the ordered magnetic field is given by the Lorentz force, while that parallel to the field is given by an Alfvén wave pressure force. The work detailed in this thesis models a low-mass star-forming region as a one-dimensional gas slab with a magnetic field lying perpendicular to the layer. Analytical, self-consistent models are formulated to study the equilibrium parallel to the background magnetic field. It is found that both short-wavelength (modelled using the WKB approximation) and large-amplitude, long-wavelength Alfvén waves can provide the necessary support parallel to the magnetic field, generating model cloud thicknesses that are consistent with the observations. The effect of damping by the linear process of ion-neutral friction is considered. It is found that the damping of the waves is not a necessary condition for the support of the cloud although it is an advantage. The possible sources of these waves are discussed. The Alfvén waves are also found to make an important contribution to the heating of a low-mass star-forming region. By modelling the dominant heating and cooling mechanisms in a molecular cloud, it is discovered that a cloud supported against its self-gravity by short-wavelength Alfvén waves will be hotter at its outer edge than in the central regions. These models successfully describe a low-mass star-forming region in equilibrium between its self-gravity, the gas pressure and an Alfvén wave pressure force. The question of the stability of such an equilibrium is considered, specifically that of an isothermal gas slab supported by short-wavelength Alfvén waves. The initial results suggest that the presence of a magnetic field and its associated Alfvén waves have a stabilising effect on the layer, and encourage further consideration of the role of Alfvén waves in low-mass star-forming regions.
1999-01-01T00:00:00ZMartin, Clare E.Low-mass star-forming regions have a lifetime which is greater than their dynamical time and must therefore be, in an average sense, in mechanical equilibrium. The work presented here proposes that an equilibrium exists between the self-gravity, gas pressure, and the magnetic field and the waves it supports. Specifically the equilibrium in the direction perpendicular to the ordered magnetic field is given by the Lorentz force, while that parallel to the field is given by an Alfvén wave pressure force. The work detailed in this thesis models a low-mass star-forming region as a one-dimensional gas slab with a magnetic field lying perpendicular to the layer. Analytical, self-consistent models are formulated to study the equilibrium parallel to the background magnetic field. It is found that both short-wavelength (modelled using the WKB approximation) and large-amplitude, long-wavelength Alfvén waves can provide the necessary support parallel to the magnetic field, generating model cloud thicknesses that are consistent with the observations. The effect of damping by the linear process of ion-neutral friction is considered. It is found that the damping of the waves is not a necessary condition for the support of the cloud although it is an advantage. The possible sources of these waves are discussed. The Alfvén waves are also found to make an important contribution to the heating of a low-mass star-forming region. By modelling the dominant heating and cooling mechanisms in a molecular cloud, it is discovered that a cloud supported against its self-gravity by short-wavelength Alfvén waves will be hotter at its outer edge than in the central regions. These models successfully describe a low-mass star-forming region in equilibrium between its self-gravity, the gas pressure and an Alfvén wave pressure force. The question of the stability of such an equilibrium is considered, specifically that of an isothermal gas slab supported by short-wavelength Alfvén waves. The initial results suggest that the presence of a magnetic field and its associated Alfvén waves have a stabilising effect on the layer, and encourage further consideration of the role of Alfvén waves in low-mass star-forming regions.Basic magnetic field configurations for solar filament channels and filamentsMackay, Duncan Hendryhttp://hdl.handle.net/10023/141882019-04-01T08:37:01Z1997-01-01T00:00:00ZThe three-dimensional magnetic structure of solar filament channels and filaments is considered. A simple analytical potential model of a filament channel is setup with line sources representing the overlying arcades and point sources the flux of the filament. A possible explanation of the distinct upper and lower bounds of a filament is given. A more detailed numerical force-free model with discrete flux sources is then developed and the effect of magnetic shear on the separatrix surface explored. Dextral channels are shown to exist for a wider range of negative values of the force-free alpha and sinistral channels for positive values of alpha. Potential models of a variety of coronal structures are then considered. The bending of a filament is modelled and a method of determining the horizontal component of a filament's magnetic field is proposed. Next, the observed opposite skew of arcades lying above switchbacks of polarity inversion lines is shown to be produced by a local flux imbalance at the corner of the switchback. Then, the magnetic structure of a particular filament in a filament channel is modelled using observations from a photospheric magnetogram. It is shown that dips in the filaments magnetic field could result from opposite polarity fragments lying below the filament. Finally, the formation of a specific filament channel and filament is modelled. The formation of the channel is shown to be due to the emergence of new flux in a sheared state. It is shown that convergence and reconnections between the new flux and old remnant flux is required for the filament to form. The field lines that represent the filament form a thin vertical sheet of flux. The changing angle of inclination of the sheet gives the appearance of twist. The method of formation is then generalised to other cases and it is shown that a hemispheric pattern consistent with the results of Martin et al. (1995) can be obtained.
1997-01-01T00:00:00ZMackay, Duncan HendryThe three-dimensional magnetic structure of solar filament channels and filaments is considered. A simple analytical potential model of a filament channel is setup with line sources representing the overlying arcades and point sources the flux of the filament. A possible explanation of the distinct upper and lower bounds of a filament is given. A more detailed numerical force-free model with discrete flux sources is then developed and the effect of magnetic shear on the separatrix surface explored. Dextral channels are shown to exist for a wider range of negative values of the force-free alpha and sinistral channels for positive values of alpha. Potential models of a variety of coronal structures are then considered. The bending of a filament is modelled and a method of determining the horizontal component of a filament's magnetic field is proposed. Next, the observed opposite skew of arcades lying above switchbacks of polarity inversion lines is shown to be produced by a local flux imbalance at the corner of the switchback. Then, the magnetic structure of a particular filament in a filament channel is modelled using observations from a photospheric magnetogram. It is shown that dips in the filaments magnetic field could result from opposite polarity fragments lying below the filament. Finally, the formation of a specific filament channel and filament is modelled. The formation of the channel is shown to be due to the emergence of new flux in a sheared state. It is shown that convergence and reconnections between the new flux and old remnant flux is required for the filament to form. The field lines that represent the filament form a thin vertical sheet of flux. The changing angle of inclination of the sheet gives the appearance of twist. The method of formation is then generalised to other cases and it is shown that a hemispheric pattern consistent with the results of Martin et al. (1995) can be obtained.Aspects of magnetic field theory in solar and laboratory plasmasLothian, Robert M.http://hdl.handle.net/10023/141832019-04-01T08:37:34Z1990-01-01T00:00:00ZUsing the Magnetohydrodynamic model, two problems in the behaviour of magnetic field structures are investigated. Firstly, the stability of tokamak equilibria to coupled tearing modes is calculated. Secondly, the equilibrium structure of a solar coronal loop is examined. The flux co-ordinate method is used to construct toroidal equilibria of the type found in large aspect ratio tokamaks. In such a field configuration, the analysis of tearing modes is complicated by the coupling of different poloidal fourier modes. The effect of coupling through elliptic shaping of plasma surfaces is calculated. For certain current profiles, this effect may cause instability. The response of coronal loops to twisting at their photospheric footpoints is investigated. Long loops are shown to have an essentially 1-D nature. This observation is used to develop a 1-D, line-tied model for such loops. This model is used to conduct an extensive survey of the non-linear twist regime, including the effects of enhanced fluid pressure. The possibility of non-equilibrium, which would provide energy for coronal heating and compact flares, is examined. When the physical variable of footpoint displacement is specified, no loss of equilibrium is found by twisting. Loss of equilibrium is found for high pressures, which we do not, however, expect to find in the corona.
1990-01-01T00:00:00ZLothian, Robert M.Using the Magnetohydrodynamic model, two problems in the behaviour of magnetic field structures are investigated. Firstly, the stability of tokamak equilibria to coupled tearing modes is calculated. Secondly, the equilibrium structure of a solar coronal loop is examined. The flux co-ordinate method is used to construct toroidal equilibria of the type found in large aspect ratio tokamaks. In such a field configuration, the analysis of tearing modes is complicated by the coupling of different poloidal fourier modes. The effect of coupling through elliptic shaping of plasma surfaces is calculated. For certain current profiles, this effect may cause instability. The response of coronal loops to twisting at their photospheric footpoints is investigated. Long loops are shown to have an essentially 1-D nature. This observation is used to develop a 1-D, line-tied model for such loops. This model is used to conduct an extensive survey of the non-linear twist regime, including the effects of enhanced fluid pressure. The possibility of non-equilibrium, which would provide energy for coronal heating and compact flares, is examined. When the physical variable of footpoint displacement is specified, no loss of equilibrium is found by twisting. Loss of equilibrium is found for high pressures, which we do not, however, expect to find in the corona.Exact solutions for axisymmetric and nonpolytropic astrophysical windsLima, João José de Faria Graça Afonsohttp://hdl.handle.net/10023/141802019-04-01T08:37:41Z1995-01-01T00:00:00ZAstrophysical outflows are common in a large variety of objects with very different length-scales. They can be almost spherical, as in the case of the solar wind, or show a high degree of anisotropy as in pre-main sequence stars, star-forming regions or even extragalactic objects. This work is aimed at finding exact solutions of the axisymmetric wind equations in which all variables depend not only on the distance to the central object but on latitude as well. The geometry of the stream/field-lines is taken as helicoidal and this seems to be a good approximation in some examples of collimated flows. From a simple hydrodynamic approach, a straightforward technique based on separation of the variables yields the most general solution of the wind equations under the above assumptions. The way the different variables depend on latitude is controlled by three anisotropy parameters which are related to typical ratios at the base of the atmosphere. The density needs to be higher at the equator than at the pole for the outflow to be able to accelerate. In these circumstances, the radial velocity always increases from equator to pole. Contrary to Parker's model of the solar wind, the solution does not pass through any critical point, since no polytropic law is assumed. However, the general behaviour is similar, with a high acceleration at the base and the velocity rapidly attaining an almost constant asymptotic value. The heating rate that sustains this rapid increase is mostly concentrated near the surface of the central object. The inclusion of the magnetic field in the analysis introduces two critical points: the Alfvenic point and an extra X -type point filtering the solution that gives a vanishing pressure at infinity. If the density anisotropy is too low the wind is unable to accelerate to large asymptotic values. The dependence of the angular velocity of the roots of the fieldlines with latitude reproduces well the observed rotation profile of photospheric magnetic features. The mass loss rate can be substantially increased if the structure of the outflow is highly anisotropic. Some applications to the solar wind are also discussed. In particular, recent results from ULYSSES (pointing out that solar speed increases with latitude while the density decreases from equator to the pole) are in good agreement with the general behaviour of the solutions presented in this work.
1995-01-01T00:00:00ZLima, João José de Faria Graça AfonsoAstrophysical outflows are common in a large variety of objects with very different length-scales. They can be almost spherical, as in the case of the solar wind, or show a high degree of anisotropy as in pre-main sequence stars, star-forming regions or even extragalactic objects. This work is aimed at finding exact solutions of the axisymmetric wind equations in which all variables depend not only on the distance to the central object but on latitude as well. The geometry of the stream/field-lines is taken as helicoidal and this seems to be a good approximation in some examples of collimated flows. From a simple hydrodynamic approach, a straightforward technique based on separation of the variables yields the most general solution of the wind equations under the above assumptions. The way the different variables depend on latitude is controlled by three anisotropy parameters which are related to typical ratios at the base of the atmosphere. The density needs to be higher at the equator than at the pole for the outflow to be able to accelerate. In these circumstances, the radial velocity always increases from equator to pole. Contrary to Parker's model of the solar wind, the solution does not pass through any critical point, since no polytropic law is assumed. However, the general behaviour is similar, with a high acceleration at the base and the velocity rapidly attaining an almost constant asymptotic value. The heating rate that sustains this rapid increase is mostly concentrated near the surface of the central object. The inclusion of the magnetic field in the analysis introduces two critical points: the Alfvenic point and an extra X -type point filtering the solution that gives a vanishing pressure at infinity. If the density anisotropy is too low the wind is unable to accelerate to large asymptotic values. The dependence of the angular velocity of the roots of the fieldlines with latitude reproduces well the observed rotation profile of photospheric magnetic features. The mass loss rate can be substantially increased if the structure of the outflow is highly anisotropic. Some applications to the solar wind are also discussed. In particular, recent results from ULYSSES (pointing out that solar speed increases with latitude while the density decreases from equator to the pole) are in good agreement with the general behaviour of the solutions presented in this work.Chromospheric effects on global solar oscillationsJohnston, Alanhttp://hdl.handle.net/10023/141732019-04-01T08:37:13Z1994-01-01T00:00:00ZA study has been made of the global solar oscillations known as p-modes. The Sun is represented by a plane-parallel stratified plasma. Solutions are found to the magnetohydrodynamic equations of motion in such a plasma, and normal mode frequencies are calculated by applying realistic boundary conditions to these solutions. The normal modes model solar p-modes. For a model consisting of an isothermal chromosphere with a uniform horizontal magnetic field, it is demonstrated that modes may form at all frequencies. Consideration is also given to the related problem of vertical propagation of fast magnetoacoustic waves in a uniform magnetic field. An investigation is carried out into the observed solar cycle variations in the frequencies of p-modes in the classical, low frequency range (1-5 mHz). A possible mechanism for the observed "turnover" effect is discussed. Through the use of a modified Bohr- Sommerfeld condition, the effect of a non-isothermal chromosphere is also considered, and a physical description of chromospheric effects on p-mode frequencies is given. The formation of modes above the acoustic cut-off frequency is investigated. The theoretically calcidated forms of frequency shift curves in this high frequency range agree well with observations. The special case of modes of degree zero is also briefly examined. A mathematical formulation for such modes is constructed, and frequency shifts are determined for a simple chromospheric model atmosphere.
1994-01-01T00:00:00ZJohnston, AlanA study has been made of the global solar oscillations known as p-modes. The Sun is represented by a plane-parallel stratified plasma. Solutions are found to the magnetohydrodynamic equations of motion in such a plasma, and normal mode frequencies are calculated by applying realistic boundary conditions to these solutions. The normal modes model solar p-modes. For a model consisting of an isothermal chromosphere with a uniform horizontal magnetic field, it is demonstrated that modes may form at all frequencies. Consideration is also given to the related problem of vertical propagation of fast magnetoacoustic waves in a uniform magnetic field. An investigation is carried out into the observed solar cycle variations in the frequencies of p-modes in the classical, low frequency range (1-5 mHz). A possible mechanism for the observed "turnover" effect is discussed. Through the use of a modified Bohr- Sommerfeld condition, the effect of a non-isothermal chromosphere is also considered, and a physical description of chromospheric effects on p-mode frequencies is given. The formation of modes above the acoustic cut-off frequency is investigated. The theoretically calcidated forms of frequency shift curves in this high frequency range agree well with observations. The special case of modes of degree zero is also briefly examined. A mathematical formulation for such modes is constructed, and frequency shifts are determined for a simple chromospheric model atmosphere.Theoretical modelling of global oscillations in solar prominencesJoarder, Parthasarathihttp://hdl.handle.net/10023/141692019-04-01T08:38:16Z1994-01-01T00:00:00ZThis thesis aims to provide a basic theoretical explanation for the oscillatory motions observed in solar quiescent prominences. The prominence is treated as a simple plasma slab embedded in a hotter and rarer uniform coronal plasma. Both the slab and its environment are permeated by a uniform magnetic field. The field lines are anchored at rigid walls placed on either side of the plasma slab and representing the photospheric line-tying effect. The magnetohydrodynamic modes of oscillation of the plasma slab are then examined for different orientations of the magnetic field with respect to the long axis of the slab. Particularly interesting in this study is the appearance of the 'string MHD' modes that are analogous to the fundamental vibrations of a mass- loaded stretched elastic string. Such modes appear whenever the magnetic field vector is inclined to the long axis of the slab, thus producing a magnetic field component in the direction transverse to the axis of the slab. Observationally, this inclination of the field is generally small. For realistic values of the angle of inclination of the magnetic field lines, the 'string Alfven' mode and an 'internal slow' mode yield periods in the range 1/2-2 hr. These modes may correspond to the observed long period (40-90 minutes) oscillations in quiescent prominences. Intermediate periodicities, in the range 8-20 min, may be associated with an 'internal Alfven' mode and a 'fast string' mode of the prominence slab. The observed short periodicities, in the range 2-5 min, may correspond to an 'internal fast' mode in prominences. Having thus established a foundation for the theoretical modelling of prominence oscillations in terms of the magnetohydrodynamic modes of oscillation of a non-gravitating plasma slab, we discuss several factors, such as the effects of gravitational stratification, the curvature of the magnetic field lines, and the fine-structures in a prominence, that may complicate a description of its oscillatory modes. Some preliminary investigations of simple magnetohydrostatic equilibrium models suggest that gravity and the curvature of the magnetic field lines play only a secondary role in determining the periods of the oscillatory modes in prominences, the basic structure of the modes being similar to that present in simple slab models.
1994-01-01T00:00:00ZJoarder, ParthasarathiThis thesis aims to provide a basic theoretical explanation for the oscillatory motions observed in solar quiescent prominences. The prominence is treated as a simple plasma slab embedded in a hotter and rarer uniform coronal plasma. Both the slab and its environment are permeated by a uniform magnetic field. The field lines are anchored at rigid walls placed on either side of the plasma slab and representing the photospheric line-tying effect. The magnetohydrodynamic modes of oscillation of the plasma slab are then examined for different orientations of the magnetic field with respect to the long axis of the slab. Particularly interesting in this study is the appearance of the 'string MHD' modes that are analogous to the fundamental vibrations of a mass- loaded stretched elastic string. Such modes appear whenever the magnetic field vector is inclined to the long axis of the slab, thus producing a magnetic field component in the direction transverse to the axis of the slab. Observationally, this inclination of the field is generally small. For realistic values of the angle of inclination of the magnetic field lines, the 'string Alfven' mode and an 'internal slow' mode yield periods in the range 1/2-2 hr. These modes may correspond to the observed long period (40-90 minutes) oscillations in quiescent prominences. Intermediate periodicities, in the range 8-20 min, may be associated with an 'internal Alfven' mode and a 'fast string' mode of the prominence slab. The observed short periodicities, in the range 2-5 min, may correspond to an 'internal fast' mode in prominences. Having thus established a foundation for the theoretical modelling of prominence oscillations in terms of the magnetohydrodynamic modes of oscillation of a non-gravitating plasma slab, we discuss several factors, such as the effects of gravitational stratification, the curvature of the magnetic field lines, and the fine-structures in a prominence, that may complicate a description of its oscillatory modes. Some preliminary investigations of simple magnetohydrostatic equilibrium models suggest that gravity and the curvature of the magnetic field lines play only a secondary role in determining the periods of the oscillatory modes in prominences, the basic structure of the modes being similar to that present in simple slab models.Magnetic surface effects on solar oscillationsJain, Rekhahttp://hdl.handle.net/10023/141532019-04-01T08:37:46Z1993-01-01T00:00:00ZThis thesis is concerned with the effects of magnetic atmospheres on solar oscillations. The behaviour of magnetohydrodynamic surface waves propagating on a single magnetic interface is discussed ignoring the effects of gravity. The effects of non-parallel propagation (where the wave vector is at an angle to the magnetic field direction) are considered. The effects of chromospheric magnetic fields on solar p- and f-modes in a stratified atmosphere are examined for three different models. In the first of these models, the chromosphere is assumed to be isothermal and permeated by a uniform and horizontal magnetic field. A dispersion relation for the p-modes trapped below such an atmosphere is derived. Asymptotic and numerical solutions for the p-modes are discussed in detail. An increase in chromospheric magnetic field strength leads to an increase in the frequency of the p-modes, whereas an increase in the chromospheric temperature leads to a decrease in the frequencies of these modes. Comparison with observational data suggests that both these effects may indeed take place. The second model is set up for magnetic fields which decrease with height in such a way that the Alfven speed remains constant. In addition to magnetic effects, the effects of non-parallel propagation on and f-modes are considered in the presence of such a non-uniform magnetic field. After deriving a very general dispersion relation, various asymptotic and numerical solutions have been obtained and the possible effects of magnetic fields and non-parallel propagation on these modes are examined. The presence of a horizontal non-uniform chromospheric field produces changes in the frequencies of the p- and f-modes, reducing the frequencies of p-modes and increasing the frequency of the f-mode. Besides depending upon magnetic field strength, frequencies also depend on both the mode's order n and its degree l. The effects of non-parallel propagation are found to be most significant for the f-mode and the low order p-modes. The magnetic structure of the chromosphere has been further generalised by combining the two models described above. In this three layer model, a dispersion relation is derived in a general manner and discussed in detail for the p-modes. The role of magnetoacoustic cut-off frequency is studied. Again, the results are qualitatively similar to those found from observation.
1993-01-01T00:00:00ZJain, RekhaThis thesis is concerned with the effects of magnetic atmospheres on solar oscillations. The behaviour of magnetohydrodynamic surface waves propagating on a single magnetic interface is discussed ignoring the effects of gravity. The effects of non-parallel propagation (where the wave vector is at an angle to the magnetic field direction) are considered. The effects of chromospheric magnetic fields on solar p- and f-modes in a stratified atmosphere are examined for three different models. In the first of these models, the chromosphere is assumed to be isothermal and permeated by a uniform and horizontal magnetic field. A dispersion relation for the p-modes trapped below such an atmosphere is derived. Asymptotic and numerical solutions for the p-modes are discussed in detail. An increase in chromospheric magnetic field strength leads to an increase in the frequency of the p-modes, whereas an increase in the chromospheric temperature leads to a decrease in the frequencies of these modes. Comparison with observational data suggests that both these effects may indeed take place. The second model is set up for magnetic fields which decrease with height in such a way that the Alfven speed remains constant. In addition to magnetic effects, the effects of non-parallel propagation on and f-modes are considered in the presence of such a non-uniform magnetic field. After deriving a very general dispersion relation, various asymptotic and numerical solutions have been obtained and the possible effects of magnetic fields and non-parallel propagation on these modes are examined. The presence of a horizontal non-uniform chromospheric field produces changes in the frequencies of the p- and f-modes, reducing the frequencies of p-modes and increasing the frequency of the f-mode. Besides depending upon magnetic field strength, frequencies also depend on both the mode's order n and its degree l. The effects of non-parallel propagation are found to be most significant for the f-mode and the low order p-modes. The magnetic structure of the chromosphere has been further generalised by combining the two models described above. In this three layer model, a dispersion relation is derived in a general manner and discussed in detail for the p-modes. The role of magnetoacoustic cut-off frequency is studied. Again, the results are qualitatively similar to those found from observation.Thermal instabilities in the solar coronaIreland, Richard C.http://hdl.handle.net/10023/141502019-04-01T08:38:40Z1995-01-01T00:00:00ZIn this thesis, several problems relating to thermal instabilities in the solar corona are examined. Chapter 1 gives a brief description of the Sun and corresponding events with particular attention focused on prominences, their formation and eruption. Various problems concerning thermal instabilities are then tackled in the later Chapters. In Chapter 2, the basic MHD equations are introduced and a physical description of the thermal instability mechanism given. The MHD equations are linearised in a uniform, infinite medium and the basic instability criteria obtained. Chapter 3 investigates the normal mode spectrum for the linearised MHD equations for a cylindrical equilibrium. This spectrum is examined for zero perpendicular thermal conduction, with both zero and non-zero scalar resistivity. Particular attention is paid to the continuous branches of this spectrum, or continuous spectra. For zero resistivity there are three types of continuous spectra present, namely the Alfven, slow and thermal continua. It is shown that when dissipation due to resistivity is included, the slow and Alfven continua are removed and the thermal continuum is shifted to a different position (where the shift is independent of the exact value of resistivity). The 'old' location of the thermal continuum is covered by a dense set of nearly singular discrete modes called a quasi-continuum, for equilibria with the thermal time scale much smaller than the Alfven time scale. This quasi-continuum is investigated numerically and the eigenfunctions are shown to have rapid spatial oscillating behaviour. These oscillations are confined to the most unstable part of the equilibrium based on the Field criterion and may be the cause of fine structure in prominences. In Chapter 4, the normal mode spectrum for the linearised MHD equations is examined for a plasma in a cylindrical equilibrium. The equations describing these normal modes are solved numerically using a finite element code. In the ideal case the Hain-Lust equation is expanded and a WKB solution obtained for large axial wave numbers. This is compared to the numerical solutions. In the non-ideal case, the ballooning equations describing localised modes are manipulated in an arcade geometry and a dispersion relation derived. It is illustrated that as the axial wave number k is increased, the fundamental thermal and Alfven modes can coalesce to form overstable magnetothermal modes. The ratio between the magnetic and thermal terms is varied and the existence of the magnetothermal modes examined. The corresponding growth rates are predicted by a WKB solution to the ballooning equations. The interaction of thermal and magnetic instabilities and the existence of these magnetothermal modes may be significant in the eruption of prominences into solar flares. Chapter 5 extends the work presented in Chapter 4 to include the effects of line-tying in a coronal arcade. The ballooning equations which were introduced in Chapter 4 are manipulated to give a dispersion relation. This relation is a quadratic in the square of the azimuthal wave number m if parallel thermal conduction is neglected and a cubic in m2 if parallel conduction is included. Rigid wall boundary conditions are applied to this dispersion relation. This dispersion relation is then solved numerically subject to these boundary conditions and the solutions plotted. Unfortunately the expression for the thermal continuum in line-tied arcades is required since the thermal continuum must play an important role in the proceedings. This calculation is left for future work. From the results obtained, it can be seen that the thermal instability can play a major part in prominence formation and destruction. The thermal instability may help create the prominence. Resistivity and perpendicular thermal conduction can cause of the observed fine scale structure. Finally, a neighbouring thermal instability may trigger a magnetic instability that causes the prominence to erupt.
1995-01-01T00:00:00ZIreland, Richard C.In this thesis, several problems relating to thermal instabilities in the solar corona are examined. Chapter 1 gives a brief description of the Sun and corresponding events with particular attention focused on prominences, their formation and eruption. Various problems concerning thermal instabilities are then tackled in the later Chapters. In Chapter 2, the basic MHD equations are introduced and a physical description of the thermal instability mechanism given. The MHD equations are linearised in a uniform, infinite medium and the basic instability criteria obtained. Chapter 3 investigates the normal mode spectrum for the linearised MHD equations for a cylindrical equilibrium. This spectrum is examined for zero perpendicular thermal conduction, with both zero and non-zero scalar resistivity. Particular attention is paid to the continuous branches of this spectrum, or continuous spectra. For zero resistivity there are three types of continuous spectra present, namely the Alfven, slow and thermal continua. It is shown that when dissipation due to resistivity is included, the slow and Alfven continua are removed and the thermal continuum is shifted to a different position (where the shift is independent of the exact value of resistivity). The 'old' location of the thermal continuum is covered by a dense set of nearly singular discrete modes called a quasi-continuum, for equilibria with the thermal time scale much smaller than the Alfven time scale. This quasi-continuum is investigated numerically and the eigenfunctions are shown to have rapid spatial oscillating behaviour. These oscillations are confined to the most unstable part of the equilibrium based on the Field criterion and may be the cause of fine structure in prominences. In Chapter 4, the normal mode spectrum for the linearised MHD equations is examined for a plasma in a cylindrical equilibrium. The equations describing these normal modes are solved numerically using a finite element code. In the ideal case the Hain-Lust equation is expanded and a WKB solution obtained for large axial wave numbers. This is compared to the numerical solutions. In the non-ideal case, the ballooning equations describing localised modes are manipulated in an arcade geometry and a dispersion relation derived. It is illustrated that as the axial wave number k is increased, the fundamental thermal and Alfven modes can coalesce to form overstable magnetothermal modes. The ratio between the magnetic and thermal terms is varied and the existence of the magnetothermal modes examined. The corresponding growth rates are predicted by a WKB solution to the ballooning equations. The interaction of thermal and magnetic instabilities and the existence of these magnetothermal modes may be significant in the eruption of prominences into solar flares. Chapter 5 extends the work presented in Chapter 4 to include the effects of line-tying in a coronal arcade. The ballooning equations which were introduced in Chapter 4 are manipulated to give a dispersion relation. This relation is a quadratic in the square of the azimuthal wave number m if parallel thermal conduction is neglected and a cubic in m2 if parallel conduction is included. Rigid wall boundary conditions are applied to this dispersion relation. This dispersion relation is then solved numerically subject to these boundary conditions and the solutions plotted. Unfortunately the expression for the thermal continuum in line-tied arcades is required since the thermal continuum must play an important role in the proceedings. This calculation is left for future work. From the results obtained, it can be seen that the thermal instability can play a major part in prominence formation and destruction. The thermal instability may help create the prominence. Resistivity and perpendicular thermal conduction can cause of the observed fine scale structure. Finally, a neighbouring thermal instability may trigger a magnetic instability that causes the prominence to erupt.Heating of turbulent solar and laboratory plasmasInverarity, Gordon W.http://hdl.handle.net/10023/141462019-04-01T08:38:17Z1995-01-01T00:00:00ZThe model of Heyvaerts and Priest (1992) for steady-state heating of the turbulent medium within a sheared solar coronal arcade structure is here developed. The energy input into the corona is calculated at the large scales of the model. At the smaller scales the effects of coronal turbulence are modelled in the form of an enhanced turbulent viscosity and magnetic diffusivity, which are related to the injected power density in the steady state. Matching the expressions for the injected and dissipated power enables the calculation of a heating power consistent with both boundary motions and turbulent effects with a minimum of arbitrary parameters - the price to be paid is that the inertial range spectrum must be prescribed and imposed at all scales. While it is capable of reproducing the observed levels of coronal heating (300 Wm⁻² 3x10⁵ erg cm⁻² s⁻ⁱ for the quiet Sun, 800 Wm⁻² (8 x 10⁵ erg cm⁻² s⁻ⁱ) for a coronal hole and 10⁴ Wm ⁻² (10⁷ erg cm⁻² s⁻ⁱ) for an active region (Withbroe and Noyes, 1977)), there are some mathematical and physical difficulties present. These are eliminated as far as is possible and it is found that the final results for heating levels differ little from the original model although there is a much greater consistency between the imposed and predicted energy power spectra. The modified approach is applied to the problems of photospheric motions twisting a coronal flux tube and of rapid motions injecting Alfven waves into an arcade. In the former case comparable levels of heating are obtained. For a driven and damped standing wave, however, desired levels of heating are only obtained when a global resonance occurs. Attempts are also made to find similar steady-state equilibria possessing flow for fusion experiments in order to apply the above procedure to investigate turbulence in laboratory plasmas. This has been hampered by the difficulty in finding simple appropriate equilibria with many scales present.
1995-01-01T00:00:00ZInverarity, Gordon W.The model of Heyvaerts and Priest (1992) for steady-state heating of the turbulent medium within a sheared solar coronal arcade structure is here developed. The energy input into the corona is calculated at the large scales of the model. At the smaller scales the effects of coronal turbulence are modelled in the form of an enhanced turbulent viscosity and magnetic diffusivity, which are related to the injected power density in the steady state. Matching the expressions for the injected and dissipated power enables the calculation of a heating power consistent with both boundary motions and turbulent effects with a minimum of arbitrary parameters - the price to be paid is that the inertial range spectrum must be prescribed and imposed at all scales. While it is capable of reproducing the observed levels of coronal heating (300 Wm⁻² 3x10⁵ erg cm⁻² s⁻ⁱ for the quiet Sun, 800 Wm⁻² (8 x 10⁵ erg cm⁻² s⁻ⁱ) for a coronal hole and 10⁴ Wm ⁻² (10⁷ erg cm⁻² s⁻ⁱ) for an active region (Withbroe and Noyes, 1977)), there are some mathematical and physical difficulties present. These are eliminated as far as is possible and it is found that the final results for heating levels differ little from the original model although there is a much greater consistency between the imposed and predicted energy power spectra. The modified approach is applied to the problems of photospheric motions twisting a coronal flux tube and of rapid motions injecting Alfven waves into an arcade. In the former case comparable levels of heating are obtained. For a driven and damped standing wave, however, desired levels of heating are only obtained when a global resonance occurs. Attempts are also made to find similar steady-state equilibria possessing flow for fusion experiments in order to apply the above procedure to investigate turbulence in laboratory plasmas. This has been hampered by the difficulty in finding simple appropriate equilibria with many scales present.WKB estimates to the critical length of twisted solar coronal loopsHerbert, Simon I.http://hdl.handle.net/10023/140922019-04-01T08:38:44Z1995-01-01T00:00:00ZThe solar corona exhibits many different phenomena, observable from the Earth or space. Magnetohydrodynamic stability theory provides a method of investigating these phenomena by using it to test proposed mathematical models. WKB is a way of approximating the solutions of second order linear homogeneous differential equations with large parameters and so together with MHD stability theory, models for solar coronal loops can be investigated. In this thesis, the problem of a line tied twisted coronal loop is studied within the framework of ideal MHD using a WKB approximation to estimate the critical length at which the various magnetic fields become unstable. The problem will be split into two halves: (i) force-free and (ii) non force-free fields. Using a finite element/Fourier method, the full MHD equations will be solved numerically and the results compared with analytical solutions.
1995-01-01T00:00:00ZHerbert, Simon I.The solar corona exhibits many different phenomena, observable from the Earth or space. Magnetohydrodynamic stability theory provides a method of investigating these phenomena by using it to test proposed mathematical models. WKB is a way of approximating the solutions of second order linear homogeneous differential equations with large parameters and so together with MHD stability theory, models for solar coronal loops can be investigated. In this thesis, the problem of a line tied twisted coronal loop is studied within the framework of ideal MHD using a WKB approximation to estimate the critical length at which the various magnetic fields become unstable. The problem will be split into two halves: (i) force-free and (ii) non force-free fields. Using a finite element/Fourier method, the full MHD equations will be solved numerically and the results compared with analytical solutions.Solar coronal stability problemsHardie, Ian S.http://hdl.handle.net/10023/140902019-04-01T08:37:51Z1993-01-01T00:00:00ZMagnetohydrodynamic stability theory provides a powerful tool for understanding and testing hypothesized mathematical and physical models of observed phenomena on the surface of the Sun. In this thesis, the problem of applying the 'correct' boundary conditions at the photospheric/coronal interface used in modelling coronal arcades is tackled. Then some aspects of the stability of coronal loops and arcades are investigated using a Fourier truncated series approximation for the equation of motion. The problem involving the boundary conditions has been the subject of a controversy for the past decade with two principal conditions suggested, the 'rigid-wall' conditions where all perturbations vanish at the interface, and 'flow-through' conditions where flows parallel to the equilibrium magnetic field take place. By modelling the photosphere and corona as two different density regions and then varying the ratio of the densities of the two regions, growth rates and eigen-functions of both ideal and resistive modes are investigated in order to follow the evolution of the modes as the density ratio is increased. In order to simplify the analysis, the 2-D equations are reduced to 1-D equations by taking a WKB approximation for the spatial variations across the field to give a localized ballooning approach with ordinary differential equations along the fieldlines. Stability of coronal loops to kink modes transformed to localized modes by increasing the poloidal wavenumber, m, is investigated. Two fields generated numerically from the Grad-Shafranov equation and three analytic fields are investigated in detail and the effect of pressure on the marginal loop length is found, both for near force-free conditions such as is found in the solar corona, and away from force-free conditions. It was found that for near force-free conditions, kink modes are the most unstable with localized modes the most stable. As pressure and pressure gradients become important, there is a reversal in the most unstable modes with localized modes the most unstable.
1993-01-01T00:00:00ZHardie, Ian S.Magnetohydrodynamic stability theory provides a powerful tool for understanding and testing hypothesized mathematical and physical models of observed phenomena on the surface of the Sun. In this thesis, the problem of applying the 'correct' boundary conditions at the photospheric/coronal interface used in modelling coronal arcades is tackled. Then some aspects of the stability of coronal loops and arcades are investigated using a Fourier truncated series approximation for the equation of motion. The problem involving the boundary conditions has been the subject of a controversy for the past decade with two principal conditions suggested, the 'rigid-wall' conditions where all perturbations vanish at the interface, and 'flow-through' conditions where flows parallel to the equilibrium magnetic field take place. By modelling the photosphere and corona as two different density regions and then varying the ratio of the densities of the two regions, growth rates and eigen-functions of both ideal and resistive modes are investigated in order to follow the evolution of the modes as the density ratio is increased. In order to simplify the analysis, the 2-D equations are reduced to 1-D equations by taking a WKB approximation for the spatial variations across the field to give a localized ballooning approach with ordinary differential equations along the fieldlines. Stability of coronal loops to kink modes transformed to localized modes by increasing the poloidal wavenumber, m, is investigated. Two fields generated numerically from the Grad-Shafranov equation and three analytic fields are investigated in detail and the effect of pressure on the marginal loop length is found, both for near force-free conditions such as is found in the solar corona, and away from force-free conditions. It was found that for near force-free conditions, kink modes are the most unstable with localized modes the most stable. As pressure and pressure gradients become important, there is a reversal in the most unstable modes with localized modes the most unstable.Instability and wave-growth within some oscillatory fluid flowsForster, Graham Keithhttp://hdl.handle.net/10023/140872019-04-01T08:37:23Z1996-01-01T00:00:00ZOscillatory fluid flows arise naturally in many systems. Whether or not these systems are stable is an important question and external periodic forcing of the flow may result in rich and complicated behaviours. Here three distinct oscillatory fluid flows are examined in detail, with the stability of each being established using a range of analytical and computational methods. The first system comprises standing surface capillary-gravity waves in second-harmonic resonance subject to Faraday excitation. Using the perturbation technique of multiple scales, the amplitude equations for the system are derived. At exact resonance, and with the absence of damping, the only fixed point of the equations is found to be the origin. A computational approach reveals that the amplitudes of the two waves remain either bounded or grow to infinity depending on initial data. With the introduction of detuning and damping into the system families of fixed points now exist and some special cases are considered. The second class of flows are unbounded time-periodic flows with fixed ellipsoidal stream surfaces, and having spatially uniform but time-periodic strain rates. Using a recently developed method based on theoretical study of the Schrodinger equation with quasi-periodic potential, a computational approach is adopted which determines the stability of the flow to three-dimensional plane wave disturbances. Results for the growth rate and winding number of the disturbance clearly reveal the regions of instability. It is found that almost all these flows are highly unstable. The third class is another set of three-dimensional time-periodic flows with spatially uniform strain rates. These flows are non-axisymmetric and have sinusoidally-fluctuating rates of strain directed along the fixed coordinate axes. The same computational method is employed and it is found that instability increases along with the non-axisymmetric nature of the flow.
1996-01-01T00:00:00ZForster, Graham KeithOscillatory fluid flows arise naturally in many systems. Whether or not these systems are stable is an important question and external periodic forcing of the flow may result in rich and complicated behaviours. Here three distinct oscillatory fluid flows are examined in detail, with the stability of each being established using a range of analytical and computational methods. The first system comprises standing surface capillary-gravity waves in second-harmonic resonance subject to Faraday excitation. Using the perturbation technique of multiple scales, the amplitude equations for the system are derived. At exact resonance, and with the absence of damping, the only fixed point of the equations is found to be the origin. A computational approach reveals that the amplitudes of the two waves remain either bounded or grow to infinity depending on initial data. With the introduction of detuning and damping into the system families of fixed points now exist and some special cases are considered. The second class of flows are unbounded time-periodic flows with fixed ellipsoidal stream surfaces, and having spatially uniform but time-periodic strain rates. Using a recently developed method based on theoretical study of the Schrodinger equation with quasi-periodic potential, a computational approach is adopted which determines the stability of the flow to three-dimensional plane wave disturbances. Results for the growth rate and winding number of the disturbance clearly reveal the regions of instability. It is found that almost all these flows are highly unstable. The third class is another set of three-dimensional time-periodic flows with spatially uniform strain rates. These flows are non-axisymmetric and have sinusoidally-fluctuating rates of strain directed along the fixed coordinate axes. The same computational method is employed and it is found that instability increases along with the non-axisymmetric nature of the flow.The effects of magnetic fields on oscillations in the solar atmosphereEvans, David J.http://hdl.handle.net/10023/140822019-04-01T08:38:44Z1990-01-01T00:00:00ZA study has been made of wave propagation in two regions of the solar atmosphere in which magnetic forces are significant. Sunspot observations indicate a rich variety of characteristic modes of oscillation roughly divided into three categories: three minute umbral oscillations, five minute umbral oscillations and penumbral waves. Outside of intense magnetic flux concentrations the oscillation spectrum is dominated by the five minute period. These waves are trapped in a cavity whose upper boundary may be affected by the magnetism of the chromosphere. A sunspot has been modelled by a uniform cylindrical flux tube. The allowable modes of oscillation are found to vary as the atmospheric parameters change with depth. Umbral three minute oscillations are interpreted as slow body modes. The umbral five minute oscillations arise through a complicated interaction with acoustic waves outside the sunspot. This drives fast body modes as well as waves simply passing through the flux tube. The former may propagate upwards and become fast surface waves. Fast and slow surface waves may explain some of the oscillations of the penumbra. The magnetic structure of the chromosphere has been modelled as an isothermal atmosphere permeated by a uniform and horizontal magnetic field. A dispersion relation for the trapped waves below such an atmosphere has been derived and both asymptotic and numerical solutions found. The effect of a uniform magnetic field is to increase the frequency of the trapped modes. A physical explanation for these changes in frequency has been put forward. Observational evidence may indicate that such effects are indeed seen. This model has been further generalised to take some account of the variation in canopy height which has been observed.
1990-01-01T00:00:00ZEvans, David J.A study has been made of wave propagation in two regions of the solar atmosphere in which magnetic forces are significant. Sunspot observations indicate a rich variety of characteristic modes of oscillation roughly divided into three categories: three minute umbral oscillations, five minute umbral oscillations and penumbral waves. Outside of intense magnetic flux concentrations the oscillation spectrum is dominated by the five minute period. These waves are trapped in a cavity whose upper boundary may be affected by the magnetism of the chromosphere. A sunspot has been modelled by a uniform cylindrical flux tube. The allowable modes of oscillation are found to vary as the atmospheric parameters change with depth. Umbral three minute oscillations are interpreted as slow body modes. The umbral five minute oscillations arise through a complicated interaction with acoustic waves outside the sunspot. This drives fast body modes as well as waves simply passing through the flux tube. The former may propagate upwards and become fast surface waves. Fast and slow surface waves may explain some of the oscillations of the penumbra. The magnetic structure of the chromosphere has been modelled as an isothermal atmosphere permeated by a uniform and horizontal magnetic field. A dispersion relation for the trapped waves below such an atmosphere has been derived and both asymptotic and numerical solutions found. The effect of a uniform magnetic field is to increase the frequency of the trapped modes. A physical explanation for these changes in frequency has been put forward. Observational evidence may indicate that such effects are indeed seen. This model has been further generalised to take some account of the variation in canopy height which has been observed.Magnetic helicity and force-free equilibria in the solar corona and in laboratory devicesDixon, Andrew Michaelhttp://hdl.handle.net/10023/140802019-04-01T08:38:53Z1988-01-01T00:00:00ZForce-free equilibria are believed to be important in both an astrophysical and a laboratory context as minimum-energy configurations (see, for example, Woltjer, 1958; Taylor, 1974). Associated is the study of magnetic helicity and its invariance. In Chapter Two of this thesis we put forward a means of heating the corona by the rotation of the foot-points of a coronal "sunspot" magnetic field anchored in the photosphere. The method adopted is essentially that of Heyvaerts and Priest (1984), employing Taylor's Hypothesis (Taylor, 1974) and a magnetic helicity evolution equation. A characteristic of the Reversed-Field Pinch device is the appearance, at high enough values of the quantity "volt-seconds over toroidal flux", of a helical distortion to the basic axi-symmetric state. In Chapter Three we look for corresponding behaviour in the "sunspot equilibrium" of the previous chapter, with limited success. However, we go on to formulate a method of calculating general axi-symmetric fields above a sunspot given the normal field component at the photosphere. Chapters Four, Five and Six are concerned with equilibrium force-free fields in a sphere. The main aim here is the calculation minimum-energy configurations having magnetic flux crossing the boundary, and so we employ "relative helicity" (Berger and Field, 1984). In Chapter Four we consider the "P1(cos𝜃)" boundary radial field, finding that the minimum-energy state is always purely symmetric. In Chapter Five we treat the "P2(cos𝜃)" boundary condition. We find in this case that a "mixed state" is theoretically possible for high enough values of the helicity. In Chapter Six, we consider a general boundary field, which we use to model point sources of magnetic flux at the boundary of a spheromak, finding that in practice an axi-symmetric configuration is always the minimum-energy state. Finally, in Chapter Seven we present an extension to the theorem of Woltjer (1958), concerning the minimization of the magnetic energy of a magnetic structure, to include the case of a free boundary subjected to external pressure forces. To illustrate the theory, we have provided three applications, the first to a finite cylindrical flux and the remainder to possible spheromak configurations.
1988-01-01T00:00:00ZDixon, Andrew MichaelForce-free equilibria are believed to be important in both an astrophysical and a laboratory context as minimum-energy configurations (see, for example, Woltjer, 1958; Taylor, 1974). Associated is the study of magnetic helicity and its invariance. In Chapter Two of this thesis we put forward a means of heating the corona by the rotation of the foot-points of a coronal "sunspot" magnetic field anchored in the photosphere. The method adopted is essentially that of Heyvaerts and Priest (1984), employing Taylor's Hypothesis (Taylor, 1974) and a magnetic helicity evolution equation. A characteristic of the Reversed-Field Pinch device is the appearance, at high enough values of the quantity "volt-seconds over toroidal flux", of a helical distortion to the basic axi-symmetric state. In Chapter Three we look for corresponding behaviour in the "sunspot equilibrium" of the previous chapter, with limited success. However, we go on to formulate a method of calculating general axi-symmetric fields above a sunspot given the normal field component at the photosphere. Chapters Four, Five and Six are concerned with equilibrium force-free fields in a sphere. The main aim here is the calculation minimum-energy configurations having magnetic flux crossing the boundary, and so we employ "relative helicity" (Berger and Field, 1984). In Chapter Four we consider the "P1(cos𝜃)" boundary radial field, finding that the minimum-energy state is always purely symmetric. In Chapter Five we treat the "P2(cos𝜃)" boundary condition. We find in this case that a "mixed state" is theoretically possible for high enough values of the helicity. In Chapter Six, we consider a general boundary field, which we use to model point sources of magnetic flux at the boundary of a spheromak, finding that in practice an axi-symmetric configuration is always the minimum-energy state. Finally, in Chapter Seven we present an extension to the theorem of Woltjer (1958), concerning the minimization of the magnetic energy of a magnetic structure, to include the case of a free boundary subjected to external pressure forces. To illustrate the theory, we have provided three applications, the first to a finite cylindrical flux and the remainder to possible spheromak configurations.MHD flows in the solar atmosphereDel Zanna, Lucahttp://hdl.handle.net/10023/140752019-04-01T08:37:17Z1997-01-01T00:00:00ZIn this thesis, different aspects of the physics of flows in the solar atmosphere are examined. These are described by means of the set of (ideal) magnetohydrodynamics (MHD) and throughout the thesis there is a progressive refinement in the mathematical methods to solve these equations. First, an analysis of symmetric MHD equilibria is presented and the difficulties that are found in solving the steady equations, both analytically and numerically, are discussed in detail. A novel method to find exact solutions in the incompressible case is presented and families of solutions are given in different geometries. Then, attention is turned to flows in coronal magnetic structures, namely quiescent prominences (closed fieldlines) and polar plumes (open fieldlines), and MHD models for these structures are developed by following two different methods: for the former a semi- analytic approach while for the latter a linearisation through a low 𝛽 assumption. In the prominence model, the effects of a subsonic flow along the fieldlines supporting the structure are studied and the results are compared both with a previous static model and with the observed flow speeds. For the plume model, flows are supposed to be transonic along the open fieldlines and their behaviour is studied for different distributions of temperature, density and magnetic flux. However, here the main goal is to demonstrate that coronal plumes are essentially magnetic features and some results of the model are compared with observations. Finally, a time dependent MHD code in spherical coordinates is presented. The aim is to study the interaction of the solar wind with the large scale coronal magnetic structures and the propagation of MHD waves. As a test in 1-D, simulations of the dynamic response of a spherically symmetric extended corona to changes at the outer pressure are studied, following a previous analytic work.
1997-01-01T00:00:00ZDel Zanna, LucaIn this thesis, different aspects of the physics of flows in the solar atmosphere are examined. These are described by means of the set of (ideal) magnetohydrodynamics (MHD) and throughout the thesis there is a progressive refinement in the mathematical methods to solve these equations. First, an analysis of symmetric MHD equilibria is presented and the difficulties that are found in solving the steady equations, both analytically and numerically, are discussed in detail. A novel method to find exact solutions in the incompressible case is presented and families of solutions are given in different geometries. Then, attention is turned to flows in coronal magnetic structures, namely quiescent prominences (closed fieldlines) and polar plumes (open fieldlines), and MHD models for these structures are developed by following two different methods: for the former a semi- analytic approach while for the latter a linearisation through a low 𝛽 assumption. In the prominence model, the effects of a subsonic flow along the fieldlines supporting the structure are studied and the results are compared both with a previous static model and with the observed flow speeds. For the plume model, flows are supposed to be transonic along the open fieldlines and their behaviour is studied for different distributions of temperature, density and magnetic flux. However, here the main goal is to demonstrate that coronal plumes are essentially magnetic features and some results of the model are compared with observations. Finally, a time dependent MHD code in spherical coordinates is presented. The aim is to study the interaction of the solar wind with the large scale coronal magnetic structures and the propagation of MHD waves. As a test in 1-D, simulations of the dynamic response of a spherically symmetric extended corona to changes at the outer pressure are studied, following a previous analytic work.The nonuniform magnetohydrodynamic nature of the solar atmosphereDe Ville, Andrewhttp://hdl.handle.net/10023/140732019-04-01T08:38:02Z1991-01-01T00:00:00ZThe nonuniform structure observed in the solar atmosphere, and in particular the corona, is thought to arise due to the interaction between the magnetic field and plasma. Using a linear theory, the nature of these interactions is investigated, and it is shown how coronal structure may be modelled in a simple way by extended standing disturbances. The effect of inertial forces in considered in both a Cartesian and cylindrical geometries, and a first correction due to gravity is calculated. The restrictions of a linear theory may be overcome by finding exact solutions. Solutions are presented which may model plasma flows in closed, partially open and open magnetic field line structures. A new method for finding particular classes of exact steady solutions in a gravitationally stratified, isothermal atmosphere is presented, along with some examples of possible solutions.
1991-01-01T00:00:00ZDe Ville, AndrewThe nonuniform structure observed in the solar atmosphere, and in particular the corona, is thought to arise due to the interaction between the magnetic field and plasma. Using a linear theory, the nature of these interactions is investigated, and it is shown how coronal structure may be modelled in a simple way by extended standing disturbances. The effect of inertial forces in considered in both a Cartesian and cylindrical geometries, and a first correction due to gravity is calculated. The restrictions of a linear theory may be overcome by finding exact solutions. Solutions are presented which may model plasma flows in closed, partially open and open magnetic field line structures. A new method for finding particular classes of exact steady solutions in a gravitationally stratified, isothermal atmosphere is presented, along with some examples of possible solutions.Aspects of solar coronal stability theoryDe Bruyne, Peter J. J.http://hdl.handle.net/10023/140712019-04-01T08:37:21Z1991-01-01T00:00:00ZSolar coronal stability theory is a powerful tool for understanding the complex behaviour of the Sun's atmosphere. It enables one to discover the driving forces behind some intriguing phenomena and to gauge the soundness of theoretical models for observed structures. In this thesis, the linear stability analysis of line-tied symmetric magnetohydrostatic equilibria is studied within the framework of ideal MHD, aimed at its application to the solar corona. Firstly, a tractable stability procedure based on a variational method is devised. It provides a necessary condition for stability to disturbances localised about a particular flux surface, and a sufficient condition for stability to all accessible perturbations that vanish at the photosphere. The tests require the minimisation of a line integral along the magnetic field lines. For 1-D equilibria, this can be performed analytically, and simple stability criteria are obtained. The necessary condition then serves as an extended Suydam criterion, incorporating the stabilising effect of line-tying. For 2-D equilibria, the minimisation requires the integration of a system of ordinary differential equations along the field lines. This stability technique is applied to arcade, loop, and prominence models, yielding tight bounds on the equilibrium parameters. Secondly, global modes in 1-D coronal loops are investigated using a normal mode method, in order to clarify their link with localised interchange modes. For nearly force-free fields it is shown that instability to localised modes implies the existence of a fast growing global kink mode driven in the neighbourhood of the radius predicted by the local analysis. This confers a new significance on the study of localised interchange modes and the associated extended Suydam criterion.
1991-01-01T00:00:00ZDe Bruyne, Peter J. J.Solar coronal stability theory is a powerful tool for understanding the complex behaviour of the Sun's atmosphere. It enables one to discover the driving forces behind some intriguing phenomena and to gauge the soundness of theoretical models for observed structures. In this thesis, the linear stability analysis of line-tied symmetric magnetohydrostatic equilibria is studied within the framework of ideal MHD, aimed at its application to the solar corona. Firstly, a tractable stability procedure based on a variational method is devised. It provides a necessary condition for stability to disturbances localised about a particular flux surface, and a sufficient condition for stability to all accessible perturbations that vanish at the photosphere. The tests require the minimisation of a line integral along the magnetic field lines. For 1-D equilibria, this can be performed analytically, and simple stability criteria are obtained. The necessary condition then serves as an extended Suydam criterion, incorporating the stabilising effect of line-tying. For 2-D equilibria, the minimisation requires the integration of a system of ordinary differential equations along the field lines. This stability technique is applied to arcade, loop, and prominence models, yielding tight bounds on the equilibrium parameters. Secondly, global modes in 1-D coronal loops are investigated using a normal mode method, in order to clarify their link with localised interchange modes. For nearly force-free fields it is shown that instability to localised modes implies the existence of a fast growing global kink mode driven in the neighbourhood of the radius predicted by the local analysis. This confers a new significance on the study of localised interchange modes and the associated extended Suydam criterion.Hysteresis and mode competition in Faraday wavesDecent, Stephen Paulhttp://hdl.handle.net/10023/140542019-04-01T08:37:39Z1996-01-01T00:00:00ZFaraday waves arise on the surface of a liquid in a container that is undergoing vertical periodic oscillations. We investigate two-dimensional Faraday waves in a long rectangular container, both theoretically and experimentally. Hysteresis occurs when both finite amplitude solutions and the flat surface solution are available. We derive a nonlinear model of a standing wave, extending the Lagrangian method of Miles (1976). The model is used to investigate hysteresis. It is found necessary to retain cubic damping, cubic forcing and the fifth-order conservative term in order to achieve agreement with experiments. The fifth-order conservative term was omitted from all previous studies of Faraday waves. Stable limit cycles are found to arise from this single-mode equation. We examine the structure of this new solution in detail, both analytically and numerically. We describe local bifurcations using a multiple time scales analysis and global bifurcations using Melnikov's method. The coefficients of linear and cubic damping are derived for a standing wave in a rectangular container by considering energy dissipation in the main body of the fluid (due to potential flow and streaming) and in boundary layers at the sidewalls and at the surface. Surface contamination, due to the presence of a thin viscoelastic surface film, creates a boundary layer at the surface which causes enhanced dissipation comparable to, or greater than, that caused by the boundary layers at the walls of the container. Three-mode interaction equations are used to model intermittency and complex modulations which are found to arise from a sideband instability mechanism similar to that of Eckhaus (1963) and Benjamin & Feir (1967). The role of cubic and fifth-order nonlinear terms on this instability mechanism is examined. Theoretical results are found to compare quite favourably with experimental data.
1996-01-01T00:00:00ZDecent, Stephen PaulFaraday waves arise on the surface of a liquid in a container that is undergoing vertical periodic oscillations. We investigate two-dimensional Faraday waves in a long rectangular container, both theoretically and experimentally. Hysteresis occurs when both finite amplitude solutions and the flat surface solution are available. We derive a nonlinear model of a standing wave, extending the Lagrangian method of Miles (1976). The model is used to investigate hysteresis. It is found necessary to retain cubic damping, cubic forcing and the fifth-order conservative term in order to achieve agreement with experiments. The fifth-order conservative term was omitted from all previous studies of Faraday waves. Stable limit cycles are found to arise from this single-mode equation. We examine the structure of this new solution in detail, both analytically and numerically. We describe local bifurcations using a multiple time scales analysis and global bifurcations using Melnikov's method. The coefficients of linear and cubic damping are derived for a standing wave in a rectangular container by considering energy dissipation in the main body of the fluid (due to potential flow and streaming) and in boundary layers at the sidewalls and at the surface. Surface contamination, due to the presence of a thin viscoelastic surface film, creates a boundary layer at the surface which causes enhanced dissipation comparable to, or greater than, that caused by the boundary layers at the walls of the container. Three-mode interaction equations are used to model intermittency and complex modulations which are found to arise from a sideband instability mechanism similar to that of Eckhaus (1963) and Benjamin & Feir (1967). The role of cubic and fifth-order nonlinear terms on this instability mechanism is examined. Theoretical results are found to compare quite favourably with experimental data.The influence of thermal and magnetic layers on solar oscillation frequenciesDaniell, Markhttp://hdl.handle.net/10023/140512019-04-01T08:37:18Z1998-01-01T00:00:00ZIn this thesis, a study is made of the global solar oscillations known as p-modes, modelled by a plane-parallel stratified plasma, within which is embedded a horizontal layered magnetic field. A magnetohydrodynamic formalism is used to investigate the models. The main aim of the thesis is to model the turnover effect in the frequency shifts of the p-modes observed over the course of the solar cycle. Radial oscillations (modes of degree zero) of the Sun are studied for several atmospheric temperature and magnetic field profiles. It is found that the turnover in frequency shifts may be obtained by an increase in the strength of the atmospheric horizontal magnetic field (assumed to be uniform), coupled with a simultaneous increase in atmospheric temperature. The effect of a thin superadiabatic layer in the upper convection zone on p-mode frequencies is also considered. For this model we study modes of general degree, and find that the observed rise and subsequent downturn in the frequency shifts can be duplicated, in the absence of a magnetic field, by simultaneously steepening the temperature gradient of the superadiabatic layer and increasing the atmospheric temperature. In the presence of a magnetic field, where the atmosphere is permeated by a uniform horizontal magnetic field, turnover is reproduced by a combination of an increase in magnetic field strength, a steepening of the temperature gradient in the superadiabatic region, and an increase in atmospheric temperature. The unstable superadiabatic layer also gives rise to convective modes, which are considered briefly. Finally, a model incorporating a magnetic layer residing at the base of the convection zone is constructed and its influence on the frequencies of p-modes assessed. By simply changing the magnetic field strength of this layer, we are unable to reproduce the observed solar cycle variations in p-mode frequencies. The buried magnetic layer supports surface and body magnetoacoustic waves, and a brief study is made of their properties.
1998-01-01T00:00:00ZDaniell, MarkIn this thesis, a study is made of the global solar oscillations known as p-modes, modelled by a plane-parallel stratified plasma, within which is embedded a horizontal layered magnetic field. A magnetohydrodynamic formalism is used to investigate the models. The main aim of the thesis is to model the turnover effect in the frequency shifts of the p-modes observed over the course of the solar cycle. Radial oscillations (modes of degree zero) of the Sun are studied for several atmospheric temperature and magnetic field profiles. It is found that the turnover in frequency shifts may be obtained by an increase in the strength of the atmospheric horizontal magnetic field (assumed to be uniform), coupled with a simultaneous increase in atmospheric temperature. The effect of a thin superadiabatic layer in the upper convection zone on p-mode frequencies is also considered. For this model we study modes of general degree, and find that the observed rise and subsequent downturn in the frequency shifts can be duplicated, in the absence of a magnetic field, by simultaneously steepening the temperature gradient of the superadiabatic layer and increasing the atmospheric temperature. In the presence of a magnetic field, where the atmosphere is permeated by a uniform horizontal magnetic field, turnover is reproduced by a combination of an increase in magnetic field strength, a steepening of the temperature gradient in the superadiabatic region, and an increase in atmospheric temperature. The unstable superadiabatic layer also gives rise to convective modes, which are considered briefly. Finally, a model incorporating a magnetic layer residing at the base of the convection zone is constructed and its influence on the frequencies of p-modes assessed. By simply changing the magnetic field strength of this layer, we are unable to reproduce the observed solar cycle variations in p-mode frequencies. The buried magnetic layer supports surface and body magnetoacoustic waves, and a brief study is made of their properties.Energy-balance models of the solar coronaWragg, M. A.http://hdl.handle.net/10023/140472019-04-01T08:37:12Z1982-01-01T00:00:00ZSolar coronal observations have shown that the corona has a highly complex structure which presumably owes its existence to the magnetic field. Models in thermal and hydrostatic equilibrium are here calculated in order to try and explain many of these observations. Coronal holes occur where open field lines reach out into space. The model of McWhirter, et al. (1975) for the inner corona in such a configuration is generalised to allow different types and magnitudes of heating as well as different area divergences and flows. It is found that hot, fast upflows cannot always exist in thermal equilibrium. The choice of boundary conditions can appreciably alter the results, and so different choices are compared. Most of the corona, especially in active regions, appears to consist of coronal loops. Subtle relations for energy balance models of such loops are found to exist between the physical parameters of a loop's length, base density, and heat input. No solution exists at coronal temperatures in certain cases, which may explain the observations of very cool loops. The effect of a loop's geometry and field line divergence on the structure is found. Results predicted from scaling laws are compared, and the uniqueness of the solution for a loop with a fixed mass is studied. The error in the predicted emission measure through assuming uniform pressure is shown to be considerable. The life-time of a loop can often be many days, suggesting the existence of a thermally stable state. A global stability analysis is performed, and it is found that a loop's stability may depend critically upon its length. Thermally isolated loops, which are the most unstable type, can be thermally stable, provided their pressure falls off sufficiently rapidly with height (due to hydrostatic equilibrium).
1982-01-01T00:00:00ZWragg, M. A.Solar coronal observations have shown that the corona has a highly complex structure which presumably owes its existence to the magnetic field. Models in thermal and hydrostatic equilibrium are here calculated in order to try and explain many of these observations. Coronal holes occur where open field lines reach out into space. The model of McWhirter, et al. (1975) for the inner corona in such a configuration is generalised to allow different types and magnitudes of heating as well as different area divergences and flows. It is found that hot, fast upflows cannot always exist in thermal equilibrium. The choice of boundary conditions can appreciably alter the results, and so different choices are compared. Most of the corona, especially in active regions, appears to consist of coronal loops. Subtle relations for energy balance models of such loops are found to exist between the physical parameters of a loop's length, base density, and heat input. No solution exists at coronal temperatures in certain cases, which may explain the observations of very cool loops. The effect of a loop's geometry and field line divergence on the structure is found. Results predicted from scaling laws are compared, and the uniqueness of the solution for a loop with a fixed mass is studied. The error in the predicted emission measure through assuming uniform pressure is shown to be considerable. The life-time of a loop can often be many days, suggesting the existence of a thermally stable state. A global stability analysis is performed, and it is found that a loop's stability may depend critically upon its length. Thermally isolated loops, which are the most unstable type, can be thermally stable, provided their pressure falls off sufficiently rapidly with height (due to hydrostatic equilibrium).Three-dimensional topology of solar coronal magnetic fieldsBrown, Daniel Stephenhttp://hdl.handle.net/10023/140362019-04-01T08:38:56Z1999-01-01T00:00:00ZThis thesis investigates the topology of the magnetic field in the solar corona. It is important have an understanding of how the highly complex coronal magnetic field behaves in order to study many fundamental coronal phenomena, such as coronal heating events, solar flares and polar plumes. The magnetic fields due to three or four discrete sources are investigated and the corresponding topological states are found. The locations of these states in parameter space is calculated and the bifurcations between states are analysed. A complete analysis has been undertaken for the three-source case and a selective one for the four-source case in order to identify new non-generic behaviour. The thesis goes on to study the topological behaviour of a coronal bright point. Different phases during the lifetime of the bright point are identified and the responsible topological behaviour due to the movement of the magnetic fragments in the photosphere is discussed.
1999-01-01T00:00:00ZBrown, Daniel StephenThis thesis investigates the topology of the magnetic field in the solar corona. It is important have an understanding of how the highly complex coronal magnetic field behaves in order to study many fundamental coronal phenomena, such as coronal heating events, solar flares and polar plumes. The magnetic fields due to three or four discrete sources are investigated and the corresponding topological states are found. The locations of these states in parameter space is calculated and the bifurcations between states are analysed. A complete analysis has been undertaken for the three-source case and a selective one for the four-source case in order to identify new non-generic behaviour. The thesis goes on to study the topological behaviour of a coronal bright point. Different phases during the lifetime of the bright point are identified and the responsible topological behaviour due to the movement of the magnetic fragments in the photosphere is discussed.External and internal magnetohydrostatic models of quiescent solar prominencesCartledge, Nicholas P.http://hdl.handle.net/10023/140292019-04-01T08:37:45Z1996-01-01T00:00:00ZQuiescent solar prominences are amongst the most interesting and yet least understood of the phenomena observed on the Sun and provide both the theorist and the observer with equally demanding challenges. The theoretical study of prominences is an important branch of solar physics as it contributes significantly to the overall understanding of the Sun and its atmosphere. One only needs to be presented with the illuminating fact that there is more mass contained in these bodies than in the remainder of the entire corona to be convinced of their importance. Although many of the physical mechanisms associated with prominence theory are important in their own right, they are also of much wider relevance for various other astrophysical phenomena. For example, radiative and magnetic instabilities are explored in detail in the context of solar prominences; yet clearly these are important processes that relate to many other branches of astrophysics. Prominences are intimately associated with solar flares which occur when a prominence loses equilibrium. Also, prominence eruptions are very important as they are closely connected with coronal mass ejections. These account for a large fraction of the total mass lost from the Sun and so are extremely important events, particularly when one considers the consequences as this plasma interacts with the Earth's environment. It is the period of global equilibrium of quiescent prominences, though, that is the focus of this thesis. Various models are proposed to help understand both the topology and supporting mechanisms of the external, coronal magnetic field, and also the internal prominence structure and the way in which the two regimes fit together. In Chapter 3 we extend a model for the equilibrium of a prominence sheet in a twisted magnetic flux-tube, given by Ridgway, Priest and Amari (1991), to incorporate a current sheet of finite height. This removes the discontinuity at the edge of the tube and provides a shear-free outer boundary which enables the tube to be matched onto a background potential field. In addition, internal prominence solutions are found by expanding the sheet to a finite width and matching suitable magnetic profiles across this region. Next we consider a global model for the magnetic field structure surrounding a polar-crown prominence. We examine potential configurations generated from typical distributions of photospheric flux, and select solutions for which there is a location of dipped magnetic field where prominence material may collect and form. Once such a configuration is available, it is necessary to construct the ensuing prominence solution. We achieve this in Chapter 4 by considering a simplified form for the photospheric field. We show that the equilibrium contains a weighted, curved prominence sheet supported in the location of dipped magnetic field. The equilibrium requires an enhanced magnetic pressure below the sheet to support the component of weight in the normal direction. The internal equilibrium of curved or inclined prominence material has not been considered previously and so we formulate, in Chapter 6, a simple one-dimensional isothermal solution for a cut across the prominence. This is developed to allow for variations along the sheet and in this way an internal solution for the curved prominence of Chapter 4 is given, which matches onto the external potential polar-crown field. Finally, in Chapter 7, we rewrite this solution in terms of its constituent internal and external components and show how the composite solution switches between the two in a region of overlap, or transition region. From this, the internal plasma properties are deduced and realistic profiles for the pressure, density and temperature are obtained.
1996-01-01T00:00:00ZCartledge, Nicholas P.Quiescent solar prominences are amongst the most interesting and yet least understood of the phenomena observed on the Sun and provide both the theorist and the observer with equally demanding challenges. The theoretical study of prominences is an important branch of solar physics as it contributes significantly to the overall understanding of the Sun and its atmosphere. One only needs to be presented with the illuminating fact that there is more mass contained in these bodies than in the remainder of the entire corona to be convinced of their importance. Although many of the physical mechanisms associated with prominence theory are important in their own right, they are also of much wider relevance for various other astrophysical phenomena. For example, radiative and magnetic instabilities are explored in detail in the context of solar prominences; yet clearly these are important processes that relate to many other branches of astrophysics. Prominences are intimately associated with solar flares which occur when a prominence loses equilibrium. Also, prominence eruptions are very important as they are closely connected with coronal mass ejections. These account for a large fraction of the total mass lost from the Sun and so are extremely important events, particularly when one considers the consequences as this plasma interacts with the Earth's environment. It is the period of global equilibrium of quiescent prominences, though, that is the focus of this thesis. Various models are proposed to help understand both the topology and supporting mechanisms of the external, coronal magnetic field, and also the internal prominence structure and the way in which the two regimes fit together. In Chapter 3 we extend a model for the equilibrium of a prominence sheet in a twisted magnetic flux-tube, given by Ridgway, Priest and Amari (1991), to incorporate a current sheet of finite height. This removes the discontinuity at the edge of the tube and provides a shear-free outer boundary which enables the tube to be matched onto a background potential field. In addition, internal prominence solutions are found by expanding the sheet to a finite width and matching suitable magnetic profiles across this region. Next we consider a global model for the magnetic field structure surrounding a polar-crown prominence. We examine potential configurations generated from typical distributions of photospheric flux, and select solutions for which there is a location of dipped magnetic field where prominence material may collect and form. Once such a configuration is available, it is necessary to construct the ensuing prominence solution. We achieve this in Chapter 4 by considering a simplified form for the photospheric field. We show that the equilibrium contains a weighted, curved prominence sheet supported in the location of dipped magnetic field. The equilibrium requires an enhanced magnetic pressure below the sheet to support the component of weight in the normal direction. The internal equilibrium of curved or inclined prominence material has not been considered previously and so we formulate, in Chapter 6, a simple one-dimensional isothermal solution for a cut across the prominence. This is developed to allow for variations along the sheet and in this way an internal solution for the curved prominence of Chapter 4 is given, which matches onto the external potential polar-crown field. Finally, in Chapter 7, we rewrite this solution in terms of its constituent internal and external components and show how the composite solution switches between the two in a region of overlap, or transition region. From this, the internal plasma properties are deduced and realistic profiles for the pressure, density and temperature are obtained.Dynamical processes in the solar atmosphereCargill, P. (Peter)http://hdl.handle.net/10023/140242019-04-01T08:38:19Z1982-01-01T00:00:00ZIt has become clear that the closed-field regions of the solar atmosphere are not static (as was once thought) but that many types of steady and unsteady flows and other dynamical, processes such as flares are continually occurring, in them. This thesis investigates some theoretical aspects of these dynamical phenomena. Steady, one-dimensional flow along a coronal loop is investigated first of all. Such a flow may be driven by a pressure difference between the foot points, and a wide range of shocked and unshocked flows are found. The presence of steady flows removes the symmetry present in most static loop models, and these models are shown to form only one class of a much wider range of dynamic solutions to the equations of motion. Thermal non-equilibrium in hot coronal loops occurs if the pressure in a loop becomes too big. The non-linear evolution of this non-equilibrium state is followed, and the loop is found to cool from of order 10[super]6 K to below 10[super]5 K in a few hours. An upflow is driven, and non-equilibrium is suggested as a means of formation of either cool loop cores or prominences. Thermal non-equilibrium is also discussed as a possible mechanism for the simple-loop flare. It is suggested that a cool equilibrium at a temperature of a few times 10[super]4 K can flare to over. 10[super]7 K if the mechanical heating in the cool loop becomes too large. The evolution is followed and the loop is found to flare to over 10[super]7 K in approximately 5 minutes. Magnetohydrodynamic shock waves have long been regarded as a potentially efficient heating mechanism. Their behaviour is re-examined here, and it is found that certain types of shock can release very large amounts of energy. These results are then applied to the heating of "post"-flare loops, for which temperatures of 10[super]7 K at the loop summit may be obtained. Finally, some solutions of the magnetostatic equation are discussed, and it is pointed out that if the gas pressure is too big then magnetostatic equilibrium will break down. It is suggested that the subsequent evolution may give rise to a surge or other mass ejection.
1982-01-01T00:00:00ZCargill, P. (Peter)It has become clear that the closed-field regions of the solar atmosphere are not static (as was once thought) but that many types of steady and unsteady flows and other dynamical, processes such as flares are continually occurring, in them. This thesis investigates some theoretical aspects of these dynamical phenomena. Steady, one-dimensional flow along a coronal loop is investigated first of all. Such a flow may be driven by a pressure difference between the foot points, and a wide range of shocked and unshocked flows are found. The presence of steady flows removes the symmetry present in most static loop models, and these models are shown to form only one class of a much wider range of dynamic solutions to the equations of motion. Thermal non-equilibrium in hot coronal loops occurs if the pressure in a loop becomes too big. The non-linear evolution of this non-equilibrium state is followed, and the loop is found to cool from of order 10[super]6 K to below 10[super]5 K in a few hours. An upflow is driven, and non-equilibrium is suggested as a means of formation of either cool loop cores or prominences. Thermal non-equilibrium is also discussed as a possible mechanism for the simple-loop flare. It is suggested that a cool equilibrium at a temperature of a few times 10[super]4 K can flare to over. 10[super]7 K if the mechanical heating in the cool loop becomes too large. The evolution is followed and the loop is found to flare to over 10[super]7 K in approximately 5 minutes. Magnetohydrodynamic shock waves have long been regarded as a potentially efficient heating mechanism. Their behaviour is re-examined here, and it is found that certain types of shock can release very large amounts of energy. These results are then applied to the heating of "post"-flare loops, for which temperatures of 10[super]7 K at the loop summit may be obtained. Finally, some solutions of the magnetostatic equation are discussed, and it is pointed out that if the gas pressure is too big then magnetostatic equilibrium will break down. It is suggested that the subsequent evolution may give rise to a surge or other mass ejection.Topological configurations of coronal magnetic fields and current sheetsBungey, Timothy N.http://hdl.handle.net/10023/140212019-04-01T08:38:37Z1996-01-01T00:00:00ZThe question of topology in the coronal magnetic field is addressed in this thesis. Magnetic reconnection, which plays a major role in many of the fascinating phenomena seen in the solar atmosphere, is likely to occur at the boundaries between different topological regions of the magnetic field. By modelling the coronal field using discrete sources of flux, to represent the concentrations seen at the photospheric surface, we study the varying topological structures present in the field. We generate a criterion for determining the presence of null points above the photospheric surface and establish that any separatrix surfaces present in the field are due to the presence of either null points, or regions where the field tangentially grazes the surface. We follow the evolution of these separatrix surfaces and, in particular, determine the existence of a well-defined separator field line in the absence of coronal null points. Finally, we look locally at the configuration of the magnetic field in the region surrounding a straight current sheet. We derive an analytical expression to describe the topology of both potential and constant-current force-free fields in the neighbourhood of a sheet, and in so doing generalise the previously known expressions.
1996-01-01T00:00:00ZBungey, Timothy N.The question of topology in the coronal magnetic field is addressed in this thesis. Magnetic reconnection, which plays a major role in many of the fascinating phenomena seen in the solar atmosphere, is likely to occur at the boundaries between different topological regions of the magnetic field. By modelling the coronal field using discrete sources of flux, to represent the concentrations seen at the photospheric surface, we study the varying topological structures present in the field. We generate a criterion for determining the presence of null points above the photospheric surface and establish that any separatrix surfaces present in the field are due to the presence of either null points, or regions where the field tangentially grazes the surface. We follow the evolution of these separatrix surfaces and, in particular, determine the existence of a well-defined separator field line in the absence of coronal null points. Finally, we look locally at the configuration of the magnetic field in the region surrounding a straight current sheet. We derive an analytical expression to describe the topology of both potential and constant-current force-free fields in the neighbourhood of a sheet, and in so doing generalise the previously known expressions.Parametric instabilities in inhomogenous plasmasBegg, Iain M.http://hdl.handle.net/10023/140162019-04-01T08:37:34Z1976-01-01T00:00:00ZThis thesis will deal with certain problems of parametric instabilities in the inhomogeneous plasma. A large amplitude, 'pump' wave can deposit some of its energy into the plasma through resonance with two lower frequency waves (which may be damped). This type of process is a parametric decay of the pump wave and has applications in many fields. We consider, predominantly, that of laser fusion, in which the pump wave is electromagnetic and incident on the plasma. The objective is to deposit as much energy as possible within the plasma. Instabilities reducing this energy input are therefore of importance and it is, mostly, to these that this thesis will turn. They are mostly scattering processes in which one of the decay modes is electromagnetic. We examine the stimulated Brillouin backscattering process (the other decay mode being an ion accoustic wave) from a reference frame in which the plasma is streaming outwards. It is found that, if this velocity is near the sound velocity, the ion acoustic wave has a frequency Doppler-shifted to zero, the electromagnetic waves then having equal frequencies. In such a situation, any reflection of the pump wave at the critical surface will enhance the initial level of the backscattered wave. We find that, allowing for this, there is considerable enhancement of backscatter from the plasma, with consequent energy loss to the pump. Since the effect is noticeably unaffected by 'off- resonance' situations, it is felt that this process could mount a barrier to possible applications. We next consider the stimulated Compton scattering process, where the pump is scattered off the 'bare' or thermal electrons in the plasma. It is found that this rather weak instability occurs predominantly only when electron plasma waves are heavily dampled. Substantial reflection only occurs for high pump powers. Whilst there is little loss to the pump energy, there is substantial perturbation to the background distribution function. However, at the high powers involved filamentation and modulation of the pump can occur with a resulting enhancement of the scattering. Finally, we consider the effect on the decay instability (photon → plasmon + phonon) of the presence of substantial filamentation of the critical surface. It is found that the growth rate is substantially reduced.
1976-01-01T00:00:00ZBegg, Iain M.This thesis will deal with certain problems of parametric instabilities in the inhomogeneous plasma. A large amplitude, 'pump' wave can deposit some of its energy into the plasma through resonance with two lower frequency waves (which may be damped). This type of process is a parametric decay of the pump wave and has applications in many fields. We consider, predominantly, that of laser fusion, in which the pump wave is electromagnetic and incident on the plasma. The objective is to deposit as much energy as possible within the plasma. Instabilities reducing this energy input are therefore of importance and it is, mostly, to these that this thesis will turn. They are mostly scattering processes in which one of the decay modes is electromagnetic. We examine the stimulated Brillouin backscattering process (the other decay mode being an ion accoustic wave) from a reference frame in which the plasma is streaming outwards. It is found that, if this velocity is near the sound velocity, the ion acoustic wave has a frequency Doppler-shifted to zero, the electromagnetic waves then having equal frequencies. In such a situation, any reflection of the pump wave at the critical surface will enhance the initial level of the backscattered wave. We find that, allowing for this, there is considerable enhancement of backscatter from the plasma, with consequent energy loss to the pump. Since the effect is noticeably unaffected by 'off- resonance' situations, it is felt that this process could mount a barrier to possible applications. We next consider the stimulated Compton scattering process, where the pump is scattered off the 'bare' or thermal electrons in the plasma. It is found that this rather weak instability occurs predominantly only when electron plasma waves are heavily dampled. Substantial reflection only occurs for high pump powers. Whilst there is little loss to the pump energy, there is substantial perturbation to the background distribution function. However, at the high powers involved filamentation and modulation of the pump can occur with a resulting enhancement of the scattering. Finally, we consider the effect on the decay instability (photon → plasmon + phonon) of the presence of substantial filamentation of the critical surface. It is found that the growth rate is substantially reduced.Magnetic annihilation and reconnectionAnderson, Craighttp://hdl.handle.net/10023/140142019-04-01T08:37:37Z1994-01-01T00:00:00ZThis thesis presents several analytical models of magnetic annihilation and reconnection and studies their properties. The models investigated are 1. Steady-state magnetic annihilation. The assumption of straight field lines reduces the resistive, viscous MHD equations to two ordinary differential equations, one for the flow and one for the magnetic field. These equations can be solved exactly (for the case of a simple stagnation-point flow) and asymptotically (for a more general stagnation-point flow). In both cases the reconnection rates can be fast due to advection effects which create large magnetic gradients. 2. Time-dependent magnetic annihilation. The assumption of straight field lines whose strength can vary with time reduces the MHD equations to two partial differential equations, one for the flow and one for the magnetic field. The time-modulated simple stagnation-point flow is shown to be an exact solution and the equation for the magnetic field is then solved on infinite and finite intervals. For the infinite interval the reconnection rates are shown to be dependent on the nature of the advected initial field. Also examined are self-similar solutions and the effect of variation of diffusivity with time. 3. Annihilation in a compressible, inviscid plasma. Here, the assumption of straight field lines and an inviscid, compressible flow reduce the MHD equations to a pair of non-linear coupled partial differential equations. Further assuming that the density only varies in one direction and the flow is of a stagnation-point type allow these equations to be solved approximately analytically and exactly numerically. It is shown that the magnetic field and reconnection rates are the same in both the compressible and incompressible cases and that the density of the plasma is greatest within the current sheet. 4. Steady-state magnetic reconnection. For an incompressible flow the MHD equations can be reduced to two coupled non-linear partial differential equations. These two equations are studied by seeking asymptotic solutions around the annihilation solution and then looking for series solutions to the first-order equations. It is found that the magnetic field always has a magnetic cusp and never possesses an x-type neutral point. 5. Reconnection in a viscous plasma. Assuming that the viscous forces dominate, the induction equation and equation of motion decouple and become linear. The magnetic field is obtained for the case of a simple stagnation-point flow. It is shown that if the inflow magnetic field is taken to be straight then the magnetic field within the region tends towards the annihilation solution as the magnetic Reynolds number increases. 6. Magnetic flipping. A previous ideal model of magnetic flipping is refined so that it becomes an exact solution of the MHD equations. In the refined model the streamlines are straight rather than curved. Assuming straight streamlines, the MHD equations reduce to two linear ordinary differential equations, one for the flow and one for the magnetic field. These are then solved exactly analytically to find a flow containing a viscous boundary layer and a magnetic field that contains an x-type neutral point. The angle between the separatrices of the field is determined by the Reynolds and magnetic Reynolds numbers. It is shown that most of the ohmic heating occurs within the viscous boundary layer.
1994-01-01T00:00:00ZAnderson, CraigThis thesis presents several analytical models of magnetic annihilation and reconnection and studies their properties. The models investigated are 1. Steady-state magnetic annihilation. The assumption of straight field lines reduces the resistive, viscous MHD equations to two ordinary differential equations, one for the flow and one for the magnetic field. These equations can be solved exactly (for the case of a simple stagnation-point flow) and asymptotically (for a more general stagnation-point flow). In both cases the reconnection rates can be fast due to advection effects which create large magnetic gradients. 2. Time-dependent magnetic annihilation. The assumption of straight field lines whose strength can vary with time reduces the MHD equations to two partial differential equations, one for the flow and one for the magnetic field. The time-modulated simple stagnation-point flow is shown to be an exact solution and the equation for the magnetic field is then solved on infinite and finite intervals. For the infinite interval the reconnection rates are shown to be dependent on the nature of the advected initial field. Also examined are self-similar solutions and the effect of variation of diffusivity with time. 3. Annihilation in a compressible, inviscid plasma. Here, the assumption of straight field lines and an inviscid, compressible flow reduce the MHD equations to a pair of non-linear coupled partial differential equations. Further assuming that the density only varies in one direction and the flow is of a stagnation-point type allow these equations to be solved approximately analytically and exactly numerically. It is shown that the magnetic field and reconnection rates are the same in both the compressible and incompressible cases and that the density of the plasma is greatest within the current sheet. 4. Steady-state magnetic reconnection. For an incompressible flow the MHD equations can be reduced to two coupled non-linear partial differential equations. These two equations are studied by seeking asymptotic solutions around the annihilation solution and then looking for series solutions to the first-order equations. It is found that the magnetic field always has a magnetic cusp and never possesses an x-type neutral point. 5. Reconnection in a viscous plasma. Assuming that the viscous forces dominate, the induction equation and equation of motion decouple and become linear. The magnetic field is obtained for the case of a simple stagnation-point flow. It is shown that if the inflow magnetic field is taken to be straight then the magnetic field within the region tends towards the annihilation solution as the magnetic Reynolds number increases. 6. Magnetic flipping. A previous ideal model of magnetic flipping is refined so that it becomes an exact solution of the MHD equations. In the refined model the streamlines are straight rather than curved. Assuming straight streamlines, the MHD equations reduce to two linear ordinary differential equations, one for the flow and one for the magnetic field. These are then solved exactly analytically to find a flow containing a viscous boundary layer and a magnetic field that contains an x-type neutral point. The angle between the separatrices of the field is determined by the Reynolds and magnetic Reynolds numbers. It is shown that most of the ohmic heating occurs within the viscous boundary layer.Plasma drift waves and instabilitiesAllan, Williamhttp://hdl.handle.net/10023/140112019-04-01T08:38:33Z1974-01-01T00:00:00ZThe work of this thesis is concerned with the investigation of the propagation of waves in a magnetized plasma containing various parameter gradients, and with the stability of ion acoustic waves in a weakly collisional plasma with a strong temperature gradient. The thesis is divided into three sections. In the first section the intention is to derive in a compact and unambiguous tensor form the dispersion relation describing the propagation of waves in a magnetized plasma containing three-dimensional density and temperature gradients, an E̲⏜ B̲ drift, and differing temperatures parallel and perpendicular to the magnetic field. This is achieved by introducing and extending the polarized co-ordinate system first proposed by Buneman in 1961, and then carrying through the standard procedure of integration along unperturbed trajectories. The "local" approximation of Krall and Rosenbluth is used in order that an analytic result may be derived. The dispersion relation obtained includes certain moment tensors whose elements may be evaluated independently of the gradients involved in the problem. These elements may then be listed and the list referred to in order to obtain the elements required for a specific problem. The second section is concerned with the use of the theory and results of J.P. Dougherty to show that in the high-frequency regime the introduction of a small amount of collisions into a plasma is sufficient to disrupt the gyro-resonances which allow the existence of Bernstein waves at multiples of the gyro-frequencies perpendicular and near- perpendicular to the magnetic field. It is shown that a collision frequency v such that (k 𝜌) ⁻² ≲ v/Ω < (k 𝜌) ⁻¹ where k 𝜌 >> 1 is sufficient to do this; k is the wave-number, 𝜌 the Larmor radius, and the gyro-frequency. It is also shown that in this case the ion-acoustic dispersion relation is valid even for propagation perpendicular to the magnetic field. In the final section the result of the second section is used to derive a dispersion relation for high-frequency wave propagation in a weakly-collisional plasma containing an electron temperature gradient. The dispersion relation is solved numerically for various electron-ion temperature ratios and electron temperature gradient drift velocities. Earlier predictions, based on analytic calculations for small temperature ratios and drift velocities, are confirmed and some new results presented. In particular, it is shown that a temperature gradient is a more effective destabilizing agent then a simple drift between ions and electrons. Dispersion plots are given, along with analytic and physical explanations of their form; finally neutral stability curves are presented. The thesis concludes with a summary of the results obtained.
1974-01-01T00:00:00ZAllan, WilliamThe work of this thesis is concerned with the investigation of the propagation of waves in a magnetized plasma containing various parameter gradients, and with the stability of ion acoustic waves in a weakly collisional plasma with a strong temperature gradient. The thesis is divided into three sections. In the first section the intention is to derive in a compact and unambiguous tensor form the dispersion relation describing the propagation of waves in a magnetized plasma containing three-dimensional density and temperature gradients, an E̲⏜ B̲ drift, and differing temperatures parallel and perpendicular to the magnetic field. This is achieved by introducing and extending the polarized co-ordinate system first proposed by Buneman in 1961, and then carrying through the standard procedure of integration along unperturbed trajectories. The "local" approximation of Krall and Rosenbluth is used in order that an analytic result may be derived. The dispersion relation obtained includes certain moment tensors whose elements may be evaluated independently of the gradients involved in the problem. These elements may then be listed and the list referred to in order to obtain the elements required for a specific problem. The second section is concerned with the use of the theory and results of J.P. Dougherty to show that in the high-frequency regime the introduction of a small amount of collisions into a plasma is sufficient to disrupt the gyro-resonances which allow the existence of Bernstein waves at multiples of the gyro-frequencies perpendicular and near- perpendicular to the magnetic field. It is shown that a collision frequency v such that (k 𝜌) ⁻² ≲ v/Ω < (k 𝜌) ⁻¹ where k 𝜌 >> 1 is sufficient to do this; k is the wave-number, 𝜌 the Larmor radius, and the gyro-frequency. It is also shown that in this case the ion-acoustic dispersion relation is valid even for propagation perpendicular to the magnetic field. In the final section the result of the second section is used to derive a dispersion relation for high-frequency wave propagation in a weakly-collisional plasma containing an electron temperature gradient. The dispersion relation is solved numerically for various electron-ion temperature ratios and electron temperature gradient drift velocities. Earlier predictions, based on analytic calculations for small temperature ratios and drift velocities, are confirmed and some new results presented. In particular, it is shown that a temperature gradient is a more effective destabilizing agent then a simple drift between ions and electrons. Dispersion plots are given, along with analytic and physical explanations of their form; finally neutral stability curves are presented. The thesis concludes with a summary of the results obtained.Aspects of current sheet theoryTur, T. J.http://hdl.handle.net/10023/140002019-04-01T08:37:27Z1977-01-01T00:00:00ZCurrent sheets are widely believed to play an important role in astrophysics when regions of magnetic flux are in motion. Several models based on the formation of current sheets have been proposed to explain such phenomena as geomagnetic storms, solar flares and prominences. In this thesis three aspects of current sheet theory are studied with particular reference to the solar flare problem. Firstly the development of two-dimensional current sheets is investigated for several simple configurations. These include converging line current sources, converging and diverging line dipole sources and a dipole of increasing moment situated in either a uniform magnetic field or a constant dipole field. These last two may be thought of as modelling the emergence of bipolar flux from beneath the photosphere, a phenomena frequently observed prior to solar flares. The length, position and shape of the current sheet is determined from the requirement that the magnetic field be frozen-into the plasma. The sheet is found to be curved, except in the symmetrical case of converging line sources. In addition, the extra energy due to the presence of the current sheet is determined. Comparison with estimates of the energy dissipated during a flare indicate that the formation of current sheets may store an adequate amount of preflare magnetic energy, provided no reconnection occurs during the formation process. A three-dimensional axi-symmetric model for current sheet formation is then considered. Two equal and co-directional dipoles approach along the axis of symmetry to form an annular current sheet between them. The equations determining the magnetic field for this configuration are reduced to a single integral equation for the current density in the sheet as a function of radial distance from the axis. A numerical method is used to solve this integral equation. The inner and outer radii of the sheet are then determined from the conditions of flux conservation as for the two-dimensional case. Finally the energetics of a current sheet that forms between newly emerging flux and an ambient field are considered. As more and more flux emerges, so the sheet rises in the solar atmosphere. The various contributions to the thermal energy balance in the sheet are approximated and the resulting equation is solved for the internal temperature of the sheet. It is found that, for certain choices of the ambient magnetic field strength and velocity, the internal temperature increases until, when the sheet reaches some critical height, no neighbouring stable state exists. The temperature then increases rapidly seeking a hotter branch of the solution curve. During this dynamic heating the threshold temperature for the onset of microinstabilities may be attained. It is suggested that this may be a suitable trigger mechanism for the recently proposed "emerging flux" model of a solar flare.
1977-01-01T00:00:00ZTur, T. J.Current sheets are widely believed to play an important role in astrophysics when regions of magnetic flux are in motion. Several models based on the formation of current sheets have been proposed to explain such phenomena as geomagnetic storms, solar flares and prominences. In this thesis three aspects of current sheet theory are studied with particular reference to the solar flare problem. Firstly the development of two-dimensional current sheets is investigated for several simple configurations. These include converging line current sources, converging and diverging line dipole sources and a dipole of increasing moment situated in either a uniform magnetic field or a constant dipole field. These last two may be thought of as modelling the emergence of bipolar flux from beneath the photosphere, a phenomena frequently observed prior to solar flares. The length, position and shape of the current sheet is determined from the requirement that the magnetic field be frozen-into the plasma. The sheet is found to be curved, except in the symmetrical case of converging line sources. In addition, the extra energy due to the presence of the current sheet is determined. Comparison with estimates of the energy dissipated during a flare indicate that the formation of current sheets may store an adequate amount of preflare magnetic energy, provided no reconnection occurs during the formation process. A three-dimensional axi-symmetric model for current sheet formation is then considered. Two equal and co-directional dipoles approach along the axis of symmetry to form an annular current sheet between them. The equations determining the magnetic field for this configuration are reduced to a single integral equation for the current density in the sheet as a function of radial distance from the axis. A numerical method is used to solve this integral equation. The inner and outer radii of the sheet are then determined from the conditions of flux conservation as for the two-dimensional case. Finally the energetics of a current sheet that forms between newly emerging flux and an ambient field are considered. As more and more flux emerges, so the sheet rises in the solar atmosphere. The various contributions to the thermal energy balance in the sheet are approximated and the resulting equation is solved for the internal temperature of the sheet. It is found that, for certain choices of the ambient magnetic field strength and velocity, the internal temperature increases until, when the sheet reaches some critical height, no neighbouring stable state exists. The temperature then increases rapidly seeking a hotter branch of the solution curve. During this dynamic heating the threshold temperature for the onset of microinstabilities may be attained. It is suggested that this may be a suitable trigger mechanism for the recently proposed "emerging flux" model of a solar flare.Thermal and resistive instabilities in the solar atmosphereSmith, E. A.http://hdl.handle.net/10023/139982019-04-01T08:38:26Z1977-01-01T00:00:00ZThe magnetic field greatly influences the plasma in the solar atmosphere and in this thesis we consider the effect of the field on the stability of the plasma. The many observations that have been made suggest that two types of field structure play a major role. Firstly a current sheet - this has field lines which change direction in a thin, current forming region, but are fairly uniform outside. We consider the case where the field strength is zero along the neutral line so that a gas pressure gradient is required across the sheet to balance the magnetic pressure gradient. Secondly a force-free field - here the magnetic force is zero, which requires the magnetic pressure to be much larger than the gas pressure. In the neutral current sheet we examine the thermal instability and the tearing-mode instability. While in the force-free magnetic arch system we look for a thermal instability which can occur when the foot points of the arch are sheared. When we investigated the thermal stability of the current sheet we found that as its length increases it passes through a series of stable equilibria until a value, L[sub]max, is reached when the sheet cools down to a max new stable equilibrium. For coronal conditions, values for L[sub]max and max cooling time are in fair agreement with the observed values for quiescent prominences. We calculate the growth rate of the tearing-mode instability in a neutral current sheet with no energy sources or sinks and find that the maximum growth rate can be significantly larger in the current sheet than in the sheared field of constant magnitude considered by others. Also the growth rate decreases when the ratio of gas to magnetic pressure is reduced. We find that the growth rate is significantly inhibited if the current sheet has a transverse magnetic field which is large enough. Lastly we examine the thermal balance in a sheared, force-free magnetic field and show that thermal instability can occur if the field is sheared enough. We assume thermal equilibrium between radiative loss and thermal conduction and we take gravity balanced by a pressure gradient. If, for example, the density at the base of the field is ten times larger than the normal coronal value, as it may be in coronal condensations, then there is instability if the shear angle is greater than 63 °. The presence of a large enough mechanical heating is found to prevent the instability occurring.
1977-01-01T00:00:00ZSmith, E. A.The magnetic field greatly influences the plasma in the solar atmosphere and in this thesis we consider the effect of the field on the stability of the plasma. The many observations that have been made suggest that two types of field structure play a major role. Firstly a current sheet - this has field lines which change direction in a thin, current forming region, but are fairly uniform outside. We consider the case where the field strength is zero along the neutral line so that a gas pressure gradient is required across the sheet to balance the magnetic pressure gradient. Secondly a force-free field - here the magnetic force is zero, which requires the magnetic pressure to be much larger than the gas pressure. In the neutral current sheet we examine the thermal instability and the tearing-mode instability. While in the force-free magnetic arch system we look for a thermal instability which can occur when the foot points of the arch are sheared. When we investigated the thermal stability of the current sheet we found that as its length increases it passes through a series of stable equilibria until a value, L[sub]max, is reached when the sheet cools down to a max new stable equilibrium. For coronal conditions, values for L[sub]max and max cooling time are in fair agreement with the observed values for quiescent prominences. We calculate the growth rate of the tearing-mode instability in a neutral current sheet with no energy sources or sinks and find that the maximum growth rate can be significantly larger in the current sheet than in the sheared field of constant magnitude considered by others. Also the growth rate decreases when the ratio of gas to magnetic pressure is reduced. We find that the growth rate is significantly inhibited if the current sheet has a transverse magnetic field which is large enough. Lastly we examine the thermal balance in a sheared, force-free magnetic field and show that thermal instability can occur if the field is sheared enough. We assume thermal equilibrium between radiative loss and thermal conduction and we take gravity balanced by a pressure gradient. If, for example, the density at the base of the field is ten times larger than the normal coronal value, as it may be in coronal condensations, then there is instability if the shear angle is greater than 63 °. The presence of a large enough mechanical heating is found to prevent the instability occurring.Numerical studies of the Fokker-Planck equationMcGowan, Alastair Davidhttp://hdl.handle.net/10023/139952019-04-01T08:37:21Z1992-01-01T00:00:00ZJorna and Wood recently developed a program that numerically solved the Fokker-Planck equation in spherical geometry. In this thesis, we describe how the original program has been redeveloped to produce a program that is an order of magnitude quicker and that has superior energy and density conservation. The revised version of the program has been used to extend the work of Jorna and Wood on thermal conduction in laser produced plasmas. It has been shown that the effect of curvature on heat flow can be described from a purely geometrical argument and that for aspect ratios similar to those found in targets, the heat flow is reduced by approximately 10%. Also, it has been shown, in contradiction with Jorna and Wood, that the inclusion of the anisotropic portion of the Rosenbluth potentials does not have a significant effect on the heat flow. Even for highly anisotropic plasmas, the inclusion of the anisotropic portion only increases the heat flow by 10%. In addition, the revised version of the program has been used to study the energy relaxation of model distributions It has been shown that the relaxation time of most non - thermal distributions depends on the detailed structure of the distribution and that the normal Spitzer collision time can under-estimate or over-estimate the time required for energy relaxation.
1992-01-01T00:00:00ZMcGowan, Alastair DavidJorna and Wood recently developed a program that numerically solved the Fokker-Planck equation in spherical geometry. In this thesis, we describe how the original program has been redeveloped to produce a program that is an order of magnitude quicker and that has superior energy and density conservation. The revised version of the program has been used to extend the work of Jorna and Wood on thermal conduction in laser produced plasmas. It has been shown that the effect of curvature on heat flow can be described from a purely geometrical argument and that for aspect ratios similar to those found in targets, the heat flow is reduced by approximately 10%. Also, it has been shown, in contradiction with Jorna and Wood, that the inclusion of the anisotropic portion of the Rosenbluth potentials does not have a significant effect on the heat flow. Even for highly anisotropic plasmas, the inclusion of the anisotropic portion only increases the heat flow by 10%. In addition, the revised version of the program has been used to study the energy relaxation of model distributions It has been shown that the relaxation time of most non - thermal distributions depends on the detailed structure of the distribution and that the normal Spitzer collision time can under-estimate or over-estimate the time required for energy relaxation.Nonlinear plasma waves and their applicationsAmin, Mohamed Ruhulhttp://hdl.handle.net/10023/139932019-04-01T08:37:24Z1999-01-01T00:00:00ZThe possibility of beat wave current drive in tokamaks is considered in this thesis in steady state 2D geometry. The problem is considered by including in the analysis the 2D toroidal inhomogeneity effect and the effect of finite spatial width of the pump microwave pulses on the beat wave excitation. Both a Langmuir beat wave as well as an obliquely propagating upper-hybrid cyclotron beat wave are considered in this study. The three wave coupled system of equations in a magnetized plasma has been derived and solved numerically for this purpose. It has been found that Langmuir type beat wave excited by two almost antiparallel pump microwaves is more efficient for action transfer than a cyclotron beat wave. It has also been found that for the same input parameters, right hand polarized pumps are more efficient than left hand polarized pump microwaves for depositing power in the beat wave. The second part of the thesis considers the relativistic excitation mechanism of a large amplitude plasma wake field by a single ultra-short laser pulse. This type of large amplitude wake field has been proposed for particle acceleration to very high energies for future generation of accelerators. The problem has been modeled self consistently in ID geometry and the relevant coupled system of equations have been solved numerically. It has been found that the shape of the laser pulse profile and the ratio of the ambient plasma frequency to the incident laser frequency play an important role for the excitation of the wake-field and the stability of the laser pulse profile.
1999-01-01T00:00:00ZAmin, Mohamed RuhulThe possibility of beat wave current drive in tokamaks is considered in this thesis in steady state 2D geometry. The problem is considered by including in the analysis the 2D toroidal inhomogeneity effect and the effect of finite spatial width of the pump microwave pulses on the beat wave excitation. Both a Langmuir beat wave as well as an obliquely propagating upper-hybrid cyclotron beat wave are considered in this study. The three wave coupled system of equations in a magnetized plasma has been derived and solved numerically for this purpose. It has been found that Langmuir type beat wave excited by two almost antiparallel pump microwaves is more efficient for action transfer than a cyclotron beat wave. It has also been found that for the same input parameters, right hand polarized pumps are more efficient than left hand polarized pump microwaves for depositing power in the beat wave. The second part of the thesis considers the relativistic excitation mechanism of a large amplitude plasma wake field by a single ultra-short laser pulse. This type of large amplitude wake field has been proposed for particle acceleration to very high energies for future generation of accelerators. The problem has been modeled self consistently in ID geometry and the relevant coupled system of equations have been solved numerically. It has been found that the shape of the laser pulse profile and the ratio of the ambient plasma frequency to the incident laser frequency play an important role for the excitation of the wake-field and the stability of the laser pulse profile.Study of solitary waves in space plasmasMamun, A. A.http://hdl.handle.net/10023/139872019-04-01T08:37:43Z1997-01-01T00:00:00ZTheoretical investigations have been made of arbitrary amplitude electrostatic solitary waves in non-thermal plasmas, which may be of relevance to ionospheric and magnetospheric plasmas, and dusty plasmas, which are most common in earth's and cometary environments as well as in planetary rings, for understanding the nonlinear features of localised electrostatic disturbances in such space plasma systems. This thesis starts with an introductory chapter where a very brief historical review of solitary waves in plasmas has been presented. The study of arbitrary amplitude electrostatic solitary waves in non-thermal plasma has considered a plasma system consisting of warm adiabatic ions and non- thermal electrons. It is found that a non-thermal electron distribution may change the nature of ion-acoustic solitary waves. If the ions are assumed to respond as a fluid to perturbations in the potential, with no significant trapping in a potential well, then a thermal plasma only supports solitary waves with a density peak. However, with a suitable distribution of non-thermal electrons, solitary waves with both density peaks and density depressions may exist. This study has also included a numerical analysis showing how these electrostatic solitary structures evolve with time. The investigation has then been extended to magnetised plasmas to study the effects of magnetic field on obliquely propagating electrostatic solitary structures. This attempt first employed the reductive perturbation method and investigated the nonlinear properties of small but finite amplitude obliquely propagating solitary waves in this magnetised non-thermal plasma model. This study is then generalised to arbitrary amplitude solitary waves by the numerical solution of the full nonlinear system of equations. This numerical method has also been utilised to present a similar study in another popular plasma model, namely the two-electron-temperature plasma model. The study of arbitrary amplitude solitary waves in a dusty plasma has considered another plasma system which consists of an inertial dust fluid and ions with Maxwellian distribution and has investigated the nonlinear properties of dust- acoustic solitary waves. A numerical study has also been made to show how these dust-acoustic solitary waves evolve with time. The effects of non-thermal and vortex-like ion distributions are then incorporated into this study. The study of arbitrary amplitude electrostatic solitary waves in this thesis has finally been concluded with some brief discussion of our results and proposal for further studies, which are expected to generalise and develop our present work to some other extents, in this versatile area of research.
1997-01-01T00:00:00ZMamun, A. A.Theoretical investigations have been made of arbitrary amplitude electrostatic solitary waves in non-thermal plasmas, which may be of relevance to ionospheric and magnetospheric plasmas, and dusty plasmas, which are most common in earth's and cometary environments as well as in planetary rings, for understanding the nonlinear features of localised electrostatic disturbances in such space plasma systems. This thesis starts with an introductory chapter where a very brief historical review of solitary waves in plasmas has been presented. The study of arbitrary amplitude electrostatic solitary waves in non-thermal plasma has considered a plasma system consisting of warm adiabatic ions and non- thermal electrons. It is found that a non-thermal electron distribution may change the nature of ion-acoustic solitary waves. If the ions are assumed to respond as a fluid to perturbations in the potential, with no significant trapping in a potential well, then a thermal plasma only supports solitary waves with a density peak. However, with a suitable distribution of non-thermal electrons, solitary waves with both density peaks and density depressions may exist. This study has also included a numerical analysis showing how these electrostatic solitary structures evolve with time. The investigation has then been extended to magnetised plasmas to study the effects of magnetic field on obliquely propagating electrostatic solitary structures. This attempt first employed the reductive perturbation method and investigated the nonlinear properties of small but finite amplitude obliquely propagating solitary waves in this magnetised non-thermal plasma model. This study is then generalised to arbitrary amplitude solitary waves by the numerical solution of the full nonlinear system of equations. This numerical method has also been utilised to present a similar study in another popular plasma model, namely the two-electron-temperature plasma model. The study of arbitrary amplitude solitary waves in a dusty plasma has considered another plasma system which consists of an inertial dust fluid and ions with Maxwellian distribution and has investigated the nonlinear properties of dust- acoustic solitary waves. A numerical study has also been made to show how these dust-acoustic solitary waves evolve with time. The effects of non-thermal and vortex-like ion distributions are then incorporated into this study. The study of arbitrary amplitude electrostatic solitary waves in this thesis has finally been concluded with some brief discussion of our results and proposal for further studies, which are expected to generalise and develop our present work to some other extents, in this versatile area of research.Steady models for magnetic reconnectionJardine, Moirahttp://hdl.handle.net/10023/139852019-04-01T08:37:50Z1989-01-01T00:00:00ZMagnetic reconnection is a fundamental physical process by which stored magnetic energy may be released. It is already known that different reconnection regimes result from changes in the nature of the plasma inflow towards the reconnection site. In this thesis, we examine both how the outflow region responds to changes both in the inflow and outflow boundary conditions and also how introducing compressibility affects the results. We find that if the inflow is converging, the outflow velocity is least, the width of the outflow region is greatest and the ratio of outflowing thermal to kinetic energy is greatest. Also, there is one free outflow parameter which would naturally be specified by the velocity of plasma leaving the reconnection site. We suggest that reverse currents seen in numerical simulations may result from the specification of an extra boundary condition. In addition, we find that the main effects of including compressibility are: to enhance convergence or divergence of the inflow; to increase the maximum reconnection rate where the inflow is converging; to increase the flow speed near the reconnection site where the inflow is diverging; to give faster, narrower outflow jets; to increase variations between regimes in the energy conversion and to increase the ratio of thermal to kinetic energy in the outflow jet.
1989-01-01T00:00:00ZJardine, MoiraMagnetic reconnection is a fundamental physical process by which stored magnetic energy may be released. It is already known that different reconnection regimes result from changes in the nature of the plasma inflow towards the reconnection site. In this thesis, we examine both how the outflow region responds to changes both in the inflow and outflow boundary conditions and also how introducing compressibility affects the results. We find that if the inflow is converging, the outflow velocity is least, the width of the outflow region is greatest and the ratio of outflowing thermal to kinetic energy is greatest. Also, there is one free outflow parameter which would naturally be specified by the velocity of plasma leaving the reconnection site. We suggest that reverse currents seen in numerical simulations may result from the specification of an extra boundary condition. In addition, we find that the main effects of including compressibility are: to enhance convergence or divergence of the inflow; to increase the maximum reconnection rate where the inflow is converging; to increase the flow speed near the reconnection site where the inflow is diverging; to give faster, narrower outflow jets; to increase variations between regimes in the energy conversion and to increase the ratio of thermal to kinetic energy in the outflow jet.Microinstabilities in high power electron cyclotron heating of plasmasMiller, Andrew Gilberthttp://hdl.handle.net/10023/139772019-04-01T08:37:41Z1991-01-01T00:00:00ZElectron cyclotron resonance heating has been successfully used in a number of experiments, firstly to raise the plasma temperature and secondly to drive currents noninductively. Recently the microwaves in tokamak experiment (MTX) has been proposed at the Lawrence Livermore Laboratory, which will involve pulsed heating at powers much higher than have previously been possible, using a Free Electron Laser (PEL). The physics of such an experiment differs greatly from the physics of experiments using less powerful but continuous operation gyrotron sources. An analytical model of the interaction between a wave and an electron is presented on the assumption that the wave amplitude experienced along the electron guiding centre changes slowly with time as it passes through the beam. This model is tested numerically by integrating the equations of motion governing the electron's motion as it interacts with the wave. Finally this model is used to predict the possible growth of instabilities in a plasma heated by a FEL. The growth rates of these waves may be large enough to act on the plasma in time scales much shorter than typical electron collision times.
1991-01-01T00:00:00ZMiller, Andrew GilbertElectron cyclotron resonance heating has been successfully used in a number of experiments, firstly to raise the plasma temperature and secondly to drive currents noninductively. Recently the microwaves in tokamak experiment (MTX) has been proposed at the Lawrence Livermore Laboratory, which will involve pulsed heating at powers much higher than have previously been possible, using a Free Electron Laser (PEL). The physics of such an experiment differs greatly from the physics of experiments using less powerful but continuous operation gyrotron sources. An analytical model of the interaction between a wave and an electron is presented on the assumption that the wave amplitude experienced along the electron guiding centre changes slowly with time as it passes through the beam. This model is tested numerically by integrating the equations of motion governing the electron's motion as it interacts with the wave. Finally this model is used to predict the possible growth of instabilities in a plasma heated by a FEL. The growth rates of these waves may be large enough to act on the plasma in time scales much shorter than typical electron collision times.A gyrokinetic analysis of electron plasma waves at resonance in magnetic field gradientsMcDonald, Darrenhttp://hdl.handle.net/10023/139752019-04-01T08:39:34Z1995-01-01T00:00:00ZTo produce nuclear fusion in a Tokamak reactor requires the heating of a plasma to a temperature of the order of 10 keV. Electron cyclotron resonant heating (ECRH), in which the plasma is heated by radio waves in resonance with the Larmor frequency of the plasma's electrons, is one scheme under consideration for achieving this. A description of such a heating scheme requires a theory to explain the propagation and absorption of high frequency waves in a plasma in the presence of a magnetic field gradient. A WKB analysis can describe some of the processes involved but a complete explanation requires the use of full wave equations. In this thesis we shall develop a technique for deriving such equations which will be shown to be simpler and more general than calculations performed by earlier workers. The technique relies on including the effect of the magnetic gradient across the Larmor orbit of the electrons in the resonance condition of the wave, the so called Gyrokinetic correction, which has been ignored in calculations by previous workers. Once derived, the equations are solved numerically and the results applied to a number of experiments currently being performed on Tokamak fusion. In addition, we shall also look at the energy loss processes of runaway electrons, which have been shown experimentally to be shorter than would be expected.
1995-01-01T00:00:00ZMcDonald, DarrenTo produce nuclear fusion in a Tokamak reactor requires the heating of a plasma to a temperature of the order of 10 keV. Electron cyclotron resonant heating (ECRH), in which the plasma is heated by radio waves in resonance with the Larmor frequency of the plasma's electrons, is one scheme under consideration for achieving this. A description of such a heating scheme requires a theory to explain the propagation and absorption of high frequency waves in a plasma in the presence of a magnetic field gradient. A WKB analysis can describe some of the processes involved but a complete explanation requires the use of full wave equations. In this thesis we shall develop a technique for deriving such equations which will be shown to be simpler and more general than calculations performed by earlier workers. The technique relies on including the effect of the magnetic gradient across the Larmor orbit of the electrons in the resonance condition of the wave, the so called Gyrokinetic correction, which has been ignored in calculations by previous workers. Once derived, the equations are solved numerically and the results applied to a number of experiments currently being performed on Tokamak fusion. In addition, we shall also look at the energy loss processes of runaway electrons, which have been shown experimentally to be shorter than would be expected.The theory of electron heating in collisonless plasma shock wavesBuckner, A. J. F.http://hdl.handle.net/10023/139732019-04-01T08:38:14Z1993-01-01T00:00:00ZEquations are derived to describe the evolution of an electron distribution function under the action of electromagnetic instabilities in a non-uniform plasma using an extension of the quasilinear theory of Kennel and Engelmann. Variations in both the electron density and temperature and the background magnetic field are taken into account. These equations are simplified in the limit of small electron beta so that an electrostatic approximation is justified. Methods are then presented which allow the solution of these equations (or, in principle, the more complex electromagnetic equations). In particular, a method of solving the kinetic dispersion relation for an arbitrary background (first-order) distribution function with the minimum of additional assumptions and approximations is described in detail. The electrostatic equations are solved for a number of different cases in order to study the action of the modified two stream instability on the electron distribution function. Throughout, realistic values of the ratios of electron to ion mass and electron plasma to cyclotron frequency ratio are used. The applications to collisionless plasma shock waves are discussed, and it is found that the modified two stream instability can produce the (relatively small) amounts of electron heating observed at quasi-perpendicular terrestrial bow shocks, and the flat-topped electron distribution functions seen to evolve. Extensions to the model which would greatly improve its applicability and accuracy, as well as the amount of computational effort required, are discussed.
1993-01-01T00:00:00ZBuckner, A. J. F.Equations are derived to describe the evolution of an electron distribution function under the action of electromagnetic instabilities in a non-uniform plasma using an extension of the quasilinear theory of Kennel and Engelmann. Variations in both the electron density and temperature and the background magnetic field are taken into account. These equations are simplified in the limit of small electron beta so that an electrostatic approximation is justified. Methods are then presented which allow the solution of these equations (or, in principle, the more complex electromagnetic equations). In particular, a method of solving the kinetic dispersion relation for an arbitrary background (first-order) distribution function with the minimum of additional assumptions and approximations is described in detail. The electrostatic equations are solved for a number of different cases in order to study the action of the modified two stream instability on the electron distribution function. Throughout, realistic values of the ratios of electron to ion mass and electron plasma to cyclotron frequency ratio are used. The applications to collisionless plasma shock waves are discussed, and it is found that the modified two stream instability can produce the (relatively small) amounts of electron heating observed at quasi-perpendicular terrestrial bow shocks, and the flat-topped electron distribution functions seen to evolve. Extensions to the model which would greatly improve its applicability and accuracy, as well as the amount of computational effort required, are discussed.Rotational flow in fluid dynamicsMurray, J. D. (James Dickson)http://hdl.handle.net/10023/139672019-11-05T10:56:04Z1955-01-01T00:00:00ZThe thesis is divided into four chapters. Chapter I gives a brief résumé of the state of rotational flow theory up to 1955. Chapter II contains a study of the constant shear flow past cylinders with various cross sections. Chapter III contains a method for obtaining the stream functions for cylinders in a variable shear flow when the latter approximates firstly to a linear vorticity distribution, and secondly to the rotational flow present in a boundary layer. Further, it illustrates the nature of the difficulties likely to be encountered in trying to obtain analytical solutions of problems where the rotation is of a more complicated nature. Finally, Chapter IV contains a relaxation solution to the two-dimensional isentropic compressible rotational flow of a gas through a channel containing a constriction, it also illustrates the complexity of the numerical work required in obtaining relaxation solutions of compressible flow problems with rotation.
1955-01-01T00:00:00ZMurray, J. D. (James Dickson)The thesis is divided into four chapters. Chapter I gives a brief résumé of the state of rotational flow theory up to 1955. Chapter II contains a study of the constant shear flow past cylinders with various cross sections. Chapter III contains a method for obtaining the stream functions for cylinders in a variable shear flow when the latter approximates firstly to a linear vorticity distribution, and secondly to the rotational flow present in a boundary layer. Further, it illustrates the nature of the difficulties likely to be encountered in trying to obtain analytical solutions of problems where the rotation is of a more complicated nature. Finally, Chapter IV contains a relaxation solution to the two-dimensional isentropic compressible rotational flow of a gas through a channel containing a constriction, it also illustrates the complexity of the numerical work required in obtaining relaxation solutions of compressible flow problems with rotation.Mode conversion of plasma wavesWoods, Anna Mariahttp://hdl.handle.net/10023/139652019-04-01T08:37:31Z1987-01-01T00:00:00ZLinear mode conversion processes are much studied in plasma physics because they determine the efficiency of any radio frequency heating scheme. Mode coupling model equations, extracted with varying degrees of rigour from the Maxwell-linearized kinetic equations, are usually fourth or higher order O.D.E's. These are solved by complicated methods to obtain transmission, conversion, reflection and absorption coefficients. Recently, Fuchs et al and Cairns and Lashmore-Davies (C.L-D.) have postulated second order O.D.E's to describe pairwise coupling events. The second order theories have reproduced results previously obtained by much more sophisticated treatments. In this thesis, we firstly examine the hybrid resonances in a cold plasma and show that they have a mode conversion interpretation in the framework of the C.L-D. model. The Budden tunnelling coefficients are recovered for this case. Next, mode conversion between the fast and slow electromagnetic waves in the lower hybrid frequency range is considered. This phenomenon determines the accessibility of the lower hybrid resonance to the slow wave, and is also of theoretical interest because the mode coupling differs in certain aspects from cases previously investigated by C.L-D. A second order approximation to the dispersion relation is used in the mode conversion region leading to Weber's equation from which transmission coefficients are then obtained in various cases. Finally, we provide justification for the use of Weber's equation. The exact fourth order system of O.D.E's for the problem is set down, and a linear transformation, which is an extension of that given by Heading, reveals the second order nature of the coupling process. Numerical solutions of the fourth order system yield transmission coefficients in excellent agreement with the second order theory, and also demonstrate that the electric field variation across the mode conversion region is well approximated, via the above transformation, by our second order theory.
1987-01-01T00:00:00ZWoods, Anna MariaLinear mode conversion processes are much studied in plasma physics because they determine the efficiency of any radio frequency heating scheme. Mode coupling model equations, extracted with varying degrees of rigour from the Maxwell-linearized kinetic equations, are usually fourth or higher order O.D.E's. These are solved by complicated methods to obtain transmission, conversion, reflection and absorption coefficients. Recently, Fuchs et al and Cairns and Lashmore-Davies (C.L-D.) have postulated second order O.D.E's to describe pairwise coupling events. The second order theories have reproduced results previously obtained by much more sophisticated treatments. In this thesis, we firstly examine the hybrid resonances in a cold plasma and show that they have a mode conversion interpretation in the framework of the C.L-D. model. The Budden tunnelling coefficients are recovered for this case. Next, mode conversion between the fast and slow electromagnetic waves in the lower hybrid frequency range is considered. This phenomenon determines the accessibility of the lower hybrid resonance to the slow wave, and is also of theoretical interest because the mode coupling differs in certain aspects from cases previously investigated by C.L-D. A second order approximation to the dispersion relation is used in the mode conversion region leading to Weber's equation from which transmission coefficients are then obtained in various cases. Finally, we provide justification for the use of Weber's equation. The exact fourth order system of O.D.E's for the problem is set down, and a linear transformation, which is an extension of that given by Heading, reveals the second order nature of the coupling process. Numerical solutions of the fourth order system yield transmission coefficients in excellent agreement with the second order theory, and also demonstrate that the electric field variation across the mode conversion region is well approximated, via the above transformation, by our second order theory.Some exact solutions in the one-dimensional unsteady motion of a gasWeir, David Gordonhttp://hdl.handle.net/10023/139642019-04-01T08:37:33Z1961-01-01T00:00:00ZIn this thesis, we present certain exact solutions of the mathematical equations governing the one-dimensional unsteady flow of a compressible fluid. In Chapter 2 we introduce the well-known simplification of the equations (1.1.10), (1.1.11) and (1.1.12) which occurs when the entropy is assumed to be constant, and conditions for parching solutions of the equations along characteristics are obtained. These results are used to generalise a problem solved by Mackie. In chapter 3 we meet the concept of a shook, and exact solutions are obtained for two problems in which shocks occur in non-uniform flows. In chapter 4 the case of waves in shallow water which has differential equations similar to those of gas flow is discussed. The results of the previous section are applied to this case and a problem attacked which permits a comparison to be made of the results obtained by this theory and a simpler linearized theory. Finally in chapter 5 we examine a method introduced by Martin for dealing with certain non-isentropic flows. Some new exact solutions of non-isentropic flows are thus obtained.
1961-01-01T00:00:00ZWeir, David GordonIn this thesis, we present certain exact solutions of the mathematical equations governing the one-dimensional unsteady flow of a compressible fluid. In Chapter 2 we introduce the well-known simplification of the equations (1.1.10), (1.1.11) and (1.1.12) which occurs when the entropy is assumed to be constant, and conditions for parching solutions of the equations along characteristics are obtained. These results are used to generalise a problem solved by Mackie. In chapter 3 we meet the concept of a shook, and exact solutions are obtained for two problems in which shocks occur in non-uniform flows. In chapter 4 the case of waves in shallow water which has differential equations similar to those of gas flow is discussed. The results of the previous section are applied to this case and a problem attacked which permits a comparison to be made of the results obtained by this theory and a simpler linearized theory. Finally in chapter 5 we examine a method introduced by Martin for dealing with certain non-isentropic flows. Some new exact solutions of non-isentropic flows are thus obtained.Stability of some free-surface flowsSmith, Frank Ian Pitthttp://hdl.handle.net/10023/139602019-04-01T08:37:30Z1969-01-01T00:00:00ZThe subject matter of this thesis is concerned with the stability of fluid flows; more particularly , with the stability of liquid films which have an interface with air. We will therefore begin by formulating the basic equations and ideas which pertain to this class of problems. Later in this chapter, a summary will be given of the topics dealt with in this dissertation.
1969-01-01T00:00:00ZSmith, Frank Ian PittThe subject matter of this thesis is concerned with the stability of fluid flows; more particularly , with the stability of liquid films which have an interface with air. We will therefore begin by formulating the basic equations and ideas which pertain to this class of problems. Later in this chapter, a summary will be given of the topics dealt with in this dissertation.The unsteady expansion of a gas into a near vacuumMcLaughlin, Raymondhttp://hdl.handle.net/10023/139562019-04-01T08:39:28Z1975-01-01T00:00:00ZThis thesis is concerned with the unsteady expansion of an initially uniform, stationary gas into a low density, stationary atmosphere, studied from the viewpoint of inviscid gasdynamics. It is found that, there are two regions in the k-𝜎 parameter space having distinct forms for the large time solution, when the atmospheric density is initially proportional to r⁻[super]k, r being the spatial coordinate, k being constant and 𝜎, the geometry index, has its usual meaning. First of all a constant asymptotic shock velocity is assumed and matched expansions, for large r, are constructed. Inner expansions, valid near the shock, are matched to zeroth and first orders with the outer expansions which are valid near the contact front. Zeroth order matching, which, yields the constant asymptotic shock velocity, is possible only in a restricted region of the k-𝜎parameter space and this situation is clarified by appealing to the similarity solutions which are extended to cover cases which have not been dealt with previously.
In the other region of the k-𝜎 parameter space the asymptotic shock velocity is proportional to r[super]∈ where ∈, a positive constant, is found from the similarity solutions as a function of k, γ ,𝜎. An attempt is made at constructing matched asymptotic expansions for large r. The inner solution can be obtained, apart from the evaluation of certain constants, to zeroth and first orders but the outer solution is inaccessible and can only be determined from the full inviscid solution. However it is shown that there exists a solution to the outer equations which matches with the inner solution up to first order. In both cases matching of the first order inner terms to the outer solution produces an eigenvalue problem, the solution of which is not attempted here. Finally full numerical solutions of the inviscid equations, one for each case, were produced using the method of backward drawn characteristics, devised by Hartree, and it will be seen that they compare most favourably with the asymptotic analysis.
1975-01-01T00:00:00ZMcLaughlin, RaymondThis thesis is concerned with the unsteady expansion of an initially uniform, stationary gas into a low density, stationary atmosphere, studied from the viewpoint of inviscid gasdynamics. It is found that, there are two regions in the k-𝜎 parameter space having distinct forms for the large time solution, when the atmospheric density is initially proportional to r⁻[super]k, r being the spatial coordinate, k being constant and 𝜎, the geometry index, has its usual meaning. First of all a constant asymptotic shock velocity is assumed and matched expansions, for large r, are constructed. Inner expansions, valid near the shock, are matched to zeroth and first orders with the outer expansions which are valid near the contact front. Zeroth order matching, which, yields the constant asymptotic shock velocity, is possible only in a restricted region of the k-𝜎parameter space and this situation is clarified by appealing to the similarity solutions which are extended to cover cases which have not been dealt with previously.
In the other region of the k-𝜎 parameter space the asymptotic shock velocity is proportional to r[super]∈ where ∈, a positive constant, is found from the similarity solutions as a function of k, γ ,𝜎. An attempt is made at constructing matched asymptotic expansions for large r. The inner solution can be obtained, apart from the evaluation of certain constants, to zeroth and first orders but the outer solution is inaccessible and can only be determined from the full inviscid solution. However it is shown that there exists a solution to the outer equations which matches with the inner solution up to first order. In both cases matching of the first order inner terms to the outer solution produces an eigenvalue problem, the solution of which is not attempted here. Finally full numerical solutions of the inviscid equations, one for each case, were produced using the method of backward drawn characteristics, devised by Hartree, and it will be seen that they compare most favourably with the asymptotic analysis.Hodograph methods applied to flow past finite wedgesMackie, A. G. (Andrew George)http://hdl.handle.net/10023/139462019-04-01T08:38:08Z1953-01-01T00:00:00Z1953-01-01T00:00:00ZMackie, A. G. (Andrew George)Two parameter integral methods in laminar boundary layer theoryLister, William Macraehttp://hdl.handle.net/10023/139442019-04-01T08:37:47Z1971-01-01T00:00:00ZThe work of this thesis is concerned, with the investigation and attempted improvement of an integral method for solving the two dimensional, incompressible laminar boundary layer equations of fluid dynamics. The method which is based on a theoretical two parameter representation of well-known boundary layer properties was first produced by Professor S. N. Curle. Its range of application, reliability and accuracy rely on four universal functions which have been derived from known exact solutions to the boundary layer equations, and are given tabulated in terms of a pressure gradient parameter 𝞴. This thesis seeks to improve these properties by making adjustments to the tabulated functions and also considers the extension of the method to certain compressible boundary layer problems. The first chapter contains the development of, and background to the method and gives a critical assessment of the existing functions. This analysis indicates that the method may be improved by supplying more data for certain ranges of 𝞴 from which the functions may be calculated; by improving the fitting process; and by the provision for small values of 𝞴 of an analytic form for a shape parameter H which the method involves.
To supply more data two new solutions for the flows u₁ = U₀ (1+𝜉) and u₁ = u₀ (𝜉+𝜉³) where 𝜉 is a non-dimensional co-ordinate in the direction of the flow, are investigated. The resulting work produces some interesting examples of the use of series expansions in boundary layer theory and these, and the results produced, are given in the second chapter. The fitting of the functions is carried out in chapter three. Polynomial models in terms of 𝞴 are fitted by least squares techniques to data from seven solutions and are adjusted to ensure an analytic form for H for small values of 𝞴. A comparison of results using new and old tables Indicates that an improvement has been made. The transformation relating certain compressible and incompressible flows is next examined and the extension of the method to such problems considered. An idea due to Stewartson for assessing the relative accuracies of methods under such circumstances indicates that the method should be highly accurate, a result confirmed by the calculation of the compressible flow u₁ = u₀ (1-𝜉) at a leading edge Mach number of four. The thesis is concluded with a review of the work carried out and the results obtained.
1971-01-01T00:00:00ZLister, William MacraeThe work of this thesis is concerned, with the investigation and attempted improvement of an integral method for solving the two dimensional, incompressible laminar boundary layer equations of fluid dynamics. The method which is based on a theoretical two parameter representation of well-known boundary layer properties was first produced by Professor S. N. Curle. Its range of application, reliability and accuracy rely on four universal functions which have been derived from known exact solutions to the boundary layer equations, and are given tabulated in terms of a pressure gradient parameter 𝞴. This thesis seeks to improve these properties by making adjustments to the tabulated functions and also considers the extension of the method to certain compressible boundary layer problems. The first chapter contains the development of, and background to the method and gives a critical assessment of the existing functions. This analysis indicates that the method may be improved by supplying more data for certain ranges of 𝞴 from which the functions may be calculated; by improving the fitting process; and by the provision for small values of 𝞴 of an analytic form for a shape parameter H which the method involves.
To supply more data two new solutions for the flows u₁ = U₀ (1+𝜉) and u₁ = u₀ (𝜉+𝜉³) where 𝜉 is a non-dimensional co-ordinate in the direction of the flow, are investigated. The resulting work produces some interesting examples of the use of series expansions in boundary layer theory and these, and the results produced, are given in the second chapter. The fitting of the functions is carried out in chapter three. Polynomial models in terms of 𝞴 are fitted by least squares techniques to data from seven solutions and are adjusted to ensure an analytic form for H for small values of 𝞴. A comparison of results using new and old tables Indicates that an improvement has been made. The transformation relating certain compressible and incompressible flows is next examined and the extension of the method to such problems considered. An idea due to Stewartson for assessing the relative accuracies of methods under such circumstances indicates that the method should be highly accurate, a result confirmed by the calculation of the compressible flow u₁ = u₀ (1-𝜉) at a leading edge Mach number of four. The thesis is concluded with a review of the work carried out and the results obtained.The evaporation kinetics of liquid helium IIHunter, George Huttonhttp://hdl.handle.net/10023/139412019-04-01T08:38:18Z1968-01-01T00:00:00ZThis work is concerned with the evaporation and condensation processes occurring when liquid helium II is in equilibrium with its saturated vapour. We define the condensation coefficient a as the fraction of atoms incident on the liquid vapour interface which cross it to form part of the liquid. Experiments to measure are described, and the results are discussed in terms of microscopic condensation processes. The measurements are made by reflecting second sound pulses from the liquid vapour surface at normal incidence and measuring the reflection coefficient. An account is given of the phenomenological theories of Osborne (1962a) and Chernikova (1964), which describe the reflection of second sound from the surface and the associated effect, its transformation into first sound in the gas. Neither of these agree with the experimental results, and Osborne's theory is modified by taking account of the conditions in the gas a small fraction of a mean free path above the surface (rather than many mean free paths above the surface, as in Osborne's original theory). Thus modified, the theory is shown to be in agreement with the measurements of the reflection coefficient. Also described are measurements made in second sound pulses generated at the interface by first sound pulses, themselves generated at the interface by second sound, propagated up the tube, and reflected from its closed and back to the surface. From the time intervals between these pulses the velocity of first sound in the vapour is deduced, and found to be in agreement with previous work. Measurements of pulse amplitude corroborate the reflection coefficient measurements, and taking the two sets of measurements together wo have concluded that a is probably 1 and not less than 0.8 between 1.0°K and 2.14°K. The microscopic processes by which condensation can take place are considered. Experiments due to beaker (unpublished, see Osborne, 1962a) and Osborne (1962b) are described, which indicate that the vapour exchanges momentum with normal fluid only. We have therefore supposed that processes in which a gas atom condenses to form excitations must conserve energy and momentum. Processes involving both bulk excitations and surface excitations are considered, but effects due to the finite lifetime of the excitations and the linewidth of the excitations spectrum are neglected. No attempt has been made to calculate the matrix elements for condensation processes, but plausible estimates have been made of their relative magnitudes. In particular, only processes involving one gas atom and one or two excitations have been considered. Using the requirements of conservation of energy and momentum, it is shown that as the temperature decreases, a decreasing fraction of the incident atom have enough energy to form two excitations, and condensation must take place by the collision of an atom with an existing excitation. A rough estimate of the collision probability for such a process leads to the conclusion that at 1°K, a should be about 0.2. This disagreement with experiment has not been resolved. Finally, some remarks are made about the implications for other work on liquid helium II, and some suggestions for future work.
1968-01-01T00:00:00ZHunter, George HuttonThis work is concerned with the evaporation and condensation processes occurring when liquid helium II is in equilibrium with its saturated vapour. We define the condensation coefficient a as the fraction of atoms incident on the liquid vapour interface which cross it to form part of the liquid. Experiments to measure are described, and the results are discussed in terms of microscopic condensation processes. The measurements are made by reflecting second sound pulses from the liquid vapour surface at normal incidence and measuring the reflection coefficient. An account is given of the phenomenological theories of Osborne (1962a) and Chernikova (1964), which describe the reflection of second sound from the surface and the associated effect, its transformation into first sound in the gas. Neither of these agree with the experimental results, and Osborne's theory is modified by taking account of the conditions in the gas a small fraction of a mean free path above the surface (rather than many mean free paths above the surface, as in Osborne's original theory). Thus modified, the theory is shown to be in agreement with the measurements of the reflection coefficient. Also described are measurements made in second sound pulses generated at the interface by first sound pulses, themselves generated at the interface by second sound, propagated up the tube, and reflected from its closed and back to the surface. From the time intervals between these pulses the velocity of first sound in the vapour is deduced, and found to be in agreement with previous work. Measurements of pulse amplitude corroborate the reflection coefficient measurements, and taking the two sets of measurements together wo have concluded that a is probably 1 and not less than 0.8 between 1.0°K and 2.14°K. The microscopic processes by which condensation can take place are considered. Experiments due to beaker (unpublished, see Osborne, 1962a) and Osborne (1962b) are described, which indicate that the vapour exchanges momentum with normal fluid only. We have therefore supposed that processes in which a gas atom condenses to form excitations must conserve energy and momentum. Processes involving both bulk excitations and surface excitations are considered, but effects due to the finite lifetime of the excitations and the linewidth of the excitations spectrum are neglected. No attempt has been made to calculate the matrix elements for condensation processes, but plausible estimates have been made of their relative magnitudes. In particular, only processes involving one gas atom and one or two excitations have been considered. Using the requirements of conservation of energy and momentum, it is shown that as the temperature decreases, a decreasing fraction of the incident atom have enough energy to form two excitations, and condensation must take place by the collision of an atom with an existing excitation. A rough estimate of the collision probability for such a process leads to the conclusion that at 1°K, a should be about 0.2. This disagreement with experiment has not been resolved. Finally, some remarks are made about the implications for other work on liquid helium II, and some suggestions for future work.Hydrodynamics of liquid helium IIGriffiths, D. J. (Derek John)http://hdl.handle.net/10023/139382019-04-01T08:37:09Z1964-01-01T00:00:00ZObservations have been made of the behaviour of a fine quartz fibre, weighted at its lower end and suspended inside a short, horizontal tunnel in which counterblow of the normal and superfluid components of liquid helium II can be produced by a heater. Section I of this thesis is an introduction to the hydrodynamic of liquid helium II. In section II the interaction with such a fibre of quantized vortex lines in the superfluid is discussed, and the effect of a short heat pulse on the fibre when it is carrying superfluid circulation in calculated approximately. The different responses of the fibre to turbulence in the normal fluid and in the superfluid are contrasted.
In section III, after a description of the apparatus and the experimental method, measurements, deduced from the response to heat pulses, of the circulation about the fibre from 1.3°K to2.1°K are reported. At all temperatures circulations of the expected order from magnitude are observed to grow and decay with time. At 1.3°K apparent circulations of up to about 1/5 quantum occur. In undisturbed helium the largest circulations are more stable than other values, persisting for up to five minutes. Measurement of the same circulation both by the heat-pulse method and by the deflection of the fibre in a steady heat current suggests that the large, persistent circulations may in fact be equal to one quantum. The sense of the observed circulations about the fibre at 1.3°K is strongly biased, this bias being probably associated with the heater geometry. In small heat current no change in the bins or persistence of circulation can be detected, but in currents above 11/2-3 mW/cm², depending on the heater, the circulation about the fibre is both more variable and of the opposite bias to that in undisturbed helium. This behaviour continues for 100 sec or more after the heat current has been switched off. At higher temperatures there are indications that the behaviour might be similar if it were possible for the helium to region its undisturbed condition after being stirred up by turbulent heat currents. In fact this seems other to be impossible, or to require many hundreds of seconds, and the situation is therefore rather confused.
In still higher heat currents measurement of superfluid circulation by heat pulses is impossible because the fibre is continuously agitated in a random way. From measurements of the rms deflections of the bob on the end of the fibre a critical heat current for the onset of such turbulence is found at 1.3°K. At higher temperatures the sensitivity is too low for the transition itself, if any, to be detected, but an upper limit to the critical heat current is given. At 1.3°K and 2.1°K the rms deflection increases monotonically with increasing heat currents, but at intermediate temperatures it is variable, because the bob is often hardly agitated for long periods during apparently supercritical heat currents. This is called quiescent behaviour.
When a supercritical heat current is a delay before agitation of the fibre begins. The delay time, which is often not very well defined, has been measured as a function of the heat current. When the current is switched off the agitation of the bob decoys in a few seconds, but at 1.3°K the circulation about the fibre is small and variable for 100 sec or more, until the persistence and bias characteristic of undisturbed helium regained. These results are discussed in section IV.
1964-01-01T00:00:00ZGriffiths, D. J. (Derek John)Observations have been made of the behaviour of a fine quartz fibre, weighted at its lower end and suspended inside a short, horizontal tunnel in which counterblow of the normal and superfluid components of liquid helium II can be produced by a heater. Section I of this thesis is an introduction to the hydrodynamic of liquid helium II. In section II the interaction with such a fibre of quantized vortex lines in the superfluid is discussed, and the effect of a short heat pulse on the fibre when it is carrying superfluid circulation in calculated approximately. The different responses of the fibre to turbulence in the normal fluid and in the superfluid are contrasted.
In section III, after a description of the apparatus and the experimental method, measurements, deduced from the response to heat pulses, of the circulation about the fibre from 1.3°K to2.1°K are reported. At all temperatures circulations of the expected order from magnitude are observed to grow and decay with time. At 1.3°K apparent circulations of up to about 1/5 quantum occur. In undisturbed helium the largest circulations are more stable than other values, persisting for up to five minutes. Measurement of the same circulation both by the heat-pulse method and by the deflection of the fibre in a steady heat current suggests that the large, persistent circulations may in fact be equal to one quantum. The sense of the observed circulations about the fibre at 1.3°K is strongly biased, this bias being probably associated with the heater geometry. In small heat current no change in the bins or persistence of circulation can be detected, but in currents above 11/2-3 mW/cm², depending on the heater, the circulation about the fibre is both more variable and of the opposite bias to that in undisturbed helium. This behaviour continues for 100 sec or more after the heat current has been switched off. At higher temperatures there are indications that the behaviour might be similar if it were possible for the helium to region its undisturbed condition after being stirred up by turbulent heat currents. In fact this seems other to be impossible, or to require many hundreds of seconds, and the situation is therefore rather confused.
In still higher heat currents measurement of superfluid circulation by heat pulses is impossible because the fibre is continuously agitated in a random way. From measurements of the rms deflections of the bob on the end of the fibre a critical heat current for the onset of such turbulence is found at 1.3°K. At higher temperatures the sensitivity is too low for the transition itself, if any, to be detected, but an upper limit to the critical heat current is given. At 1.3°K and 2.1°K the rms deflection increases monotonically with increasing heat currents, but at intermediate temperatures it is variable, because the bob is often hardly agitated for long periods during apparently supercritical heat currents. This is called quiescent behaviour.
When a supercritical heat current is a delay before agitation of the fibre begins. The delay time, which is often not very well defined, has been measured as a function of the heat current. When the current is switched off the agitation of the bob decoys in a few seconds, but at 1.3°K the circulation about the fibre is small and variable for 100 sec or more, until the persistence and bias characteristic of undisturbed helium regained. These results are discussed in section IV.Approximate methods in high speed flowBurnside, Robert R.http://hdl.handle.net/10023/139312019-04-01T08:37:48Z1962-01-01T00:00:00ZIn many problems arising in the theory of compressible flow, the equations characterising the solution of the system are so intractable that recourse must be made to some approximate method which allows the essential features of the flow to be preserved, whilst to some degree, simplifying the mathematics. It is with certain methods of this type that this thesis is concerned.
In the subsequent work, we shall assume that the effects due to viscosity and heat conduction are so small as to be negligible. These assumptions may be shown to be largely valid except in those domains of the flow-field where the modified system of equations predicts regions in which the solution is in general multivalued. In the modified system, however, such ‘regions’ are avoided by the introduction of mathematical discontinuities and, assuming that the jump conditions across them can be determines, are sufficient to provide single-valued solutions valid everywhere, except at the discontinuity. The methods to be presented are formulated in the plane consisting of one space variable and one time variable.
1962-01-01T00:00:00ZBurnside, Robert R.In many problems arising in the theory of compressible flow, the equations characterising the solution of the system are so intractable that recourse must be made to some approximate method which allows the essential features of the flow to be preserved, whilst to some degree, simplifying the mathematics. It is with certain methods of this type that this thesis is concerned.
In the subsequent work, we shall assume that the effects due to viscosity and heat conduction are so small as to be negligible. These assumptions may be shown to be largely valid except in those domains of the flow-field where the modified system of equations predicts regions in which the solution is in general multivalued. In the modified system, however, such ‘regions’ are avoided by the introduction of mathematical discontinuities and, assuming that the jump conditions across them can be determines, are sufficient to provide single-valued solutions valid everywhere, except at the discontinuity. The methods to be presented are formulated in the plane consisting of one space variable and one time variable.Aspects of natural convention and of non-linear hydridynamic stabilityUsher, J. R.http://hdl.handle.net/10023/139222019-04-01T08:37:05Z1974-01-01T00:00:00ZIn Part I of this thesis, steady and time-dependent, natural-convection similarity flows with mass transfer are discussed. Similarity flows for natural convection on families of two-dimensional bodies with closed lower ends are enumerated, when both a temperature distribution and a suction velocity distribution are prescribed at the body surface. For steady similarity flow on a heated vertical flat plate, with mass transfer at the surface, a numerical procedure is introduced for determining the velocity and temperature profiles. These results are presented in Figs. 2 and 3. Other similarity flows may be found by the same method.
A simplification, valid for “strong” suction, is discussed. An extension of Mangler’s transformation [1948] is given which reduces the equations governing axisymmetric flow to those for two-dimensional flow in steady natural convection.
In Part II non-linear resonant instability in parallel shear flows is discussed. A.D.D.Craik’s (see Usher and Craik [I]) modified version of Bateman’s [1956] variational formulation for viscous flows is employed to derive the second-order interaction equations governing the temporal evolution of a resonant wave triad in a sheer flow. (An extension of Craik’s variational formulation to free surface flows is presented but is not required in the subsequent analysis for the resonance problem). This problem was treated previously using a ‘direct’ approach (employing the Navier-Stokes equations) by Craik [1971]. The major advantage of the present method over the ‘direct’ method is the substantial reduction in algebraic complexity. Also, a justification of the validity of Craik’s previous analysis is given.
For this same resonance problem, third-order interaction equations are derived by the *direct* method since, to this order of approximation, little advantage is to be gained from the variational formulation. The resonance theory is thereby developed to the same order of approximation as the non-resonant third-order theory of Stuart [1960, 1962].
An asymptotic analysis for large Reynolds numbers reveals that the magnitudes of the third-order interaction coefficients – like certain of those at second-order – are remarkably large. Such results lead to a discussion of the regions of validity of the perturbation analysis. Also some light is shed on the roles played by resonance and three-dimensionality in the non-linear instability of shear flows.
1974-01-01T00:00:00ZUsher, J. R.In Part I of this thesis, steady and time-dependent, natural-convection similarity flows with mass transfer are discussed. Similarity flows for natural convection on families of two-dimensional bodies with closed lower ends are enumerated, when both a temperature distribution and a suction velocity distribution are prescribed at the body surface. For steady similarity flow on a heated vertical flat plate, with mass transfer at the surface, a numerical procedure is introduced for determining the velocity and temperature profiles. These results are presented in Figs. 2 and 3. Other similarity flows may be found by the same method.
A simplification, valid for “strong” suction, is discussed. An extension of Mangler’s transformation [1948] is given which reduces the equations governing axisymmetric flow to those for two-dimensional flow in steady natural convection.
In Part II non-linear resonant instability in parallel shear flows is discussed. A.D.D.Craik’s (see Usher and Craik [I]) modified version of Bateman’s [1956] variational formulation for viscous flows is employed to derive the second-order interaction equations governing the temporal evolution of a resonant wave triad in a sheer flow. (An extension of Craik’s variational formulation to free surface flows is presented but is not required in the subsequent analysis for the resonance problem). This problem was treated previously using a ‘direct’ approach (employing the Navier-Stokes equations) by Craik [1971]. The major advantage of the present method over the ‘direct’ method is the substantial reduction in algebraic complexity. Also, a justification of the validity of Craik’s previous analysis is given.
For this same resonance problem, third-order interaction equations are derived by the *direct* method since, to this order of approximation, little advantage is to be gained from the variational formulation. The resonance theory is thereby developed to the same order of approximation as the non-resonant third-order theory of Stuart [1960, 1962].
An asymptotic analysis for large Reynolds numbers reveals that the magnitudes of the third-order interaction coefficients – like certain of those at second-order – are remarkably large. Such results lead to a discussion of the regions of validity of the perturbation analysis. Also some light is shed on the roles played by resonance and three-dimensionality in the non-linear instability of shear flows.Ion dynamics in collisionless shock wavesSherwell, Davidhttp://hdl.handle.net/10023/139172019-04-01T08:37:38Z1976-01-01T00:00:00ZIn a laminar model of a collisionless magnetosonic shock wave, ion equations of motion are integrated through shock-like profiles. Conservation relations and Maxwell's equations allow a self-consistent determination of unknown downstream ion distribution functions fᵢ, ion temperature Tᵢ, and electric potential jump 𝛷. Favourable comparison of model Tᵢ, 𝛷.
Favourable comparison of model Tᵢ, 𝛷 , with experiment establishes (at low 𝛽 ≲ O.3, 𝛽=8 π N
[sub] l k (T[sub]e₂+Tᵢ[sub]l)/B₁²)
importance of laminar ion dynamics. Heating is due to distortion of Maxwellian distributions when entropy is conserved; in particular shock dynamics is dominated by a fast "tail" of reflected ions. The solutions for fᵢ are considered. The "stability" of the model to its assumptions (linear profiles, shock thickness (L[sub]s)) is shown. When reflections occur a self-consistent length emerges. The solutions Tᵢ, 𝛷 are extensively studied at various Mach numbers for different values of 𝛽. Laminar ion heating is very efficient and at high 𝛽 can exceed proper conservation levels due to ion reflections; at high 𝛽(≥ 𝛽 *) the electric potential is unable to slow the ions to conservation levels. The model predicts significant reflected ion currents in the plane of the shock. The boundary 𝛽 * is determined. Then laminar ion dynamics on the scale of the electron heating length (~10 C/w[sub]p ₑ) cannot occur for 𝛽 > 𝛽 *. Dependence on L[sub]s and T ₑ₁,/Tᵢ₁ is considered. The nature of non-laminar 𝛽 >𝛽* shocks is considered. Collisions are found to be important in laboratory shocks, and are efficient in slowing the reflected ions. In the absence of collisions, ion instabilities must be considered. It is shown that turbulent slowing of the fast ions cannot take place in L[sub]s alone. Further it is shown possible to construct a shock so that non-laminar mechanisms cannot occur significantly. Then the laminar model is re-instated. A decoupling of ion and electron heating lengths is proposed. Reflection heating in the Earth's Bow Shock (𝛽>𝛽*) is modelled, and is comparable with experiment.
1976-01-01T00:00:00ZSherwell, DavidIn a laminar model of a collisionless magnetosonic shock wave, ion equations of motion are integrated through shock-like profiles. Conservation relations and Maxwell's equations allow a self-consistent determination of unknown downstream ion distribution functions fᵢ, ion temperature Tᵢ, and electric potential jump 𝛷. Favourable comparison of model Tᵢ, 𝛷.
Favourable comparison of model Tᵢ, 𝛷 , with experiment establishes (at low 𝛽 ≲ O.3, 𝛽=8 π N
[sub] l k (T[sub]e₂+Tᵢ[sub]l)/B₁²)
importance of laminar ion dynamics. Heating is due to distortion of Maxwellian distributions when entropy is conserved; in particular shock dynamics is dominated by a fast "tail" of reflected ions. The solutions for fᵢ are considered. The "stability" of the model to its assumptions (linear profiles, shock thickness (L[sub]s)) is shown. When reflections occur a self-consistent length emerges. The solutions Tᵢ, 𝛷 are extensively studied at various Mach numbers for different values of 𝛽. Laminar ion heating is very efficient and at high 𝛽 can exceed proper conservation levels due to ion reflections; at high 𝛽(≥ 𝛽 *) the electric potential is unable to slow the ions to conservation levels. The model predicts significant reflected ion currents in the plane of the shock. The boundary 𝛽 * is determined. Then laminar ion dynamics on the scale of the electron heating length (~10 C/w[sub]p ₑ) cannot occur for 𝛽 > 𝛽 *. Dependence on L[sub]s and T ₑ₁,/Tᵢ₁ is considered. The nature of non-laminar 𝛽 >𝛽* shocks is considered. Collisions are found to be important in laboratory shocks, and are efficient in slowing the reflected ions. In the absence of collisions, ion instabilities must be considered. It is shown that turbulent slowing of the fast ions cannot take place in L[sub]s alone. Further it is shown possible to construct a shock so that non-laminar mechanisms cannot occur significantly. Then the laminar model is re-instated. A decoupling of ion and electron heating lengths is proposed. Reflection heating in the Earth's Bow Shock (𝛽>𝛽*) is modelled, and is comparable with experiment.Investigations on classical symmetries theory of quantizationGuest, P. B.http://hdl.handle.net/10023/139132019-11-27T17:23:33Z1972-01-01T00:00:00ZThe thesis divides naturally into two parts. Part I raises, and in some cases answers, questions concerning symmetry in classical mechanics. The main result (Theorem 6.4) shows that the assumption of the existence of a realization puts an upper limit on the rank of the algebra.
The heart of the thesis (covering three-quarters of the volume) is section II on the quantization of classical systems. §1 lists axioms desirable in any quantization rule for the 'functions of the q's'. The momentum observables are introduced in §2 prior to their quantization in §4. §5 essentially shows how conventional quantum mechanics fits into this scheme of things. By progressive specialization from a general manifold to a vector space, from a general quantization scheme to one which is linear on the linear momentum functions, and finally to an entirely well-behaved (admissible) quantization rule, into which conventional quantum mechanics fits nicely, we obtain in §7-§9 results which become progressively more and more powerful. The final theorem (Theorem 9.2) is perhaps the most significant of all. This result states that there exists a class of functions, which contains all functions of the q's and functions of the p's and all momentum observables and which is closed with respect to any linear canonical transformation L; a rule A assigning a unique self-adjoint operator to each such function f; a unitary operator WL corresponding to L and an equation
𝛬(𝑓 ∘ 𝐿) = 𝑊[sub]𝐿⁻ 𝛬 𝑓 𝑊[sub]𝐿
1972-01-01T00:00:00ZGuest, P. B.The thesis divides naturally into two parts. Part I raises, and in some cases answers, questions concerning symmetry in classical mechanics. The main result (Theorem 6.4) shows that the assumption of the existence of a realization puts an upper limit on the rank of the algebra.
The heart of the thesis (covering three-quarters of the volume) is section II on the quantization of classical systems. §1 lists axioms desirable in any quantization rule for the 'functions of the q's'. The momentum observables are introduced in §2 prior to their quantization in §4. §5 essentially shows how conventional quantum mechanics fits into this scheme of things. By progressive specialization from a general manifold to a vector space, from a general quantization scheme to one which is linear on the linear momentum functions, and finally to an entirely well-behaved (admissible) quantization rule, into which conventional quantum mechanics fits nicely, we obtain in §7-§9 results which become progressively more and more powerful. The final theorem (Theorem 9.2) is perhaps the most significant of all. This result states that there exists a class of functions, which contains all functions of the q's and functions of the p's and all momentum observables and which is closed with respect to any linear canonical transformation L; a rule A assigning a unique self-adjoint operator to each such function f; a unitary operator WL corresponding to L and an equation
𝛬(𝑓 ∘ 𝐿) = 𝑊[sub]𝐿⁻ 𝛬 𝑓 𝑊[sub]𝐿An algebraic formulation of asmptotically separable quantum mechanicsMcLean, R. G. Derekhttp://hdl.handle.net/10023/139092019-04-01T08:37:14Z1984-01-01T00:00:00ZThis thesis explores the possibility of an algebraic formulation of non-relativistic quantum theory in which certain paradoxes associated with non-locality may be resolved. It is shown that the localisation of a free quantum mechanical wave function at large time coincides approximately with the localisation of an ensemble of classical particles having the same momentum range. This result is used to give a formal definition of spatially separating states and spatially separating particles. We then study certain C*-algebras on which expectation values converge in an infinite time limit. By considering such algebras which contain local observables it is possible to introduce states at infinity as limits of states described by wave functions. In such a state at infinity there is zero probability of a position measurement finding the system in any bounded region in configuration space. It is shown that a C*-algebra exists on which any coherent superposition of spatially separating states will converge in an infinite time limit to a mixture of disjoint states. This allows us to obtain an asymptotic resolution of de Broglie's paradox and the Einstein, Podolsy and Rosen paradox. These results are obtained for the simplest types of quantum systems i.e. a one particle system without spin having configuration space IRⁿ and a system consisting of two such particles which may be distinguished from each other.
1984-01-01T00:00:00ZMcLean, R. G. DerekThis thesis explores the possibility of an algebraic formulation of non-relativistic quantum theory in which certain paradoxes associated with non-locality may be resolved. It is shown that the localisation of a free quantum mechanical wave function at large time coincides approximately with the localisation of an ensemble of classical particles having the same momentum range. This result is used to give a formal definition of spatially separating states and spatially separating particles. We then study certain C*-algebras on which expectation values converge in an infinite time limit. By considering such algebras which contain local observables it is possible to introduce states at infinity as limits of states described by wave functions. In such a state at infinity there is zero probability of a position measurement finding the system in any bounded region in configuration space. It is shown that a C*-algebra exists on which any coherent superposition of spatially separating states will converge in an infinite time limit to a mixture of disjoint states. This allows us to obtain an asymptotic resolution of de Broglie's paradox and the Einstein, Podolsy and Rosen paradox. These results are obtained for the simplest types of quantum systems i.e. a one particle system without spin having configuration space IRⁿ and a system consisting of two such particles which may be distinguished from each other.Geometrical and topological properties of fractal percolationOrzechowski, Mark E.http://hdl.handle.net/10023/139072020-07-08T10:09:13Z1998-01-01T00:00:00ZThe basic 'fractal percolation' process was first proposed by Mandelbrot in 1974 and takes the following form. Let M ≥2 and P ∈ [0,1]; we start with the unit square C₀ = [0,1]²; Divide C₀ into M² equal closed squares, each of side-length M⁻¹ , in the natural way and retain each of these squares with probability p, or else remove it with probability 1 - p. We let C₁ be the union of those squares retained. The process is now repeated within each square of C₁ to give a new set C₂⊆C₁, consisting of squares of side-length M⁻². Iterating the construction in the obvious way, we obtain a decreasing sequence of sets C₀⊇ C₁ ⊇ C₂ ⊇ … with limit C[sub]∞ = ∩[sub]n≥₁C[sub]n.
The set C[sub]∞ is an example of a random Cantor set, and is typically highly intricate in nature. It may be empty, dust-like or highly connected, depending on the value of p; percolation is said to occur if C[sub]∞ contains large connected components linking opposite sides of the unit square.
In this thesis we shall investigate some of the geometrical and topological properties of C[sub]∞ that hold either almost surely (with probability 1) or with non-zero probability. In particular, the following results are established. We obtain (almost sure) lower and upper bounds on the box-counting dimension of the 'straightest' crossings in C[sub]∞ whenever percolation occurs; we also look at the distribution of the sizes of the connected components and the probability of percolation. In the three-dimensional version of the process, we establish the existence of two distinct phases of percolation, corresponding to the occurrence of paths and surfaces (or 'sheets') in the limit set, and study the limiting behaviour of the phase transition to sheet percolation as M → ∞. We also consider the results of some computer simulations of fractal percolation and present a number of generalisations of the basic process and other closely related constructions.
1998-01-01T00:00:00ZOrzechowski, Mark E.The basic 'fractal percolation' process was first proposed by Mandelbrot in 1974 and takes the following form. Let M ≥2 and P ∈ [0,1]; we start with the unit square C₀ = [0,1]²; Divide C₀ into M² equal closed squares, each of side-length M⁻¹ , in the natural way and retain each of these squares with probability p, or else remove it with probability 1 - p. We let C₁ be the union of those squares retained. The process is now repeated within each square of C₁ to give a new set C₂⊆C₁, consisting of squares of side-length M⁻². Iterating the construction in the obvious way, we obtain a decreasing sequence of sets C₀⊇ C₁ ⊇ C₂ ⊇ … with limit C[sub]∞ = ∩[sub]n≥₁C[sub]n.
The set C[sub]∞ is an example of a random Cantor set, and is typically highly intricate in nature. It may be empty, dust-like or highly connected, depending on the value of p; percolation is said to occur if C[sub]∞ contains large connected components linking opposite sides of the unit square.
In this thesis we shall investigate some of the geometrical and topological properties of C[sub]∞ that hold either almost surely (with probability 1) or with non-zero probability. In particular, the following results are established. We obtain (almost sure) lower and upper bounds on the box-counting dimension of the 'straightest' crossings in C[sub]∞ whenever percolation occurs; we also look at the distribution of the sizes of the connected components and the probability of percolation. In the three-dimensional version of the process, we establish the existence of two distinct phases of percolation, corresponding to the occurrence of paths and surfaces (or 'sheets') in the limit set, and study the limiting behaviour of the phase transition to sheet percolation as M → ∞. We also consider the results of some computer simulations of fractal percolation and present a number of generalisations of the basic process and other closely related constructions.Graph directed self-conformal multifractalsCole, Julianhttp://hdl.handle.net/10023/139032019-04-01T08:39:36Z1999-01-01T00:00:00ZIn this thesis we study the multifractal structure of graph directed self-conformal measures. We begin by introducing a number of notions from geometric measure theory. In particular, several notions of dimension, graph directed iterated function schemes, and the thermodynamic formalism. We then give an historical introduction to multifractal analysis. Finally, we develop our own contribution to multifractal analysis. Our own contribution to multifractal analysis can be broken into three parts; the proof of two multifractal density theorems, the calculation of the multifractal spectrum of self-conformal measures coded by graph directed iterated function schemes, and the introduction of a relative multifractal formalism together with an investigation of the relative multifractal structure of one graph directed self-conformal measure with respect to another. Specifically, in Chapter 5 we show that by interpreting the multifractal Hausdorff and packing measures Olsen introduced in [0195] as Henstock-Thomson variation measures we are able to obtain two stronger density theorems than those obtained by Olsen. In Chapter 6 we give full details of the calculation of the multifractal spectrum of graph directed self-conformal measures satisfying the strong open set condition and show that the multifractal Hausdorff and packing measures introduced by Olsen in [0195] take positive and finite values at the critical dimension provided that the self-conformal measures satisfy the strong separation condition. In Chapter 7 we formalise the idea of performing multifractal analysis with respect to an arbitrary reference measure by developing a formalism for the multifractal analysis of one measure with respect to another. This formalism is based on the ideas of the 'multifractal formalism' as first introduced by Halsey et. al. [HJKPS86] and closely parallels Olsen's formal treatment of this formalism in [0195]. In Chapter 8 we illustrate our relative multifractal formalism by investigating the relative multifractal structure of one graph directed self-conformal measure with respect to another where the two measures are based on the same graph directed self-conformal iterated function scheme which satisfies the strong open set condition.
1999-01-01T00:00:00ZCole, JulianIn this thesis we study the multifractal structure of graph directed self-conformal measures. We begin by introducing a number of notions from geometric measure theory. In particular, several notions of dimension, graph directed iterated function schemes, and the thermodynamic formalism. We then give an historical introduction to multifractal analysis. Finally, we develop our own contribution to multifractal analysis. Our own contribution to multifractal analysis can be broken into three parts; the proof of two multifractal density theorems, the calculation of the multifractal spectrum of self-conformal measures coded by graph directed iterated function schemes, and the introduction of a relative multifractal formalism together with an investigation of the relative multifractal structure of one graph directed self-conformal measure with respect to another. Specifically, in Chapter 5 we show that by interpreting the multifractal Hausdorff and packing measures Olsen introduced in [0195] as Henstock-Thomson variation measures we are able to obtain two stronger density theorems than those obtained by Olsen. In Chapter 6 we give full details of the calculation of the multifractal spectrum of graph directed self-conformal measures satisfying the strong open set condition and show that the multifractal Hausdorff and packing measures introduced by Olsen in [0195] take positive and finite values at the critical dimension provided that the self-conformal measures satisfy the strong separation condition. In Chapter 7 we formalise the idea of performing multifractal analysis with respect to an arbitrary reference measure by developing a formalism for the multifractal analysis of one measure with respect to another. This formalism is based on the ideas of the 'multifractal formalism' as first introduced by Halsey et. al. [HJKPS86] and closely parallels Olsen's formal treatment of this formalism in [0195]. In Chapter 8 we illustrate our relative multifractal formalism by investigating the relative multifractal structure of one graph directed self-conformal measure with respect to another where the two measures are based on the same graph directed self-conformal iterated function scheme which satisfies the strong open set condition.Parametric models of surfacesRobertson, Stewart A. (Stewart Alexander)http://hdl.handle.net/10023/138972019-04-01T08:37:29Z1957-01-01T00:00:00Z1957-01-01T00:00:00ZRobertson, Stewart A. (Stewart Alexander)Finite difference techniques of improved accuracyLambert, J. D.http://hdl.handle.net/10023/138882019-04-01T08:37:46Z1963-01-01T00:00:00ZIt is the major purpose of this thesis to propose finite difference techniques of improved accuracy for the numerical solution of ordinary differential equations, and for the numerical evaluation of definite integrals, the former problem being discussed in Chapter II, and the latter in Chapter IV. In Chapter III the stability of the formulae evolved in Chapter II is studied.
1963-01-01T00:00:00ZLambert, J. D.It is the major purpose of this thesis to propose finite difference techniques of improved accuracy for the numerical solution of ordinary differential equations, and for the numerical evaluation of definite integrals, the former problem being discussed in Chapter II, and the latter in Chapter IV. In Chapter III the stability of the formulae evolved in Chapter II is studied.Polynomial interpolation on a triangular regionYahaya, Daudhttp://hdl.handle.net/10023/138872019-04-01T08:38:55Z1994-01-01T00:00:00ZIt is well known that given f there is a unique polynomial of degree at most n which interpolates f on the standard triangle with uniform nodes (i, j), i, j ≥ 0, i + j ≤n. This leads us to the study of polynomial interpolation on a "triangular" domain with the nodes,
S = {([i], [j]): i, j ≥ 0, i + j ≤n}, [k] = [k][sub]q = (1-qᵏ)/(1-q), q > 0, which includes the standard triangle as a special case. In Chapter 2 of this thesis we derive a forward difference formula (of degree at most n) in the x and y directions for the interpolating polynomial P[sub]n on S. We also construct a Lagrange form of an interpolating polynomial which uses hyperbolas (although its coefficients are of degree up to 2n) and discuss a Neville-Aitken algorithm. In Chapter 3 we derive the Newton formula for the interpolating polynomial P[sub]n on the set of distinct points {(xᵢ, y[sub]j): i, j ≥ 0, i + j ≤n}. In particular if xᵢ = [i][sub]p and y[sub]j = [j]q, we show that Newton's form of P[sub]n reduces to a forward difference formula. We show further that we can express the interpolating polynomial on S itself in a Lagrange form and although its coefficients Ln/ij are not as simple as those of the first Lagrange form, they all have degree n. Moreover, Ln/ij can all be expressed in terms of Lm/0,0, 0 ≤ m ≤ n. In Chapter 4 we show that P[sub]n has a limit when both p, q → 0. We then verify that the interpolation properties of the limit form depend on the appropriate partial derivatives of f(x, y). In Chapter 5 we study integration rules I[sub]n of interpolatory type on the triangle S[sub] = {(x, y): 0 ≤ x ≤y ≤ [n]). For 1 ≤ n ≤5, we calculate the weights wn/ij for I[sub]n in terms of the parameter q and study certain general properties which govern wn/ij on S[sub]n. Finally, Chapter 6 deals with the behaviour of the Lebesgue functions 𝜆[sub]n(x, y; q) and the corresponding Lebesgue constant. We prove a property concerning where 𝜆[sub]n takes the value 1 at points other than the interpolation nodes. We also analyse the discontinuity of the directional derivative of 𝜆[sub]n on S[sub]n.
1994-01-01T00:00:00ZYahaya, DaudIt is well known that given f there is a unique polynomial of degree at most n which interpolates f on the standard triangle with uniform nodes (i, j), i, j ≥ 0, i + j ≤n. This leads us to the study of polynomial interpolation on a "triangular" domain with the nodes,
S = {([i], [j]): i, j ≥ 0, i + j ≤n}, [k] = [k][sub]q = (1-qᵏ)/(1-q), q > 0, which includes the standard triangle as a special case. In Chapter 2 of this thesis we derive a forward difference formula (of degree at most n) in the x and y directions for the interpolating polynomial P[sub]n on S. We also construct a Lagrange form of an interpolating polynomial which uses hyperbolas (although its coefficients are of degree up to 2n) and discuss a Neville-Aitken algorithm. In Chapter 3 we derive the Newton formula for the interpolating polynomial P[sub]n on the set of distinct points {(xᵢ, y[sub]j): i, j ≥ 0, i + j ≤n}. In particular if xᵢ = [i][sub]p and y[sub]j = [j]q, we show that Newton's form of P[sub]n reduces to a forward difference formula. We show further that we can express the interpolating polynomial on S itself in a Lagrange form and although its coefficients Ln/ij are not as simple as those of the first Lagrange form, they all have degree n. Moreover, Ln/ij can all be expressed in terms of Lm/0,0, 0 ≤ m ≤ n. In Chapter 4 we show that P[sub]n has a limit when both p, q → 0. We then verify that the interpolation properties of the limit form depend on the appropriate partial derivatives of f(x, y). In Chapter 5 we study integration rules I[sub]n of interpolatory type on the triangle S[sub] = {(x, y): 0 ≤ x ≤y ≤ [n]). For 1 ≤ n ≤5, we calculate the weights wn/ij for I[sub]n in terms of the parameter q and study certain general properties which govern wn/ij on S[sub]n. Finally, Chapter 6 deals with the behaviour of the Lebesgue functions 𝜆[sub]n(x, y; q) and the corresponding Lebesgue constant. We prove a property concerning where 𝜆[sub]n takes the value 1 at points other than the interpolation nodes. We also analyse the discontinuity of the directional derivative of 𝜆[sub]n on S[sub]n.Some contributions to the theory and application of polynomial approximationPhillips, G. M. (George McArtney)http://hdl.handle.net/10023/138832019-04-01T08:38:17Z1969-01-01T00:00:00ZThe fundamental theorem, as far as this work is concerned, is Weierstrass' theorem (1885) on the approximability of continuous functions by polynomials. Since the time of Weierstrass (1815-97) and his equally important contemporary Chebyshev (1821-94), the topic of approximation has grown enormously into a subject of considerable interest to both pure and applied mathematicians. The subject matter of this thesis, being exclusively concerned with polynomial approximations to a single-valued, function of one real variable, is on the side of 'applied' side of approximation theory. The first chapter lists the definitions and theorems required subsequently. Chapter is devoted to estimates for the maximum error in minimax polynomial approximations. Extensions of this are used to obtain crude error estimates for cubic spline approximations. The following chapter extends the minimax results to deal also with best L[sub]p polynomial approximations, which include beat least squares (L₂) and best modulus of integral (L₁) approximations as special cases. Chapter 4 is different in character. It is on the practical problem of approximating to convex or nearly convex data.
1969-01-01T00:00:00ZPhillips, G. M. (George McArtney)The fundamental theorem, as far as this work is concerned, is Weierstrass' theorem (1885) on the approximability of continuous functions by polynomials. Since the time of Weierstrass (1815-97) and his equally important contemporary Chebyshev (1821-94), the topic of approximation has grown enormously into a subject of considerable interest to both pure and applied mathematicians. The subject matter of this thesis, being exclusively concerned with polynomial approximations to a single-valued, function of one real variable, is on the side of 'applied' side of approximation theory. The first chapter lists the definitions and theorems required subsequently. Chapter is devoted to estimates for the maximum error in minimax polynomial approximations. Extensions of this are used to obtain crude error estimates for cubic spline approximations. The following chapter extends the minimax results to deal also with best L[sub]p polynomial approximations, which include beat least squares (L₂) and best modulus of integral (L₁) approximations as special cases. Chapter 4 is different in character. It is on the practical problem of approximating to convex or nearly convex data.Some consequences of symmetry in strong Stieltjes distributionsBracciali, Cleonice Fátima Braccialihttp://hdl.handle.net/10023/138812019-04-01T08:39:33Z1998-01-01T00:00:00ZThe main purpose of this work is to study a class of strong Stieltjes distributions 𝜓(t), defined on an interval (a, b) ⊆ (0, ∞), where 0 < 𝛽 < b ≤ ∞ and a = 𝛽²/b which satisfy the symmetric property
(dψ(t))/t[super]ω=-(dψ(β^2/t))/((β^2/t)[super]ω), tε (a,b), 2ωε𝓩
We investigate the consequences of this symmetric property on the orthogonal L-polynomials related to distributions ψ(t)and which are the denominators of the two-point Pade approximants for the power series that arise in the moment problem. We examine relations involving the coefficients of the continued fractions that correspond to these power series. We also study the consequences of the symmetry on the associated quadrature formulae.
1998-01-01T00:00:00ZBracciali, Cleonice Fátima BraccialiThe main purpose of this work is to study a class of strong Stieltjes distributions 𝜓(t), defined on an interval (a, b) ⊆ (0, ∞), where 0 < 𝛽 < b ≤ ∞ and a = 𝛽²/b which satisfy the symmetric property
(dψ(t))/t[super]ω=-(dψ(β^2/t))/((β^2/t)[super]ω), tε (a,b), 2ωε𝓩
We investigate the consequences of this symmetric property on the orthogonal L-polynomials related to distributions ψ(t)and which are the denominators of the two-point Pade approximants for the power series that arise in the moment problem. We examine relations involving the coefficients of the continued fractions that correspond to these power series. We also study the consequences of the symmetry on the associated quadrature formulae.Some aspects of the Jacobian conjecture: (the geometry of automorphisms of ℂ²)Ali, A. Hamid A. Hussainhttp://hdl.handle.net/10023/138782019-04-01T08:38:06Z1987-01-01T00:00:00ZWe consider the affine varieties which arise by considering invertible polynomial maps from ℂ² to itself of less than or equal to a given-degree. These varieties arise naturally in the investigation of the long-standing Jacobian Conjecture. We start with some calculations in the lower degree cases. These calculations provide a proof of the Jacobian conjecture in these cases and suggest how the investigation in the higher degree cases should proceed. We then show how invertible polynomial maps can be decomposed as products of what we call triangular maps and we are able to prove a uniqueness result which gives a stronger version of Jung's theorem [j] which is one of the most important results in this area. Our proof also gives a new derivation of Jung's theorem from Segre's lemma. We give a different decomposition of an invertible polynomial map as a composition of "irreducible maps" and we are able to write down standard forms for these irreducibles. We use these standard forms to give a description of the structure of the varieties of invertible maps. We consider some interesting group actions on our varieties and show how these fit in with the structure we describe. Finally, we look at the problem of identifying polynomial maps of finite order. Our description of the structure of the above varieties allows us to solve this problem completely and we are able to show that the only elements of finite order are those which arise from conjugating linear elements of finite order.
1987-01-01T00:00:00ZAli, A. Hamid A. HussainWe consider the affine varieties which arise by considering invertible polynomial maps from ℂ² to itself of less than or equal to a given-degree. These varieties arise naturally in the investigation of the long-standing Jacobian Conjecture. We start with some calculations in the lower degree cases. These calculations provide a proof of the Jacobian conjecture in these cases and suggest how the investigation in the higher degree cases should proceed. We then show how invertible polynomial maps can be decomposed as products of what we call triangular maps and we are able to prove a uniqueness result which gives a stronger version of Jung's theorem [j] which is one of the most important results in this area. Our proof also gives a new derivation of Jung's theorem from Segre's lemma. We give a different decomposition of an invertible polynomial map as a composition of "irreducible maps" and we are able to write down standard forms for these irreducibles. We use these standard forms to give a description of the structure of the varieties of invertible maps. We consider some interesting group actions on our varieties and show how these fit in with the structure we describe. Finally, we look at the problem of identifying polynomial maps of finite order. Our description of the structure of the above varieties allows us to solve this problem completely and we are able to show that the only elements of finite order are those which arise from conjugating linear elements of finite order.On Riesz summabilityShawyer, Brucehttp://hdl.handle.net/10023/138252019-04-01T08:38:34Z1963-01-01T00:00:00ZThe thesis is divided into four chapters. The first contains notation and fundamental results. The others contain a number of theorems on Riesz summability, ordinary in the second, absolute in the third and strong in the fourth. The substance of chapter II has appeared in the Proceedings of the Glasgow Mathermatical Association [2].
1963-01-01T00:00:00ZShawyer, BruceThe thesis is divided into four chapters. The first contains notation and fundamental results. The others contain a number of theorems on Riesz summability, ordinary in the second, absolute in the third and strong in the fourth. The substance of chapter II has appeared in the Proceedings of the Glasgow Mathermatical Association [2].Global optimization using interval arithmeticMohd, Ismail Binhttp://hdl.handle.net/10023/138242019-04-01T08:37:07Z1987-01-01T00:00:00ZThis thesis contains a description of algorithm, MW, for bounding the global minimizers and globally minimum value of a twice continuously differentiable function f :Rⁿ → R¹ R1 in a compact sub-interval of Rⁿ. The algorithm MW is similar to the algorithm of Hansen (Han-80a] in that interval arithmetic is used together with certain of Hansen's ideas, but is different from Hansen's algorithm in that MW bounds the Kuhn Tucker points corresponding to the global minimizers of f in the given sab-interval. The Kuhn Tucker points are bounded with prescribed precision by using either of the algorithms KMSW [SheW-85c] or MAP [SheW-85b]. Numerical results which are obtained from Triplex [BaCM-82a] [MorC-83a] implementations of H and MW axe presented.
1987-01-01T00:00:00ZMohd, Ismail BinThis thesis contains a description of algorithm, MW, for bounding the global minimizers and globally minimum value of a twice continuously differentiable function f :Rⁿ → R¹ R1 in a compact sub-interval of Rⁿ. The algorithm MW is similar to the algorithm of Hansen (Han-80a] in that interval arithmetic is used together with certain of Hansen's ideas, but is different from Hansen's algorithm in that MW bounds the Kuhn Tucker points corresponding to the global minimizers of f in the given sab-interval. The Kuhn Tucker points are bounded with prescribed precision by using either of the algorithms KMSW [SheW-85c] or MAP [SheW-85b]. Numerical results which are obtained from Triplex [BaCM-82a] [MorC-83a] implementations of H and MW axe presented.Modifications of some algorithms for unconstrained optimizationMirnia-Harikandi, Khttp://hdl.handle.net/10023/138222019-04-01T08:37:04Z1979-01-01T00:00:00ZThis thesis contains an account of several modifications to two algorithms for unconstrained optimization, both of which are due to Gill and Murray. Chapter One contains a brief survey of unconstrained optimization and contains also some results which are used subsequently. Chapter Two contains an account of some work on iterative procedures for the solution of operator equations in Banach spaces due to Wolfe (1978a) in which it is suggested that it may be possible, in certain circumstances, to use high-order iterative procedures rather than Newton's method, thereby obtaining computational advantages. In Chapter Three the Newton-type algorithm of Gill and Murray (1974) is described and the ideas contained in Chapter Two are used to construct some modifications of this algorithm. Chapter Four contains some algorithms for the numerical estimation of both full and b and-type Hessian matrices. These algorithms may be used in conjunction with the optimization algorithms which are described in Chapters Three and Five. In Chapter Five the least-squares algorithm of Gill and Murray (1976) is described and the ideas contained in Chapter Two are used to construct some modifications of this algorithm. Chapter Six contains the computational results which were obtained by using the algorithms which are described in Chapters Three, Four and Five to solve the test problems which are listed in Appendices One and Two.
1979-01-01T00:00:00ZMirnia-Harikandi, KThis thesis contains an account of several modifications to two algorithms for unconstrained optimization, both of which are due to Gill and Murray. Chapter One contains a brief survey of unconstrained optimization and contains also some results which are used subsequently. Chapter Two contains an account of some work on iterative procedures for the solution of operator equations in Banach spaces due to Wolfe (1978a) in which it is suggested that it may be possible, in certain circumstances, to use high-order iterative procedures rather than Newton's method, thereby obtaining computational advantages. In Chapter Three the Newton-type algorithm of Gill and Murray (1974) is described and the ideas contained in Chapter Two are used to construct some modifications of this algorithm. Chapter Four contains some algorithms for the numerical estimation of both full and b and-type Hessian matrices. These algorithms may be used in conjunction with the optimization algorithms which are described in Chapters Three and Five. In Chapter Five the least-squares algorithm of Gill and Murray (1976) is described and the ideas contained in Chapter Two are used to construct some modifications of this algorithm. Chapter Six contains the computational results which were obtained by using the algorithms which are described in Chapters Three, Four and Five to solve the test problems which are listed in Appendices One and Two.Formal methods for deriving Green-type transitional and uniform asymptotic expansions from differential equationsJorna, Siebehttp://hdl.handle.net/10023/138212019-04-01T08:38:51Z1965-01-01T00:00:00ZIn the present work, we develop and illustrate powerful, but straightforward, formal methods for deriving asymptotic expansions from differential equations. In the second chapter, the ‘inverse Frobenius method’ for deriving Stokes expansions is exemplified. The main body of this thesis, however, consists of the development of the new Green-Liouville-Melin transform method, and its detailed application to modified Bessel functions, parabolic cylinder functions, Whittaker functions, Poiseuille functions, confluent hypergeometric functions, and also to periodic Mathieu functions and oblate spheroidal wave functions, all with at least one parameter large⁺. The wide scope of the method is evinced by the fact that treatment of the essentially eigenvalue problem posed by the two last-named cases requires no additional techniques. This method, as will be explained in detail in chapter 3, yields Green-type, transitional and uniform expansions.
The transitional expansions found in this way are usually of a simpler form than those derived by alternative processes (e.g. perturbation theory). To state an example, the asymptotic expansions for the periodic Mathieu functions ce(z,h) and se(z,h) valid near |z| = 1/2π that have been obtained in earlier work contain the complicated parabolic cylinder functions (c.f. Meixner 1948, Sips 1949, Dingle and Müller 1962). By contrast, our methods yield expansions of comparable applicability, but involving only elementary functions. To demonstrate their usefulness, we have fed these expansions into a digital computer and obtained extensive tables for ce(z,h) and se(z,h) in the range 50°≤ z ≤90° . Extracts from these tables and comparisons with correct results are given in §8.71.
Following the chapters on the introduction and applications of the Mellin transform technique, there is some preliminary work on a new method for determining the general term in Green-type expansions. The method is illustrated by detailed calculations for modified Bessel and parabolic cylinder functions.
In the final chapter, we present certain suggestions for further work.
1965-01-01T00:00:00ZJorna, SiebeIn the present work, we develop and illustrate powerful, but straightforward, formal methods for deriving asymptotic expansions from differential equations. In the second chapter, the ‘inverse Frobenius method’ for deriving Stokes expansions is exemplified. The main body of this thesis, however, consists of the development of the new Green-Liouville-Melin transform method, and its detailed application to modified Bessel functions, parabolic cylinder functions, Whittaker functions, Poiseuille functions, confluent hypergeometric functions, and also to periodic Mathieu functions and oblate spheroidal wave functions, all with at least one parameter large⁺. The wide scope of the method is evinced by the fact that treatment of the essentially eigenvalue problem posed by the two last-named cases requires no additional techniques. This method, as will be explained in detail in chapter 3, yields Green-type, transitional and uniform expansions.
The transitional expansions found in this way are usually of a simpler form than those derived by alternative processes (e.g. perturbation theory). To state an example, the asymptotic expansions for the periodic Mathieu functions ce(z,h) and se(z,h) valid near |z| = 1/2π that have been obtained in earlier work contain the complicated parabolic cylinder functions (c.f. Meixner 1948, Sips 1949, Dingle and Müller 1962). By contrast, our methods yield expansions of comparable applicability, but involving only elementary functions. To demonstrate their usefulness, we have fed these expansions into a digital computer and obtained extensive tables for ce(z,h) and se(z,h) in the range 50°≤ z ≤90° . Extracts from these tables and comparisons with correct results are given in §8.71.
Following the chapters on the introduction and applications of the Mellin transform technique, there is some preliminary work on a new method for determining the general term in Green-type expansions. The method is illustrated by detailed calculations for modified Bessel and parabolic cylinder functions.
In the final chapter, we present certain suggestions for further work.Presentations of linear groupsWilliams, Peter D.http://hdl.handle.net/10023/138142019-04-01T08:39:38Z1983-01-01T00:00:00ZIf d(M) denotes the rank of the Schur multiplicator of a finite group G, then a group is efficient if -def G = d(M). Efficient presentations of the simple groups PSL(2,p), p an odd prime > 3, were obtained by J.G. Sunday.This raised the question of whether or not all finite simple groups are efficient.
In this thesis, we investigate the deficiency of the groups PSL(2,pⁿ). J.A. Todd gave presentations for PSL(2,pⁿ) which use large numbers of generators and relations. Starting with these, we obtain, at best, deficiency -1 presentations for PSL(2,2ⁿ) (= SL(2,2ⁿ)) and deficiency -6 presentations for PSL(2,pⁿ), p an odd prime. If pⁿ = -1(mod 4), the latter can be reduced to a deficiency -4 presentation. Efficient presentations for PSL(2,25), PSL(2,27) and PSL(2,49) are obtained.
The Behr-Mennicke presentation for PSL(2,p) is one of the most fundamental in the sense that it forms the basis for others, such as those given by Sunday, Zassenhaus and Sidki. Behr and Mennicke derived their presentation indirectly, and it would be desirable to have a more direct proof. The groups G[sub]p(a) are defined as
< U, R, S | U³ = (UR)² = (US)² = Sᵖ = Rᵗ = (SaRU)³= 1, Sᵃ²R = RS >
where a ε GF(p)* and a²ᵗ = 1 (mod p) . We show that G[sub]p (2) is isomorphic with the Behr-Mennicke presentation for PSL(2,p), p > 3. Conditions are found to discover when Gp (a) is isomorphic with PSL(2,p) and, under these conditions, this provides a direct proof of the Behr-Mennicke presentations. For any odd positive integer m, we show that the groups SL(2,ℤ (m)) and PSL(2,ℤ(m)) are efficient.
1983-01-01T00:00:00ZWilliams, Peter D.If d(M) denotes the rank of the Schur multiplicator of a finite group G, then a group is efficient if -def G = d(M). Efficient presentations of the simple groups PSL(2,p), p an odd prime > 3, were obtained by J.G. Sunday.This raised the question of whether or not all finite simple groups are efficient.
In this thesis, we investigate the deficiency of the groups PSL(2,pⁿ). J.A. Todd gave presentations for PSL(2,pⁿ) which use large numbers of generators and relations. Starting with these, we obtain, at best, deficiency -1 presentations for PSL(2,2ⁿ) (= SL(2,2ⁿ)) and deficiency -6 presentations for PSL(2,pⁿ), p an odd prime. If pⁿ = -1(mod 4), the latter can be reduced to a deficiency -4 presentation. Efficient presentations for PSL(2,25), PSL(2,27) and PSL(2,49) are obtained.
The Behr-Mennicke presentation for PSL(2,p) is one of the most fundamental in the sense that it forms the basis for others, such as those given by Sunday, Zassenhaus and Sidki. Behr and Mennicke derived their presentation indirectly, and it would be desirable to have a more direct proof. The groups G[sub]p(a) are defined as
< U, R, S | U³ = (UR)² = (US)² = Sᵖ = Rᵗ = (SaRU)³= 1, Sᵃ²R = RS >
where a ε GF(p)* and a²ᵗ = 1 (mod p) . We show that G[sub]p (2) is isomorphic with the Behr-Mennicke presentation for PSL(2,p), p > 3. Conditions are found to discover when Gp (a) is isomorphic with PSL(2,p) and, under these conditions, this provides a direct proof of the Behr-Mennicke presentations. For any odd positive integer m, we show that the groups SL(2,ℤ (m)) and PSL(2,ℤ(m)) are efficient.A study of the infinite dimensional linear and symplectic groupsArrell, David G.http://hdl.handle.net/10023/138102019-04-01T08:39:29Z1979-01-01T00:00:00ZBy a linear group we shall mean essentially a group of invertible matrices over a ring. Thus, we include in our class of linear groups the 'classical' geometric groups. These are the general linear group, GL[sub]n(F), the orthogonal groups, 0[sub]n (F) and the syraplectic groups Sp[sub]n(F). The normal and subnormal subgroup structure of these groups is well known and has been the subject of much investigation since the turn of the century. We study here the normal and subnormal structure of some of their infinite dimensional counterparts, namely, the infinite dimensional linear group GL(Ω,R), for arbitrary rings R, and the infinite dimensional syraplectic group Sp(Ω,R), for commutative rings R with identity. We shall see that a key role in the classification of the normal and subnormal subgroups of GL(Ω,R) and Sp(Ω,R) is played by the 'elementary' normal subgroups E(Ω,R) and ESp(Ω,R). We shall also see that, in the case of the infinite dimensional linear group, the normal subgroup structure depends very much upon the way in which R is generated as a right R-module. We shall also give a presentation for the 'elementary' subgroup E(Ω,R) when R is a division ring.
1979-01-01T00:00:00ZArrell, David G.By a linear group we shall mean essentially a group of invertible matrices over a ring. Thus, we include in our class of linear groups the 'classical' geometric groups. These are the general linear group, GL[sub]n(F), the orthogonal groups, 0[sub]n (F) and the syraplectic groups Sp[sub]n(F). The normal and subnormal subgroup structure of these groups is well known and has been the subject of much investigation since the turn of the century. We study here the normal and subnormal structure of some of their infinite dimensional counterparts, namely, the infinite dimensional linear group GL(Ω,R), for arbitrary rings R, and the infinite dimensional syraplectic group Sp(Ω,R), for commutative rings R with identity. We shall see that a key role in the classification of the normal and subnormal subgroups of GL(Ω,R) and Sp(Ω,R) is played by the 'elementary' normal subgroups E(Ω,R) and ESp(Ω,R). We shall also see that, in the case of the infinite dimensional linear group, the normal subgroup structure depends very much upon the way in which R is generated as a right R-module. We shall also give a presentation for the 'elementary' subgroup E(Ω,R) when R is a division ring.A variable input boundary problem in contaminant transportWarner, G. C.http://hdl.handle.net/10023/138012019-04-01T08:37:13Z1997-01-01T00:00:00ZThis thesis considers the large-time behaviour of the equation
(∂(u+uᵖ) )/( ∂t ) + Q(t) ∂u/( ∂x) = ∂²u/∂x² p>0, r≥ -1
With 0 ≤ 𝓍 < ∞, t ≥ 0 and Q (t) ~ tʳ, t ∞. This equation models, after suitable scalings are introduced, the one-dimensional flow of a solute through a porous medium with the solute undergoing adsorption by the solid matrix. We consider two models for the contaminant input at 𝓍= 0, the first being continuous input and the second being an initial pulse of contaminant which terminates after a finite time. Thus the total mass of the solute both adsorbed and in solution is considered to be dependent on time. It is found that the asymptotic solution depends crucially on both p and r. In finding the asymptotic solution, a similarity variable is introduced which for p ≥ 1 may involve spatial translation. We also have that when p < 1 interfaces appear and hence we have bounded support, whilst for p≥1 we do not. The principal role of r is to determine the balance between diffusion and convection effects. In the continuous input case this balance is independent of p, whilst in the pulse problem p is also involved in determining the balance.
1997-01-01T00:00:00ZWarner, G. C.This thesis considers the large-time behaviour of the equation
(∂(u+uᵖ) )/( ∂t ) + Q(t) ∂u/( ∂x) = ∂²u/∂x² p>0, r≥ -1
With 0 ≤ 𝓍 < ∞, t ≥ 0 and Q (t) ~ tʳ, t ∞. This equation models, after suitable scalings are introduced, the one-dimensional flow of a solute through a porous medium with the solute undergoing adsorption by the solid matrix. We consider two models for the contaminant input at 𝓍= 0, the first being continuous input and the second being an initial pulse of contaminant which terminates after a finite time. Thus the total mass of the solute both adsorbed and in solution is considered to be dependent on time. It is found that the asymptotic solution depends crucially on both p and r. In finding the asymptotic solution, a similarity variable is introduced which for p ≥ 1 may involve spatial translation. We also have that when p < 1 interfaces appear and hence we have bounded support, whilst for p≥1 we do not. The principal role of r is to determine the balance between diffusion and convection effects. In the continuous input case this balance is independent of p, whilst in the pulse problem p is also involved in determining the balance.The prediction of thermal phase-change boundaries and associated temperature distributionsWood, A. S.http://hdl.handle.net/10023/137972019-11-27T17:04:04Z1984-01-01T00:00:00ZThe past three decades have seen a fast expanding interest in thermal problems exhibiting a change of phase, more commonly known as Stefan problems. With the rapid advance in computer technology the use and expansion of numerical simulation schemes has been responsible for large advances in this field. The increasing size of computers has led to more sophisticated and complex numerical solutions becoming feasible from a computational point of view. On the other hand, part of this interest has developed from industrial quarters where a knowledge of the location of a melting/freezing boundary may be of critical importance for certain processes. Much experimental work has been completed in this field. However, it is still useful to be able to obtain quick, accurate numerical solutions to such problems and it is with this in mind that this thesis is presented. Ideas from both of the above areas of interest are treated. In the first case a simple to program and computationally efficient numerical scheme is proposed for solving one dimensional Stefan problems and its merits are discussed in relation to several of the more common existing solution schemes. This scheme is then modified to cater for a two dimensional problem which crudely imitates a possible heating configuration in some industrial processes. The problem, with its attendant difficulties, is first approximated by a 'test' problem which is constructed so as to admit an analytic solution. This allows assessment of the numerical procedure in two dimensions. In the course of this work a pseudo-analytic solution was obtained for the original two dimensional problem. Finally, in collaboration with the British Gas Corporation, a complex industrial freezing problem is discussed concerning the flow of liquid through an enclosed channel. Some simplifying assumptions are proposed to reduce the problem to a form for which a relatively simple numerical scheme may be adopted. Several simulations are completed to examine the effect of varying physical parameters on the solution and in particular to test for situations of blockage or steady-state.
1984-01-01T00:00:00ZWood, A. S.The past three decades have seen a fast expanding interest in thermal problems exhibiting a change of phase, more commonly known as Stefan problems. With the rapid advance in computer technology the use and expansion of numerical simulation schemes has been responsible for large advances in this field. The increasing size of computers has led to more sophisticated and complex numerical solutions becoming feasible from a computational point of view. On the other hand, part of this interest has developed from industrial quarters where a knowledge of the location of a melting/freezing boundary may be of critical importance for certain processes. Much experimental work has been completed in this field. However, it is still useful to be able to obtain quick, accurate numerical solutions to such problems and it is with this in mind that this thesis is presented. Ideas from both of the above areas of interest are treated. In the first case a simple to program and computationally efficient numerical scheme is proposed for solving one dimensional Stefan problems and its merits are discussed in relation to several of the more common existing solution schemes. This scheme is then modified to cater for a two dimensional problem which crudely imitates a possible heating configuration in some industrial processes. The problem, with its attendant difficulties, is first approximated by a 'test' problem which is constructed so as to admit an analytic solution. This allows assessment of the numerical procedure in two dimensions. In the course of this work a pseudo-analytic solution was obtained for the original two dimensional problem. Finally, in collaboration with the British Gas Corporation, a complex industrial freezing problem is discussed concerning the flow of liquid through an enclosed channel. Some simplifying assumptions are proposed to reduce the problem to a form for which a relatively simple numerical scheme may be adopted. Several simulations are completed to examine the effect of varying physical parameters on the solution and in particular to test for situations of blockage or steady-state.Compressible boundary layers with sharp pressure gradientsReader-Harris, Michael Johnhttp://hdl.handle.net/10023/137952019-04-01T08:37:51Z1981-01-01T00:00:00ZThe work of this thesis was undertaken as a C.A.S.E. award project in collaboration with Rolls-Royce to examine compressible laminar boundary layers with sharp adverse pressure-gradients. Much of the work is devoted to the solution of two important particular problems. The first flow considered is that along a semi-infinite flat plate with uniform pressure when X < X₀ and with the pressure for X > X₀ being so chosen that the boundary layer is just on the point of separation for all X > X₀. Immediately downstream of X₀ there is a sharp pressure rise to which the flow reacts mainly in a thin inner sublayer; so inner and outer asymptotic expansions are derived and matched for the stream function and a function of the temperature. Throughout the thesis the ratio of the viscosity to the absolute temperature is taken to be a function of x, the distance along the wall, alone, and the Illingworth-Stewartson transformation is applied. The Prandtl number, σ, is taken to be of order unity and detailed results are presented for σ= 1 and 0.72. The second flow considered is that along a finite flat plate where the transformed external velocity U₁(X) is chosen such that
U₁(X) = u₀(-X/L)[super]ε, where O< ε <<1,
is the transformed length of the plate and X represents transformed distance downstream from the trailing edge. The skin friction, position of separation and heat transfer right up to separation are determined. On the basis of these two solutions, another solution which is not presented in detail, and a solution (due to Curie) to a fourth sharp pressure gradient problem, a general Stratford-type method for computing compressible boundary layers is derived, which may be used to predict the position of separation, skin friction, heat transfer, displacement and momentum thicknesses for a compressible boundary layer with an unfavourable pressure gradient. In all this work techniques of series analysis are used to good effect. This led us to look at another boundary-layer problem in which such techniques could be used, one in which two parallel infinite disks are initially rotating with angular velocity Ω about a common axis in incompressible fluid, the appropriate Reynolds number being very large. Suddenly the angular velocity of one of the disks is reversed. A new examination of this problem is presented in the appendix to the thesis.
1981-01-01T00:00:00ZReader-Harris, Michael JohnThe work of this thesis was undertaken as a C.A.S.E. award project in collaboration with Rolls-Royce to examine compressible laminar boundary layers with sharp adverse pressure-gradients. Much of the work is devoted to the solution of two important particular problems. The first flow considered is that along a semi-infinite flat plate with uniform pressure when X < X₀ and with the pressure for X > X₀ being so chosen that the boundary layer is just on the point of separation for all X > X₀. Immediately downstream of X₀ there is a sharp pressure rise to which the flow reacts mainly in a thin inner sublayer; so inner and outer asymptotic expansions are derived and matched for the stream function and a function of the temperature. Throughout the thesis the ratio of the viscosity to the absolute temperature is taken to be a function of x, the distance along the wall, alone, and the Illingworth-Stewartson transformation is applied. The Prandtl number, σ, is taken to be of order unity and detailed results are presented for σ= 1 and 0.72. The second flow considered is that along a finite flat plate where the transformed external velocity U₁(X) is chosen such that
U₁(X) = u₀(-X/L)[super]ε, where O< ε <<1,
is the transformed length of the plate and X represents transformed distance downstream from the trailing edge. The skin friction, position of separation and heat transfer right up to separation are determined. On the basis of these two solutions, another solution which is not presented in detail, and a solution (due to Curie) to a fourth sharp pressure gradient problem, a general Stratford-type method for computing compressible boundary layers is derived, which may be used to predict the position of separation, skin friction, heat transfer, displacement and momentum thicknesses for a compressible boundary layer with an unfavourable pressure gradient. In all this work techniques of series analysis are used to good effect. This led us to look at another boundary-layer problem in which such techniques could be used, one in which two parallel infinite disks are initially rotating with angular velocity Ω about a common axis in incompressible fluid, the appropriate Reynolds number being very large. Suddenly the angular velocity of one of the disks is reversed. A new examination of this problem is presented in the appendix to the thesis.The numerical solution of boundary value problems in partial differential equationsKeast, Patrickhttp://hdl.handle.net/10023/137932019-04-01T08:38:55Z1967-01-01T00:00:00Z1967-01-01T00:00:00ZKeast, PatrickOn the fast and accurate computer solution of partial differential systemsHill, Michael T.http://hdl.handle.net/10023/137912019-04-01T08:37:26Z1974-01-01T00:00:00ZTwo methods are presented for use on an electronic computer for the solution of partial differential systems. The first is concerned with accurate solutions of differential equations. It is equally applicable to ordinary differential equations and partial differential equations, and can be used for parabolic, hyperbolic or elliptic systems, and also for non-linear and mixed systems. It can be used in conjunction with existing schemes. Conversely, the method can be used as a very fast method of obtaining a rough solution of the system. It has an additional advantage over traditional higher order methods in that it does not require extra boundary conditions. The second method is concerned with the acceleration of the convergence rate in the solution of hyperbolic systems. The number of iterations has been reduced from tens of thousands with the traditional Lax-Wendroff methods to the order of twenty iterations. Analyses for both the differential and the difference systems are presented. Again the method is easily added to existing programs. The two methods may be used together to give one fast and accurate method.
1974-01-01T00:00:00ZHill, Michael T.Two methods are presented for use on an electronic computer for the solution of partial differential systems. The first is concerned with accurate solutions of differential equations. It is equally applicable to ordinary differential equations and partial differential equations, and can be used for parabolic, hyperbolic or elliptic systems, and also for non-linear and mixed systems. It can be used in conjunction with existing schemes. Conversely, the method can be used as a very fast method of obtaining a rough solution of the system. It has an additional advantage over traditional higher order methods in that it does not require extra boundary conditions. The second method is concerned with the acceleration of the convergence rate in the solution of hyperbolic systems. The number of iterations has been reduced from tens of thousands with the traditional Lax-Wendroff methods to the order of twenty iterations. Analyses for both the differential and the difference systems are presented. Again the method is easily added to existing programs. The two methods may be used together to give one fast and accurate method.Alternating direction methods for hyperbolic systemsGourlay, A. R.http://hdl.handle.net/10023/137882019-04-01T08:37:57Z1966-01-01T00:00:00Z1966-01-01T00:00:00ZGourlay, A. R.Alternating direction implicit methods for partial differential equationsFairweather, Graemehttp://hdl.handle.net/10023/137842019-04-01T08:39:31Z1966-01-01T00:00:00Z1966-01-01T00:00:00ZFairweather, GraemeThe use of non-polynomial interpolants in the numerical solution of ordinary differential equationsShaw, Brianhttp://hdl.handle.net/10023/137832019-04-01T08:38:31Z1966-01-01T00:00:00Z1966-01-01T00:00:00ZShaw, BrianFinite difference methods for non-linear hyperbolic systemsMorris, John Ll.http://hdl.handle.net/10023/137822020-02-20T11:58:55Z1968-01-01T00:00:00Z1968-01-01T00:00:00ZMorris, John Ll.Inner product quadrature formulasGribble, Julian de Gruchyhttp://hdl.handle.net/10023/137802019-04-01T08:38:01Z1979-01-01T00:00:00ZWe investigate an approach to approximating the integral
(0.1) ⨍[sub]R w(x)f(x)g(x)dx ≡ I (f;g),
where R is a region in one-dimensional Euclidean space, and w a weight function. Since (0.1) may be regarded as a continuous bi-linear functional in f and g we approximate it by a discrete bi-linear functional, which we term an Inner Product Quadrature Formula (I.P.Q.F.).
(0.2) Q(f;g) ≡ f̲ᵀAg̲,
Where f̲ᵀ = (Sₒ(f), . . . , s[sub]m(f))ᵀ
g̲ᵀ = (Tₒ(g), . . . , T[sub]n(g)) ᵀ
A = (aᵢ[sub]j)ᵐi=o,ⁿj=0,
And a[sub]i[sub]j are real numbers, 𝛴 ᵐi=0 𝛴ⁿj =0 |aᵢ [sub]j | > 0
The so-called elementary functionals {Sᵢ}ᵐi=0 and {T[sub]j}ⁿj=0 are two sets of linearly independent linear functionals, acting f and g respectively, defined over a certain subspace of functions to which f and g belong. The simplest example of these functionals is function evaluation at a given point.
The matrix A is determined by requiring (0.2) to be exact for certain classes of functions f and g, say
F𝜀𝛷𝛾 ≡ {𝛷₀, . . . , 𝛷ᵧ}, 𝛾≥0
G𝜀𝛹𝛿 ≡ {𝛹₀, . . . , 𝛹[sub] 𝛿} 𝛿≥0
In Chapter 1 we introduce the concept of I.P.Q.F. in more detail and make some general comments about approaches available when examining numerical integration. After explaining in some detail why we feel I.P.Q.F. are a useful tool in §2.1, we proceed in the remainder of Chapter 2 to investigate various conditions which may be placed on 𝛷ᵞ, 𝛹[super] 𝛿 {Sᵢ}ᵐi=0 and {T[sub]j}ⁿj=0 in order to guarantee the existence of I.P.Q.F. exact when F𝜀𝛷𝛾 and G𝜀𝛹𝛿.
In particular we investigate the question of maximizing 𝛾+ 𝛿. In the case where 𝛷ᵢ and 𝛹[sub]j are the standard monomials of degree i and j respectively, some results have already been published in B.I.T. (1977) p. 392-408. We investigate various choices of 𝛷ᵢ and 𝛹[sub]j :
(a) {𝛷ᵢ}ᵐ⁺¹ I = 0 (i.e. 𝛾 = m+1) and {𝛹[sub]j}ᵐ[sub]j = 0 (i.e. 𝛿 = m) being Tchebychev sets (§2.7),
(b) {𝛷ᵢ}²ᵐ⁺¹ I = 0 (i.e. 𝛾 = 2m+1) being a Tchebychev set and 𝛹[super]𝛿 contains only one function (i.e. 𝛿 = 0) (§2.6)
(c) 𝛷ᵢ ≡ (𝛷[sub]l)ⁱ, i=0,1, . . . and 𝛷ᵢ = 𝛹ᵢ, i= 0, 1, … (§2.8).
In Chapter 3 we consider the question of compounding I.P.Q.F. both in the classical sense, and, briefly, by examining spline functions, regarding them as providing a link between an I.P.Q.F on one hand and a compounded I.P.Q.F. on the other. Various methods of theoretically estimating the errors involved are considered in Chapter M-. In the fifth Chapter we examine various ways in which the concept of I.P.Q.F. might (or might not) be extended. Finally, we make some brief comments about the possible applications of I.P.Q.F., and give a few examples.
1979-01-01T00:00:00ZGribble, Julian de GruchyWe investigate an approach to approximating the integral
(0.1) ⨍[sub]R w(x)f(x)g(x)dx ≡ I (f;g),
where R is a region in one-dimensional Euclidean space, and w a weight function. Since (0.1) may be regarded as a continuous bi-linear functional in f and g we approximate it by a discrete bi-linear functional, which we term an Inner Product Quadrature Formula (I.P.Q.F.).
(0.2) Q(f;g) ≡ f̲ᵀAg̲,
Where f̲ᵀ = (Sₒ(f), . . . , s[sub]m(f))ᵀ
g̲ᵀ = (Tₒ(g), . . . , T[sub]n(g)) ᵀ
A = (aᵢ[sub]j)ᵐi=o,ⁿj=0,
And a[sub]i[sub]j are real numbers, 𝛴 ᵐi=0 𝛴ⁿj =0 |aᵢ [sub]j | > 0
The so-called elementary functionals {Sᵢ}ᵐi=0 and {T[sub]j}ⁿj=0 are two sets of linearly independent linear functionals, acting f and g respectively, defined over a certain subspace of functions to which f and g belong. The simplest example of these functionals is function evaluation at a given point.
The matrix A is determined by requiring (0.2) to be exact for certain classes of functions f and g, say
F𝜀𝛷𝛾 ≡ {𝛷₀, . . . , 𝛷ᵧ}, 𝛾≥0
G𝜀𝛹𝛿 ≡ {𝛹₀, . . . , 𝛹[sub] 𝛿} 𝛿≥0
In Chapter 1 we introduce the concept of I.P.Q.F. in more detail and make some general comments about approaches available when examining numerical integration. After explaining in some detail why we feel I.P.Q.F. are a useful tool in §2.1, we proceed in the remainder of Chapter 2 to investigate various conditions which may be placed on 𝛷ᵞ, 𝛹[super] 𝛿 {Sᵢ}ᵐi=0 and {T[sub]j}ⁿj=0 in order to guarantee the existence of I.P.Q.F. exact when F𝜀𝛷𝛾 and G𝜀𝛹𝛿.
In particular we investigate the question of maximizing 𝛾+ 𝛿. In the case where 𝛷ᵢ and 𝛹[sub]j are the standard monomials of degree i and j respectively, some results have already been published in B.I.T. (1977) p. 392-408. We investigate various choices of 𝛷ᵢ and 𝛹[sub]j :
(a) {𝛷ᵢ}ᵐ⁺¹ I = 0 (i.e. 𝛾 = m+1) and {𝛹[sub]j}ᵐ[sub]j = 0 (i.e. 𝛿 = m) being Tchebychev sets (§2.7),
(b) {𝛷ᵢ}²ᵐ⁺¹ I = 0 (i.e. 𝛾 = 2m+1) being a Tchebychev set and 𝛹[super]𝛿 contains only one function (i.e. 𝛿 = 0) (§2.6)
(c) 𝛷ᵢ ≡ (𝛷[sub]l)ⁱ, i=0,1, . . . and 𝛷ᵢ = 𝛹ᵢ, i= 0, 1, … (§2.8).
In Chapter 3 we consider the question of compounding I.P.Q.F. both in the classical sense, and, briefly, by examining spline functions, regarding them as providing a link between an I.P.Q.F on one hand and a compounded I.P.Q.F. on the other. Various methods of theoretically estimating the errors involved are considered in Chapter M-. In the fifth Chapter we examine various ways in which the concept of I.P.Q.F. might (or might not) be extended. Finally, we make some brief comments about the possible applications of I.P.Q.F., and give a few examples.Interval methods for non-linear systemsShearer, J. M.http://hdl.handle.net/10023/137792019-04-01T08:38:35Z1986-01-01T00:00:00ZIn numerical mathematics, there is a need for methods which provide a user with the solution to his problem without requiring him to understand the mathematics underlying the method of solution. Such a method involves computable tests to determine whether or not a solution exists in a given region, and whether, if it exists, such a solution may be found by using the given method. Two valuable tools for the implementation of such methods are interval mathematics and symbolic computation. In. practice all computers have memories of finite size and cannot perform exact arithmetic. Therefore, in addition to the error which is inherent in a given numerical method, namely truncation error, there is also the error due to rounding. Using interval arithmetic, computable tests which guarantee the existence of a solution to a given problem in a given region, and the convergence of a particular iterative method to this solution, become practically realizable. This is not possible using real arithmetic due to the accumulation of rounding error on a computer. The advent of packages which allow symbolic computations to be carried out on a given computer is an important advance for computational numerical mathematics. In particular, the ability to compute derivatives automatically removes the need for a user to supply them, thus eliminating a major source of error in the use of methods requiring first or higher derivatives. In this thesis some methods which use interval arithmetic and symbolic computation for the solution of systems of nonlinear algebraic equations are presented. Some algorithms based on the symmetric single-step algorithm are described. These methods however do not possess computable existence, uniqueness, and convergence tests. Algorithms which do possess such tests, based on the Krawczyk-Moore algorithm are also presented. A simple package which allows symbolic computations to be carried out is described. Several applications for such a package are given. In particular, an interval form of Brown's method is presented.
1986-01-01T00:00:00ZShearer, J. M.In numerical mathematics, there is a need for methods which provide a user with the solution to his problem without requiring him to understand the mathematics underlying the method of solution. Such a method involves computable tests to determine whether or not a solution exists in a given region, and whether, if it exists, such a solution may be found by using the given method. Two valuable tools for the implementation of such methods are interval mathematics and symbolic computation. In. practice all computers have memories of finite size and cannot perform exact arithmetic. Therefore, in addition to the error which is inherent in a given numerical method, namely truncation error, there is also the error due to rounding. Using interval arithmetic, computable tests which guarantee the existence of a solution to a given problem in a given region, and the convergence of a particular iterative method to this solution, become practically realizable. This is not possible using real arithmetic due to the accumulation of rounding error on a computer. The advent of packages which allow symbolic computations to be carried out on a given computer is an important advance for computational numerical mathematics. In particular, the ability to compute derivatives automatically removes the need for a user to supply them, thus eliminating a major source of error in the use of methods requiring first or higher derivatives. In this thesis some methods which use interval arithmetic and symbolic computation for the solution of systems of nonlinear algebraic equations are presented. Some algorithms based on the symmetric single-step algorithm are described. These methods however do not possess computable existence, uniqueness, and convergence tests. Algorithms which do possess such tests, based on the Krawczyk-Moore algorithm are also presented. A simple package which allows symbolic computations to be carried out is described. Several applications for such a package are given. In particular, an interval form of Brown's method is presented.Statistical problems in measuring surface ozone and modelling its patternsHutchison, Paul Stewarthttp://hdl.handle.net/10023/137732019-04-01T08:37:10Z1996-01-01T00:00:00ZThe thesis examines ground level air pollution data supplied by ITE Bush, Penicuik, Midlothian, Scotland. There is a brief examination of sulphur dioxide concentration data, but the Thesis is primarily concerned with ozone. The diurnal behaviour of ozone is the major topic, and a new methodology of classification of 'ozone days' is introduced and discussed. In chapter 2, the inverse Gaussian distribution is considered and rejected as a possible alternative to the standard approach of using the lognormal as a model for the frequency distribution of observed sulphur dioxide concentrations. In chapter 3, the behaviour of digital gas pollution analysers is investigated by making use of data obtained from two such machines operating side by side. A time series model of the differences between the readings obtained from the two machines is considered, and possible effects on modelling discussed. In chapter 4, the changes in the diurnal behaviour of ozone over a year are examined. A new approach involving a distortion of the time axis is shown to give diurnal ozone curves more homogeneous properties and have beneficial effects for modelling purposes. Chapter 5 extends the analysis of the diurnal behaviour of ozone begun in chapter 4 by considering individual 'ozone days' and attempting to classify them as one of several typical 'types' of day. The time distortion method introduced in chapter 4 is used, and a new classification methodology is introduced for considering data of this type. The statistical properties of this method are discussed in chapter 6.
1996-01-01T00:00:00ZHutchison, Paul StewartThe thesis examines ground level air pollution data supplied by ITE Bush, Penicuik, Midlothian, Scotland. There is a brief examination of sulphur dioxide concentration data, but the Thesis is primarily concerned with ozone. The diurnal behaviour of ozone is the major topic, and a new methodology of classification of 'ozone days' is introduced and discussed. In chapter 2, the inverse Gaussian distribution is considered and rejected as a possible alternative to the standard approach of using the lognormal as a model for the frequency distribution of observed sulphur dioxide concentrations. In chapter 3, the behaviour of digital gas pollution analysers is investigated by making use of data obtained from two such machines operating side by side. A time series model of the differences between the readings obtained from the two machines is considered, and possible effects on modelling discussed. In chapter 4, the changes in the diurnal behaviour of ozone over a year are examined. A new approach involving a distortion of the time axis is shown to give diurnal ozone curves more homogeneous properties and have beneficial effects for modelling purposes. Chapter 5 extends the analysis of the diurnal behaviour of ozone begun in chapter 4 by considering individual 'ozone days' and attempting to classify them as one of several typical 'types' of day. The time distortion method introduced in chapter 4 is used, and a new classification methodology is introduced for considering data of this type. The statistical properties of this method are discussed in chapter 6.A study of character recognition using geometric moments under conditions of simple and non-simple lossTucker, N. D.http://hdl.handle.net/10023/137682019-04-01T08:37:08Z1974-01-01T00:00:00ZThe theory of Loss Functions Is a fundamental part of Statistical Decision Theory and of Pattern Recognition. However It is a subject which few have studied In detail. This thesis is an attempt to develop a simple character recognition process In which losses may be Implemented when and where necessary. After a brief account of the history of Loss Functions and an Introduction to elementary Decision Theory, some examples have been constructed to demonstrate how various decision boundaries approximate to the optimal boundary and what Increase In loss would be associated with these sub-optimal boundaries. The results show that the Euclidean and Hamming distance discriminants can be sufficiently close approximations that the decision process may be legitimately simplified by the use of these linear boundaries. Geometric moments were adopted for the computer simulation of the recognition process because each moment is closely related to the symmetry and structure of a character, unlike many other features. The theory of Moments is discussed, in particular their geometrical properties. A brief description of the programs used in the simulation follows. Two different data sets were investigated, the first being hand-drawn capitals and the second machine-scanned lower case type script. This latter set was in the form of a message, which presented interesting programming problems in itself. The results from the application of different discriminants to these sets under conditions of simple loss are analysed and the recognition efficiencies are found to vary between about 30% and. 99% depending on the number of moments being used and the type of discriminant. Next certain theoretical problems are studied. The relations between the rejection rate, the error rate and the rejection threshold are discussed both theoretically and practically. Also an attempt is made to predict theoretically the variation of efficiency with the number of moments used in the discrimination. This hypothesis is then tested on the data already calculated and shown to be true within reasonable limits. A discussion of moment ordering by defining their re-solving powers is undertaken and it seems likely that the moments normally used unordered are among the most satisfactory. Finally, some time is devoted towards methods of improving recognition efficiency. Information content is discussed along with the possibilities inherent in the use of digraph and trigraph probabilities. A breakdown of the errors in the recognition system adopted here is presented along with suggestions to improve the technique. The execution time of the different decision mechanisms is then inspected and a refined 2-Stage method is produced. Lastly the various methods by which a decision mechanism might be improved are united under a common loss matrix, formed by a product of matrices each of which represents a particular facet of the recognition problem.
1974-01-01T00:00:00ZTucker, N. D.The theory of Loss Functions Is a fundamental part of Statistical Decision Theory and of Pattern Recognition. However It is a subject which few have studied In detail. This thesis is an attempt to develop a simple character recognition process In which losses may be Implemented when and where necessary. After a brief account of the history of Loss Functions and an Introduction to elementary Decision Theory, some examples have been constructed to demonstrate how various decision boundaries approximate to the optimal boundary and what Increase In loss would be associated with these sub-optimal boundaries. The results show that the Euclidean and Hamming distance discriminants can be sufficiently close approximations that the decision process may be legitimately simplified by the use of these linear boundaries. Geometric moments were adopted for the computer simulation of the recognition process because each moment is closely related to the symmetry and structure of a character, unlike many other features. The theory of Moments is discussed, in particular their geometrical properties. A brief description of the programs used in the simulation follows. Two different data sets were investigated, the first being hand-drawn capitals and the second machine-scanned lower case type script. This latter set was in the form of a message, which presented interesting programming problems in itself. The results from the application of different discriminants to these sets under conditions of simple loss are analysed and the recognition efficiencies are found to vary between about 30% and. 99% depending on the number of moments being used and the type of discriminant. Next certain theoretical problems are studied. The relations between the rejection rate, the error rate and the rejection threshold are discussed both theoretically and practically. Also an attempt is made to predict theoretically the variation of efficiency with the number of moments used in the discrimination. This hypothesis is then tested on the data already calculated and shown to be true within reasonable limits. A discussion of moment ordering by defining their re-solving powers is undertaken and it seems likely that the moments normally used unordered are among the most satisfactory. Finally, some time is devoted towards methods of improving recognition efficiency. Information content is discussed along with the possibilities inherent in the use of digraph and trigraph probabilities. A breakdown of the errors in the recognition system adopted here is presented along with suggestions to improve the technique. The execution time of the different decision mechanisms is then inspected and a refined 2-Stage method is produced. Lastly the various methods by which a decision mechanism might be improved are united under a common loss matrix, formed by a product of matrices each of which represents a particular facet of the recognition problem.A study of the work and methods of Henry Briggs, with special reference to the early history of interpolationWaterson, Andrewhttp://hdl.handle.net/10023/137602019-04-01T08:38:37Z1941-01-01T00:00:00Z1941-01-01T00:00:00ZWaterson, AndrewTransformations in regression, estimation, testing and modellingParker, Imeldahttp://hdl.handle.net/10023/137592019-04-01T08:38:15Z1988-01-01T00:00:00ZTransformation is a powerful tool for model building. In regression the response variable is transformed in order to achieve the usual assumptions of normality, constant variance and additivity of effects. Here the normality assumption is replaced by the Laplace distributional assumption, appropriate when more large errors occur than would be expected if the errors were normally distributed. The parametric model is enlarged to include a transformation parameter and a likelihood procedure is adopted for estimating this parameter simultaneously with other parameters of interest. Diagnostic methods are described for assessing the influence of individual observations on the choice of transformation. Examples are presented. In distribution methodology the independent responses are transformed in order that a distributional assumption is satisfied for the transformed data. Here the interest is in the family of distributions which are not dependent on an unknown shape parameter. The gamma distribution (known order), with special case the exponential distribution, is a member of this family. An information number approach is proposed for transforming a known distribution to the gamma distribution (known order). The approach provides an insight into the large-sample behaviour of the likelihood procedure considered by Draper and Guttman (1968) for investigating transformations of data which allow the transformed observations to follow a gamma distribution. The information number approach is illustrated for three examples end the improvement towards the gamma distribution introduced by transformation is measured numerically and graphically. A graphical procedure is proposed for the general case of investigating transformations of data which allow the transformed observations to follow a distribution dependent on unknown threshold and scale parameters. The procedure is extended to include model testing and estimation for any distribution which with the aid of a power transformation can be put in the simple form of a distribution that is not dependent on an unknown shape parameter. The procedure is based on a ratio, R(y), which is constructed from the power transformation. Also described is a ratio-based technique for estimating the threshold parameter in important parametric models, including the three-parameter Weibull and lognormal distributions. Ratio estimation for the weibull distribution is assessed and compared with the modified maximum likelihood estimation of Cohen and Whitten (1982) in terms of bias and root mean squared error, by means of a simulation study. The methods are illustrated with several examples and extend naturally to singly Type 1 and Type 2 censored data.
1988-01-01T00:00:00ZParker, ImeldaTransformation is a powerful tool for model building. In regression the response variable is transformed in order to achieve the usual assumptions of normality, constant variance and additivity of effects. Here the normality assumption is replaced by the Laplace distributional assumption, appropriate when more large errors occur than would be expected if the errors were normally distributed. The parametric model is enlarged to include a transformation parameter and a likelihood procedure is adopted for estimating this parameter simultaneously with other parameters of interest. Diagnostic methods are described for assessing the influence of individual observations on the choice of transformation. Examples are presented. In distribution methodology the independent responses are transformed in order that a distributional assumption is satisfied for the transformed data. Here the interest is in the family of distributions which are not dependent on an unknown shape parameter. The gamma distribution (known order), with special case the exponential distribution, is a member of this family. An information number approach is proposed for transforming a known distribution to the gamma distribution (known order). The approach provides an insight into the large-sample behaviour of the likelihood procedure considered by Draper and Guttman (1968) for investigating transformations of data which allow the transformed observations to follow a gamma distribution. The information number approach is illustrated for three examples end the improvement towards the gamma distribution introduced by transformation is measured numerically and graphically. A graphical procedure is proposed for the general case of investigating transformations of data which allow the transformed observations to follow a distribution dependent on unknown threshold and scale parameters. The procedure is extended to include model testing and estimation for any distribution which with the aid of a power transformation can be put in the simple form of a distribution that is not dependent on an unknown shape parameter. The procedure is based on a ratio, R(y), which is constructed from the power transformation. Also described is a ratio-based technique for estimating the threshold parameter in important parametric models, including the three-parameter Weibull and lognormal distributions. Ratio estimation for the weibull distribution is assessed and compared with the modified maximum likelihood estimation of Cohen and Whitten (1982) in terms of bias and root mean squared error, by means of a simulation study. The methods are illustrated with several examples and extend naturally to singly Type 1 and Type 2 censored data.Parameterisation-invariant versions of Wald testsLarsen, Pia Veldthttp://hdl.handle.net/10023/137502019-04-01T08:37:58Z1999-01-01T00:00:00ZAlthough Wald tests form one of the major classes of hypothesis tests, they suffer from the well-known major drawback that they are not invariant under reparameterisation. This thesis uses the differential-geometric concept of a yoke to introduce one-parameter families of geometric Wald statistics, which are parameterisation-invariant statistics in the spirit of the traditional Wald statistics. Both the geometric Wald statistics based on the expected likelihood yokes and those based on the observed likelihood yokes are investigated. Bartlett-type adjustments of the geometric Wald statistics are obtained, in order to improve the accuracy of the chi-squared approximations to their distributions under the null hypothesis.
1999-01-01T00:00:00ZLarsen, Pia VeldtAlthough Wald tests form one of the major classes of hypothesis tests, they suffer from the well-known major drawback that they are not invariant under reparameterisation. This thesis uses the differential-geometric concept of a yoke to introduce one-parameter families of geometric Wald statistics, which are parameterisation-invariant statistics in the spirit of the traditional Wald statistics. Both the geometric Wald statistics based on the expected likelihood yokes and those based on the observed likelihood yokes are investigated. Bartlett-type adjustments of the geometric Wald statistics are obtained, in order to improve the accuracy of the chi-squared approximations to their distributions under the null hypothesis.Estimating the parameters in mixtures of circular and spherical distributionsKoutbeiy, Majdi Aminehttp://hdl.handle.net/10023/137482019-04-01T08:39:27Z1990-01-01T00:00:00ZIn this thesis we compare various methods for estimating the unknown parameters in mixtures of circular and spherical distributions. We study the von Mises distribution on the circle and the Fisher distribution on the sphere. We propose a new method of estimation based on the characteristic function and compare it with the classical methods based on maximum likelihood and moments. Thus far these methods have only been successfully applied to distributions on the line. Here we show that the extension to circular and spherical distributions is reasonably straightforward and convergence to the final estimates is fairly rapid. We apply these methods to various simulated and real data sets and show that the results obtained for the mixture of two von Mises distributions are satisfactory but generally depend on the sample size and method of estimation used. However, results obtained for the mixture of two Fisher distributions show that maximum likelihood performs best overall.
1990-01-01T00:00:00ZKoutbeiy, Majdi AmineIn this thesis we compare various methods for estimating the unknown parameters in mixtures of circular and spherical distributions. We study the von Mises distribution on the circle and the Fisher distribution on the sphere. We propose a new method of estimation based on the characteristic function and compare it with the classical methods based on maximum likelihood and moments. Thus far these methods have only been successfully applied to distributions on the line. Here we show that the extension to circular and spherical distributions is reasonably straightforward and convergence to the final estimates is fairly rapid. We apply these methods to various simulated and real data sets and show that the results obtained for the mixture of two von Mises distributions are satisfactory but generally depend on the sample size and method of estimation used. However, results obtained for the mixture of two Fisher distributions show that maximum likelihood performs best overall.Statistical aspects of the population regulation of migrating brown trout "Salmo trutta" in a Lake District streamFryer, Robert Johnhttp://hdl.handle.net/10023/137462019-04-01T08:38:39Z1990-01-01T00:00:00ZStatistical aspects of the population regulation of a migratory brown trout population are investigated. The life cycle of the trout, the study area and the sampling routine are described in Chapter 1. Models of numerical changes in fish populations are reviewed in Chapter 2. Measures that assess the nonlinear behaviour of nonlinear regression models are described in Chapter 3. The additive error Ricker model describes the relationship between the number of 0+ parr in May/June and the number of eggs. The nonlinear behaviour of the model is investigated in Chapter 4. The parameter effects nonlinearity of the model is reduced by a reparameterisation. Chapter 5 investigates the effect of errors in the egg variable on the distributions of the least squares estimators of the additive error and the multiplicative error Ricker models. The errors-in-variables considerably increase the variances of the least squares estimators. Models of the relationships between the numbers of 0+ parr in August/September, the number of 1+ parr, the egg production of a year class and the number of eggs are developed in Chapter 6. These models account for the effect of summer drought on survival. Survival is density dependent during the first summer of the life cycle and density independent thereafter. Standard measures of nonlinearity can seriously underestimate the nonlinear behaviour of piecewise linear change-point models. New measures of nonlinearity appropriate for piecewise linear change-point models are developed in Chapter 7. Chapter 8 develops a model of the growth of brown trout fed on maximum rations as a function of time, body weight and water temperature. Chapter 9 develops a model that relates the survival rate of 0+ parr between May/June and August/September to the length distribution of the trout in May/June. The results of the Thesis are discussed in Chapter 10.
1990-01-01T00:00:00ZFryer, Robert JohnStatistical aspects of the population regulation of a migratory brown trout population are investigated. The life cycle of the trout, the study area and the sampling routine are described in Chapter 1. Models of numerical changes in fish populations are reviewed in Chapter 2. Measures that assess the nonlinear behaviour of nonlinear regression models are described in Chapter 3. The additive error Ricker model describes the relationship between the number of 0+ parr in May/June and the number of eggs. The nonlinear behaviour of the model is investigated in Chapter 4. The parameter effects nonlinearity of the model is reduced by a reparameterisation. Chapter 5 investigates the effect of errors in the egg variable on the distributions of the least squares estimators of the additive error and the multiplicative error Ricker models. The errors-in-variables considerably increase the variances of the least squares estimators. Models of the relationships between the numbers of 0+ parr in August/September, the number of 1+ parr, the egg production of a year class and the number of eggs are developed in Chapter 6. These models account for the effect of summer drought on survival. Survival is density dependent during the first summer of the life cycle and density independent thereafter. Standard measures of nonlinearity can seriously underestimate the nonlinear behaviour of piecewise linear change-point models. New measures of nonlinearity appropriate for piecewise linear change-point models are developed in Chapter 7. Chapter 8 develops a model of the growth of brown trout fed on maximum rations as a function of time, body weight and water temperature. Chapter 9 develops a model that relates the survival rate of 0+ parr between May/June and August/September to the length distribution of the trout in May/June. The results of the Thesis are discussed in Chapter 10.Reliability theory in operational researchAl-Baidhani, Fadil Ajabhttp://hdl.handle.net/10023/137452019-04-01T08:38:42Z1991-01-01T00:00:00ZThis thesis is concerned principally with the problem of estimating the parameters of the Weibull and Beta distributions using several different techniques. These distributions are used in the area of reliability testing and it is important to achieve the best estimates possible of the parameters involved. After considering several accepted methods of estimating the relevant parameters, it is considered that the best method depends on the aim of the analysis, and on the value of the shape parameter 𝛽. For estimating the two-parameter Weibull distribution, it is recommended that Generalized Least Squares (GLS) is the best method to use for values of 𝛽 between 0.5 and 30. However, Maximum Likelihood Estimator (MLE) is a good method for estimating quantiles.
On this basis, the three-parameter Weibull distribution is investigated. The traditional parametrization is compared with a new parametrization developed in this work. By considering parameter effects and intrinsic curvature it is shown that the new parametrization results in a linear effect of the shape parameter. Also it has advantages in quantile estimation because of its ability to provide estimates for a wider range of data sets.
A less frequently used distribution in the field of reliability is the Beta distribution. The lack of frequency of its use is partly due to the difficulty in estimating its parameters. A simple, applicable method is developed here of estimating these parameters. This 'group method' involves estimating the two ends of the distribution. It is shown that this procedure can be used, together with other methods of estimating the two- parameter Beta distribution successfully to estimate the four-parameter Beta distribution.
1991-01-01T00:00:00ZAl-Baidhani, Fadil AjabThis thesis is concerned principally with the problem of estimating the parameters of the Weibull and Beta distributions using several different techniques. These distributions are used in the area of reliability testing and it is important to achieve the best estimates possible of the parameters involved. After considering several accepted methods of estimating the relevant parameters, it is considered that the best method depends on the aim of the analysis, and on the value of the shape parameter 𝛽. For estimating the two-parameter Weibull distribution, it is recommended that Generalized Least Squares (GLS) is the best method to use for values of 𝛽 between 0.5 and 30. However, Maximum Likelihood Estimator (MLE) is a good method for estimating quantiles.
On this basis, the three-parameter Weibull distribution is investigated. The traditional parametrization is compared with a new parametrization developed in this work. By considering parameter effects and intrinsic curvature it is shown that the new parametrization results in a linear effect of the shape parameter. Also it has advantages in quantile estimation because of its ability to provide estimates for a wider range of data sets.
A less frequently used distribution in the field of reliability is the Beta distribution. The lack of frequency of its use is partly due to the difficulty in estimating its parameters. A simple, applicable method is developed here of estimating these parameters. This 'group method' involves estimating the two ends of the distribution. It is shown that this procedure can be used, together with other methods of estimating the two- parameter Beta distribution successfully to estimate the four-parameter Beta distribution.Inference for plant-captureAshbridge, Jonathanhttp://hdl.handle.net/10023/137412019-04-01T08:38:39Z1998-01-01T00:00:00ZWhen investigating the dynamics of an animal population, a primary objective is to obtain reasonable estimates of abundance or population size. This thesis concentrates on the problem of obtaining point estimates of abundance from capture-recapture data and on how such estimation can be improved by using the method of plant-capture. Plant-capture constitutes a natural generalisation of capture-recapture. In a plant-capture study a pre-marked population of known size is added to the target population of unknown size. The capture-recapture experiment is then carried out on the augmented population. Chapter 1 considers the addition of planted individuals to target populations which behave according to the standard capture-recapture model M₀. Chapter 2 investigates an analogous model based on sampling in continuous time. In each of these chapters, distributional results are derived under the assumption that the behaviour of the plants is indistinguishable from that of members of the target population. Maximum likelihood estimators and other new estimators are proposed for each model. The results suggest that the use of plants is beneficial, and furthermore that the new estimators perform more satisfactorily than the maximum likelihood estimators. Chapter 3 introduces, initially in the absence of plants, a new class of estimators, described as coverage adjusted estimators, for the standard capture-recapture model M[sub]h. These new estimators are shown, through simulation and real life data, to compare favourably with estimators that have previously been proposed. Plant-capture versions of these new estimators are then derived and the usefulness of the plants is demonstrated through simulation. Chapter 4 describes how the approach taken in chapter 3 can be modified to produce a new estimator for the analogous continuous time model. This estimator is then shown through simulation to be preferable to estimators that have previously been proposed.
1998-01-01T00:00:00ZAshbridge, JonathanWhen investigating the dynamics of an animal population, a primary objective is to obtain reasonable estimates of abundance or population size. This thesis concentrates on the problem of obtaining point estimates of abundance from capture-recapture data and on how such estimation can be improved by using the method of plant-capture. Plant-capture constitutes a natural generalisation of capture-recapture. In a plant-capture study a pre-marked population of known size is added to the target population of unknown size. The capture-recapture experiment is then carried out on the augmented population. Chapter 1 considers the addition of planted individuals to target populations which behave according to the standard capture-recapture model M₀. Chapter 2 investigates an analogous model based on sampling in continuous time. In each of these chapters, distributional results are derived under the assumption that the behaviour of the plants is indistinguishable from that of members of the target population. Maximum likelihood estimators and other new estimators are proposed for each model. The results suggest that the use of plants is beneficial, and furthermore that the new estimators perform more satisfactorily than the maximum likelihood estimators. Chapter 3 introduces, initially in the absence of plants, a new class of estimators, described as coverage adjusted estimators, for the standard capture-recapture model M[sub]h. These new estimators are shown, through simulation and real life data, to compare favourably with estimators that have previously been proposed. Plant-capture versions of these new estimators are then derived and the usefulness of the plants is demonstrated through simulation. Chapter 4 describes how the approach taken in chapter 3 can be modified to produce a new estimator for the analogous continuous time model. This estimator is then shown through simulation to be preferable to estimators that have previously been proposed.The asymptotic distribution and robustness of the likelihood ratio and score test statisticsEmberson, E. A.http://hdl.handle.net/10023/137382019-04-01T08:38:29Z1995-01-01T00:00:00ZCordeiro & Ferrari (1991) use the asymptotic expansion of Harris (1985) for the moment generating function of the score statistic to produce a generalization of Bartlett adjustment for application to the score statistic. It is shown here that Harris's expansion is not invariant under reparameterization and an invariant expansion is derived using a method based on the expected likelihood yoke. A necessary and sufficient condition for the existence of a generalized Bartlett adjustment for an arbitrary statistic is given in terms of its moment generating function. Generalized Bartlett adjustments to the likelihood ratio and score test statistics are derived in the case where the interest parameter is one-dimensional under the assumption of a mis-specified model, where the true distribution is not assumed to be that under the null hypothesis.
1995-01-01T00:00:00ZEmberson, E. A.Cordeiro & Ferrari (1991) use the asymptotic expansion of Harris (1985) for the moment generating function of the score statistic to produce a generalization of Bartlett adjustment for application to the score statistic. It is shown here that Harris's expansion is not invariant under reparameterization and an invariant expansion is derived using a method based on the expected likelihood yoke. A necessary and sufficient condition for the existence of a generalized Bartlett adjustment for an arbitrary statistic is given in terms of its moment generating function. Generalized Bartlett adjustments to the likelihood ratio and score test statistics are derived in the case where the interest parameter is one-dimensional under the assumption of a mis-specified model, where the true distribution is not assumed to be that under the null hypothesis.On the equivalence of Markov Algorithms and Turing machines and some consequent resultsPapathanassiou, Eleftherioshttp://hdl.handle.net/10023/137362019-04-01T08:38:06Z1979-01-01T00:00:00ZTuring Machines and Markov Algorithms are, and were designed to be, the most powerful devices possible in the field of abstract automata: by their means any and every computable function can be computed.
Because of their equal, indeed maximal, strength, it was naturally assumed that these devices should be equivalent. Nonetheless a formal, exact proof of this universally presumed equivalence was lacking.
The present dissertation rectifies that omission by developing the desired complete, rigorous proof of the equivalence between Turing Machines and Markov Algorithms. The demonstration is being conducted in a constructionist way: for any given Markov Algorithm it is shown that a Turing Machine can be constructed capable of performing exactly what the Algorithm can do and nothing more, and vice versa.
The proof consists in the theoretical construction, given an arbitrary Markov Algorithm, of a Turing Machine behaving in exactly the same way as the Algorithm for all possible inputs; and conversely. Furthermore, the proof is given concrete shape by designing a computer program which can actually carry out the said theoretical constructions.
The equivalence between TM and MA as proven in the first part of our thesis, is being used in the second part for establishing some important consequent results: Thus the equivalence of Deterministic and Nondeterministic MA, of TM and Type 0 Grammars, and of Labelled and Unlabelled MA is concisely shown, and the use of TM as recognizers for type 1 and 3 grammars exclusively is exhibited. It is interesting that, by utilizing the equivalence of TM and MA, it was made possible that the proofs of these latter results be based on primitive principles.
1979-01-01T00:00:00ZPapathanassiou, EleftheriosTuring Machines and Markov Algorithms are, and were designed to be, the most powerful devices possible in the field of abstract automata: by their means any and every computable function can be computed.
Because of their equal, indeed maximal, strength, it was naturally assumed that these devices should be equivalent. Nonetheless a formal, exact proof of this universally presumed equivalence was lacking.
The present dissertation rectifies that omission by developing the desired complete, rigorous proof of the equivalence between Turing Machines and Markov Algorithms. The demonstration is being conducted in a constructionist way: for any given Markov Algorithm it is shown that a Turing Machine can be constructed capable of performing exactly what the Algorithm can do and nothing more, and vice versa.
The proof consists in the theoretical construction, given an arbitrary Markov Algorithm, of a Turing Machine behaving in exactly the same way as the Algorithm for all possible inputs; and conversely. Furthermore, the proof is given concrete shape by designing a computer program which can actually carry out the said theoretical constructions.
The equivalence between TM and MA as proven in the first part of our thesis, is being used in the second part for establishing some important consequent results: Thus the equivalence of Deterministic and Nondeterministic MA, of TM and Type 0 Grammars, and of Labelled and Unlabelled MA is concisely shown, and the use of TM as recognizers for type 1 and 3 grammars exclusively is exhibited. It is interesting that, by utilizing the equivalence of TM and MA, it was made possible that the proofs of these latter results be based on primitive principles.Some contributions to the theory of mathematical programmingSaksena, Chandra P.http://hdl.handle.net/10023/137342019-04-01T08:39:22Z1970-01-01T00:00:00ZAs stated earlier the Simplex Method (or its variations e.g. Dual Simplex Method) has thus far been the most effective and widely used general method for the solution of linear programming problems. The Simplex Method in its various forms starts initially with a basic feasible solution and continues its moves in different iterations within the feasible region till it finds the optimal solution. The only other notable variation of the Simplex Method, namely the Dual Simplex Method, on the other hand, by virtue of the special formulation of the linear programming problem, starts with an in-feasible solution and continues to move in the in-feasible region till it finds the optimal solution at which it enters the feasible region. In other respects both the Simplex and the Dual Simplex Methods follow essentially the same principle for obtaining the optimal solution. The rigorous mathematical features have been widely discussed in the literature [12, 16, 34, 35, 38, 68, 77] and only those formal aspects of this topic which are closely related to the subject of this thesis will be outlined.
The Multiplex Method, though reported in the literature [30, 15, 69, 71, 29, 32], is not so well known and has also not been widely coded on electronic computers. It had earlier been programmed for the English Electric’s Computer ‘DEUCE’ by the author [72] and Ferranti’s ‘MERCURY’ by Ole-John Dahl in 1960 [15]. Later both the above mentioned computers were obsolete and the efforts presently concentrate on coding it for UNIVAC 1100 and IBM 360. The Multiplex Method, as such, has been included in the present thesis and discussed in some detail in chapter 2. The flow diagram and the algorithm for the method is given in section 2.4, chapter 2.
The main body of the thesis consists of developing a new linear programming method which has been called the Bounding Hyperplane Method – Part I. This is explained in detail in chapter 3. The method could initially start with either a basic feasible or in-feasible point and in its subsequent moves it may either alternate between the feasible and the in-feasible regions or get restricted to either of them depending upon the problem. It is applicable as a new phase which we call phase 0 to the Simple Method, particularly in situations where an initial basic feasible point is not available. In such cases it either results in a feasible point at the end of phase 0 or else yields a ‘better’ in-feasible point for phase 1 operations of the Simplex Method. Moreover, it is found that the number of iterations required to reach either the former by the application of phase 0 or the latter by the application of first phase 0 and then phase 1 are, in general, less than those required by following phase 1 alone. This is explained with illustrations in Chapter 6. Even when applied alone the method, in general, yields the optimal solution in fewer iterations as compared with the Simplex Method. This is illustrated with examples in chapter 3.
We also develop and illustrate a powerful but straight-forward method whereby we first find the solution to the equality constraints and (if the former does not yield an inconsistent solution point) then the transformations to the latter are obtained from the equality solution tableau corresponding to the former. This results in reducing the iteration time appreciably for each iteration of the method. It has been called the B.H.P.M. – part II and is discussed in chapter 4.
To estimate the time taken by the B.H.P and the Simplex Method, the two codes (written in Fortran) have been run on a number of problems taken from the literature. The results have been summarised in chapter 7.
Finally, the suggestions for further research towards i. the extensions of the B.H.P.M. to the quadratic programming problem where the function in (1.1.1) is positive semi-definite, and (ii) the accuracy of computations in linear programming, in general, are discussed in sections 8.1 and 8.2 respectively of chapter 8.
1970-01-01T00:00:00ZSaksena, Chandra P.As stated earlier the Simplex Method (or its variations e.g. Dual Simplex Method) has thus far been the most effective and widely used general method for the solution of linear programming problems. The Simplex Method in its various forms starts initially with a basic feasible solution and continues its moves in different iterations within the feasible region till it finds the optimal solution. The only other notable variation of the Simplex Method, namely the Dual Simplex Method, on the other hand, by virtue of the special formulation of the linear programming problem, starts with an in-feasible solution and continues to move in the in-feasible region till it finds the optimal solution at which it enters the feasible region. In other respects both the Simplex and the Dual Simplex Methods follow essentially the same principle for obtaining the optimal solution. The rigorous mathematical features have been widely discussed in the literature [12, 16, 34, 35, 38, 68, 77] and only those formal aspects of this topic which are closely related to the subject of this thesis will be outlined.
The Multiplex Method, though reported in the literature [30, 15, 69, 71, 29, 32], is not so well known and has also not been widely coded on electronic computers. It had earlier been programmed for the English Electric’s Computer ‘DEUCE’ by the author [72] and Ferranti’s ‘MERCURY’ by Ole-John Dahl in 1960 [15]. Later both the above mentioned computers were obsolete and the efforts presently concentrate on coding it for UNIVAC 1100 and IBM 360. The Multiplex Method, as such, has been included in the present thesis and discussed in some detail in chapter 2. The flow diagram and the algorithm for the method is given in section 2.4, chapter 2.
The main body of the thesis consists of developing a new linear programming method which has been called the Bounding Hyperplane Method – Part I. This is explained in detail in chapter 3. The method could initially start with either a basic feasible or in-feasible point and in its subsequent moves it may either alternate between the feasible and the in-feasible regions or get restricted to either of them depending upon the problem. It is applicable as a new phase which we call phase 0 to the Simple Method, particularly in situations where an initial basic feasible point is not available. In such cases it either results in a feasible point at the end of phase 0 or else yields a ‘better’ in-feasible point for phase 1 operations of the Simplex Method. Moreover, it is found that the number of iterations required to reach either the former by the application of phase 0 or the latter by the application of first phase 0 and then phase 1 are, in general, less than those required by following phase 1 alone. This is explained with illustrations in Chapter 6. Even when applied alone the method, in general, yields the optimal solution in fewer iterations as compared with the Simplex Method. This is illustrated with examples in chapter 3.
We also develop and illustrate a powerful but straight-forward method whereby we first find the solution to the equality constraints and (if the former does not yield an inconsistent solution point) then the transformations to the latter are obtained from the equality solution tableau corresponding to the former. This results in reducing the iteration time appreciably for each iteration of the method. It has been called the B.H.P.M. – part II and is discussed in chapter 4.
To estimate the time taken by the B.H.P and the Simplex Method, the two codes (written in Fortran) have been run on a number of problems taken from the literature. The results have been summarised in chapter 7.
Finally, the suggestions for further research towards i. the extensions of the B.H.P.M. to the quadratic programming problem where the function in (1.1.1) is positive semi-definite, and (ii) the accuracy of computations in linear programming, in general, are discussed in sections 8.1 and 8.2 respectively of chapter 8.Involutive automorphisms and real forms of Kac-Moody algebrasClarke, Stefanhttp://hdl.handle.net/10023/137312019-04-01T08:39:33Z1996-01-01T00:00:00ZInvolutive automorphisms of complex affine Kac-Moody algebras (in particular, their conjugacy classes within the group of all automorphisms) and their compact real forms are studied, using the matrix formulation which was developed by Cornwell. The initial study of the a⁽¹⁾ series of affine untwisted Kac-Moody algebras is extended to include the complex affine untwisted Kac-Moody algebras B⁽¹⁾, C⁽¹⁾ and D⁽¹⁾. From the information obtained, explicit bases for real forms of these Kac-Moody algebras are then constructed. A scheme for naming some real forms is suggested. Further work is included which examines the involutive automorphisms and the real forms of A₂⁽²⁾and the algebra G⁽¹⁾₂ (which is based upon an exceptional simple Lie algebra). The work involving the algebra A₂⁽²⁾is part of work towards extending the matrix formulation to twisted Kac-Moody algebras. The analysis also acts as a practical test of this method, and from it we may infer different ways of using the formulation to eventually obtain a complete picture of the conjugacy classes of the involutive automorphisms of all the affine Kac-Moody algebras.
1996-01-01T00:00:00ZClarke, StefanInvolutive automorphisms of complex affine Kac-Moody algebras (in particular, their conjugacy classes within the group of all automorphisms) and their compact real forms are studied, using the matrix formulation which was developed by Cornwell. The initial study of the a⁽¹⁾ series of affine untwisted Kac-Moody algebras is extended to include the complex affine untwisted Kac-Moody algebras B⁽¹⁾, C⁽¹⁾ and D⁽¹⁾. From the information obtained, explicit bases for real forms of these Kac-Moody algebras are then constructed. A scheme for naming some real forms is suggested. Further work is included which examines the involutive automorphisms and the real forms of A₂⁽²⁾and the algebra G⁽¹⁾₂ (which is based upon an exceptional simple Lie algebra). The work involving the algebra A₂⁽²⁾is part of work towards extending the matrix formulation to twisted Kac-Moody algebras. The analysis also acts as a practical test of this method, and from it we may infer different ways of using the formulation to eventually obtain a complete picture of the conjugacy classes of the involutive automorphisms of all the affine Kac-Moody algebras.Subalgebras of free nilpotent and polynilpotent lie algebrasBoral, Melihhttp://hdl.handle.net/10023/137292019-04-01T08:39:28Z1977-01-01T00:00:00ZIn this thesis we study subalgebras in free nilpotent and polynilpotent Lie algebras. Chapter 1 sets up the notation and includes definitions and elementary properties of free and certain reduced free Lie algebras that we use throughout this thesis. We also describe a Hall basis of a free Lie algebra as in [4] and a basis for a free polynilpotent Lie algebra which was developed in [24].
In Chapter 2 we first consider the class of nilpotency of subalbebras of free nilpotent Lie algebras starting with two-generator subalgebras. Then we study those subalgebras in a free nilpotent Lie algebra which, are themselves free nilpotent. We give necessary and sufficient conditions in the case of two-generator subalgebras.
Chapter 3 extends the results obtained in Chapter 2 to the polynilpotent case. First we look at two-generator subalgebras of a free polynilpotent Lie algebra. Then we consider more general subalgebras. Finally we study those subalgebras which are themselves free polynilpotent and give necessary and sufficient conditions for two-generator subalgebras to be free polynilpotent.
In Chapter 4 we first study certain properties of ideals in free, free nilpotent and free polynilpotent Lie algebras and establish the fact that in a free polynilpotent Lie algebra a nonzero ideal which is finitely-generated as a subalgebra must be equal to the whole algebra. Then we consider the quotient Lie algebra of a lower central term of a free Lie algebra by a term of the lower central series of an ideal. We then generalize the results to cover the free nilpotent and free polynilpotent cases. In the last section of Chapter 4 we consider ideals of free nilpotent (and later polynilpotent) Lie algebras as free nilpotent (polynilpotent) subalgebras and establish the fact that in most non-trivial cases such an ideal cannot be free nilpotent (polynilpotent).
In the last chapter we consider the m+k-th term of the lower central series of a free Lie algebra as a subalgebra of the m-th term for m ⩽ k and generalize the results proved in [25]. We give reasons for the failure of these results in the case m > k.
1977-01-01T00:00:00ZBoral, MelihIn this thesis we study subalgebras in free nilpotent and polynilpotent Lie algebras. Chapter 1 sets up the notation and includes definitions and elementary properties of free and certain reduced free Lie algebras that we use throughout this thesis. We also describe a Hall basis of a free Lie algebra as in [4] and a basis for a free polynilpotent Lie algebra which was developed in [24].
In Chapter 2 we first consider the class of nilpotency of subalbebras of free nilpotent Lie algebras starting with two-generator subalgebras. Then we study those subalgebras in a free nilpotent Lie algebra which, are themselves free nilpotent. We give necessary and sufficient conditions in the case of two-generator subalgebras.
Chapter 3 extends the results obtained in Chapter 2 to the polynilpotent case. First we look at two-generator subalgebras of a free polynilpotent Lie algebra. Then we consider more general subalgebras. Finally we study those subalgebras which are themselves free polynilpotent and give necessary and sufficient conditions for two-generator subalgebras to be free polynilpotent.
In Chapter 4 we first study certain properties of ideals in free, free nilpotent and free polynilpotent Lie algebras and establish the fact that in a free polynilpotent Lie algebra a nonzero ideal which is finitely-generated as a subalgebra must be equal to the whole algebra. Then we consider the quotient Lie algebra of a lower central term of a free Lie algebra by a term of the lower central series of an ideal. We then generalize the results to cover the free nilpotent and free polynilpotent cases. In the last section of Chapter 4 we consider ideals of free nilpotent (and later polynilpotent) Lie algebras as free nilpotent (polynilpotent) subalgebras and establish the fact that in most non-trivial cases such an ideal cannot be free nilpotent (polynilpotent).
In the last chapter we consider the m+k-th term of the lower central series of a free Lie algebra as a subalgebra of the m-th term for m ⩽ k and generalize the results proved in [25]. We give reasons for the failure of these results in the case m > k.Finite difference solutions of the Von Mises equationThomson, John Younghttp://hdl.handle.net/10023/137272019-04-01T08:38:50Z1958-01-01T00:00:00ZPrandtl in 1904 discovered that the flow of a fluid over a thin obstacle can be adequately represented by an approximate set of equations, much simpler than the complex Navier-Stokes equations which govern the motion of fluid.
A particularly simple for of these equations, for the two-dimensional steady flow of a fluid past a flat plate, are the Von Mises Boundary layer equations. Unfortunately the Von Mises transformation introduces a singularity at the plate and this discouraged the use of the equations as a means for obtaining numerical solutions of boundary layer problems in incompressible and compressible flow.
In this thesis, we show that this difficulty can be overcome and the Von Mises equations are used as a basis for a finite difference evaluation of the velocity and temperature in the boundary layer adjacent to a flat plate, particular attention being given to conditions near the plate and more especially to the separation point.
In the section on compressible flow, the calculations also yield a check on certain common simplifying assumptions.
1958-01-01T00:00:00ZThomson, John YoungPrandtl in 1904 discovered that the flow of a fluid over a thin obstacle can be adequately represented by an approximate set of equations, much simpler than the complex Navier-Stokes equations which govern the motion of fluid.
A particularly simple for of these equations, for the two-dimensional steady flow of a fluid past a flat plate, are the Von Mises Boundary layer equations. Unfortunately the Von Mises transformation introduces a singularity at the plate and this discouraged the use of the equations as a means for obtaining numerical solutions of boundary layer problems in incompressible and compressible flow.
In this thesis, we show that this difficulty can be overcome and the Von Mises equations are used as a basis for a finite difference evaluation of the velocity and temperature in the boundary layer adjacent to a flat plate, particular attention being given to conditions near the plate and more especially to the separation point.
In the section on compressible flow, the calculations also yield a check on certain common simplifying assumptions.Semigroups of singular endomorphisms of vector spaceDawlings, Robert J. H.http://hdl.handle.net/10023/137252019-04-01T08:38:41Z1980-01-01T00:00:00ZIn 1967, J. A. Erdős showed, using a matrix theory approach that the semigroup Sing[sub]n of singular endomorphisms of an n-dimensional vector space is generated by the set E of idempotent endomorphisms of rank n - 1. This thesis gives an alternative proof using a linear algebra and semigroup theory approach. It is also shown that not all the elements of E are needed to generate Sing[sub]n. Necessary conditions for a subset of E to generate found; these conditions are shown to be sufficient if the vector space is defined over a finite field. In this case, the minimum order of all subsets of E that generate Sing[sub]n is found. The problem of determining the number of subsets of E that generate Sing[sub]n and have this minimum order is considered; it is completely solved when the vector space is two-dimensional. From the proof given by Erdős, it could be deduced that any element of Sing[sub]n could be expressed as the product of, at most, 2n elements of E. It is shown here that this bound may be reduced to n, and that this is best possible. It is also shown that, if E⁺ is the set of all idempotent of Singn, then (E⁺)ⁿ⁻¹ is strictly contained in Sing[sub]n. Finally, it is shown that Erdős's result cannot be extended to the semigroup Sing of continuous singular endomorphisms of a separable Hilbert space. The sub semigroup of Sing generated by the idempotent of Sing is determined and is, clearly, strictly contained in Sing.
1980-01-01T00:00:00ZDawlings, Robert J. H.In 1967, J. A. Erdős showed, using a matrix theory approach that the semigroup Sing[sub]n of singular endomorphisms of an n-dimensional vector space is generated by the set E of idempotent endomorphisms of rank n - 1. This thesis gives an alternative proof using a linear algebra and semigroup theory approach. It is also shown that not all the elements of E are needed to generate Sing[sub]n. Necessary conditions for a subset of E to generate found; these conditions are shown to be sufficient if the vector space is defined over a finite field. In this case, the minimum order of all subsets of E that generate Sing[sub]n is found. The problem of determining the number of subsets of E that generate Sing[sub]n and have this minimum order is considered; it is completely solved when the vector space is two-dimensional. From the proof given by Erdős, it could be deduced that any element of Sing[sub]n could be expressed as the product of, at most, 2n elements of E. It is shown here that this bound may be reduced to n, and that this is best possible. It is also shown that, if E⁺ is the set of all idempotent of Singn, then (E⁺)ⁿ⁻¹ is strictly contained in Sing[sub]n. Finally, it is shown that Erdős's result cannot be extended to the semigroup Sing of continuous singular endomorphisms of a separable Hilbert space. The sub semigroup of Sing generated by the idempotent of Sing is determined and is, clearly, strictly contained in Sing.Formal languages and idempotent semigroupsSezinando, Helena Maria da Encarnaçãohttp://hdl.handle.net/10023/137242019-04-01T08:38:43Z1991-01-01T00:00:00ZThe structure of the lattice 𝗟𝗕 of varieties of idempotent semigroups or bands (as universal algebras) was determined by Birjukov, Fennemore and Gerhard. Wis- math determined the structure of a related lattice: the lattice LBM of varieties of band monoids. In the first two parts we study several questions about these varieties.
In Part I we compute the cardinalities of the Green classes of the free objects in each variety of 𝗟𝗕 [𝗟𝗕𝗠]. These cardinalities constitute a useful piece of information in the study of several questions about these varieties and some of the conclusions obtained here are used in parts II and III.
Part II concerns expansions of bands [band monoids]. More precisely, we compute here the cut-down to generators of the Rhodes expansions of the free objects in the varieties of 𝗟𝗕. We define Rhodes expansion of a monoid, its cut-down to generators and we compute the cut-down to generators of the Rhodes expansions of the free objects in the varieties of 𝗟𝗕𝗠.
In Part III we deal with Eilenberg varieties of band monoids. The last chapter is particularly concerned with the description of the varieties of languages corresponding to these varieties.
1991-01-01T00:00:00ZSezinando, Helena Maria da EncarnaçãoThe structure of the lattice 𝗟𝗕 of varieties of idempotent semigroups or bands (as universal algebras) was determined by Birjukov, Fennemore and Gerhard. Wis- math determined the structure of a related lattice: the lattice LBM of varieties of band monoids. In the first two parts we study several questions about these varieties.
In Part I we compute the cardinalities of the Green classes of the free objects in each variety of 𝗟𝗕 [𝗟𝗕𝗠]. These cardinalities constitute a useful piece of information in the study of several questions about these varieties and some of the conclusions obtained here are used in parts II and III.
Part II concerns expansions of bands [band monoids]. More precisely, we compute here the cut-down to generators of the Rhodes expansions of the free objects in the varieties of 𝗟𝗕. We define Rhodes expansion of a monoid, its cut-down to generators and we compute the cut-down to generators of the Rhodes expansions of the free objects in the varieties of 𝗟𝗕𝗠.
In Part III we deal with Eilenberg varieties of band monoids. The last chapter is particularly concerned with the description of the varieties of languages corresponding to these varieties.Random-walk theory and statistical mechanics of lattice systemsNieto, Alberto Robledohttp://hdl.handle.net/10023/137222019-04-01T08:39:26Z1974-01-01T00:00:00ZIt has been found elsewhere that when approximate relations for the two-particle correlation functions of classical statistical mechanics, such as the Percus-Yevick and the mean-spherical approximations, are applied to the lattice gas models with nearest-neighbour interactions simple expressions are obtained for the total correlation function in terms of the lattice Green's function. Since many of the properties of random walks on a lattice can be described by the lattice Green's function, it follows that these systems, at least when treated under these approximations, may be analysed in terms of the language of random walks.
Here the theory of random walks on lattices is appropriately extended to show that the relationship between the correlation functions and the lattice Green's function is not dependent upon the employment of these approximations, nor to a particular range or form of the potential function. Instead, this relationship is shown to be an alternative form of the Ornstein-Zernike relation between the direct and total correlation functions. The direct correlation function is directly related to the probability of a single step, whereas the total correlation function is given by the first-passage- time probabilities of the random walks. Thermodynamic properties, such as the isothermal compressibility, are also interpreted in terms of random-walk concepts.
The random-walk formulation is then extended to include the study of ordered phases in lattice-gas models and hence order-disorder transitions in these systems. Also, an asymptotic form for the lattice Green's function is derived to illustrate how the form of decay of the total correlation function at large distances depends on the properties of the potential function.
To show that the random-walk interpretation of the Ornstein-Zernike relation is not restricted to lattice systems, we define analogous random-walk functions for continuous space. These lead to a straight-forward generalization of most expressions for discrete space-; the relationship between the continuous-space total correlation and Green's functions has the same form as that for the lattice systems. We also explore the possibility of obtaining random-walk properties of a (lattice or continuous-space) system, not from the existing approximate expressions for the direct correlation function, but instead from a generalised Ornstein-Zernike relation based on a maximum principle of classical statistical mechanics.
Finally, we choose a few specific lattice-gas models to show how the method describes their different properties, such as the behaviour of the total correlation function or that of an order- disorder phase transition.
1974-01-01T00:00:00ZNieto, Alberto RobledoIt has been found elsewhere that when approximate relations for the two-particle correlation functions of classical statistical mechanics, such as the Percus-Yevick and the mean-spherical approximations, are applied to the lattice gas models with nearest-neighbour interactions simple expressions are obtained for the total correlation function in terms of the lattice Green's function. Since many of the properties of random walks on a lattice can be described by the lattice Green's function, it follows that these systems, at least when treated under these approximations, may be analysed in terms of the language of random walks.
Here the theory of random walks on lattices is appropriately extended to show that the relationship between the correlation functions and the lattice Green's function is not dependent upon the employment of these approximations, nor to a particular range or form of the potential function. Instead, this relationship is shown to be an alternative form of the Ornstein-Zernike relation between the direct and total correlation functions. The direct correlation function is directly related to the probability of a single step, whereas the total correlation function is given by the first-passage- time probabilities of the random walks. Thermodynamic properties, such as the isothermal compressibility, are also interpreted in terms of random-walk concepts.
The random-walk formulation is then extended to include the study of ordered phases in lattice-gas models and hence order-disorder transitions in these systems. Also, an asymptotic form for the lattice Green's function is derived to illustrate how the form of decay of the total correlation function at large distances depends on the properties of the potential function.
To show that the random-walk interpretation of the Ornstein-Zernike relation is not restricted to lattice systems, we define analogous random-walk functions for continuous space. These lead to a straight-forward generalization of most expressions for discrete space-; the relationship between the continuous-space total correlation and Green's functions has the same form as that for the lattice systems. We also explore the possibility of obtaining random-walk properties of a (lattice or continuous-space) system, not from the existing approximate expressions for the direct correlation function, but instead from a generalised Ornstein-Zernike relation based on a maximum principle of classical statistical mechanics.
Finally, we choose a few specific lattice-gas models to show how the method describes their different properties, such as the behaviour of the total correlation function or that of an order- disorder phase transition.Contributions to the theory of Ockham algebrasFang, Jiehttp://hdl.handle.net/10023/137202019-04-01T08:37:58Z1997-01-01T00:00:00ZIn the first part of this thesis we consider particular ordered sets (connected and of small height) and determine the cardinality of the corresponding dual MS - algebra and of its set of fixed points.
The remainder of the thesis is devoted to a study of congruences of Ockham algebras and a generalised variety K𝜔 of Ockham algebras that contains all of the Berman varieties K[sub]p,[sub]q. In particular we consider the congruences [sub]i(i = 1, 2,...) defined on an Ockham algebra (L; f) by
(x, y) ∊ [sub]i ⇔ fⁱ(x)= fⁱ(y)
and show that (L; f) ∊ K𝜔 is subdirectly irreducible if and only if the lattice of congruences of L reduces to the chain
𝜔 = 𝝫₀ ≤ 𝝫₁≤ 𝝫₂≤ … ≤𝝫𝜔<𝞲
Where 𝝫𝜔 = ⌵ [sub]i≥0𝝫i. Finally we obtain a characterisation of the finite simple Ockham algebras.
1997-01-01T00:00:00ZFang, JieIn the first part of this thesis we consider particular ordered sets (connected and of small height) and determine the cardinality of the corresponding dual MS - algebra and of its set of fixed points.
The remainder of the thesis is devoted to a study of congruences of Ockham algebras and a generalised variety K𝜔 of Ockham algebras that contains all of the Berman varieties K[sub]p,[sub]q. In particular we consider the congruences [sub]i(i = 1, 2,...) defined on an Ockham algebra (L; f) by
(x, y) ∊ [sub]i ⇔ fⁱ(x)= fⁱ(y)
and show that (L; f) ∊ K𝜔 is subdirectly irreducible if and only if the lattice of congruences of L reduces to the chain
𝜔 = 𝝫₀ ≤ 𝝫₁≤ 𝝫₂≤ … ≤𝝫𝜔<𝞲
Where 𝝫𝜔 = ⌵ [sub]i≥0𝝫i. Finally we obtain a characterisation of the finite simple Ockham algebras.The descent algebras of Coxeter groupsVan WIlligenburg, Stephaniehttp://hdl.handle.net/10023/137132019-04-01T08:38:05Z1997-01-01T00:00:00ZA descent algebra is a subalgebra of the group algebra of a Coxeter group. They were first defined over a field of characteristic zero. In this thesis, the main areas of research to be addressed are;
1. The formulation of a rule for multiplying two elements of descent algebra of the Coxeter groups of type D.
2. The identification of properties exhibited by descent algebras over a field of prime characteristic.
In addressing the first, a framework which exploits the specific properties of Coxeter groups is set up. With this framework, a new justification is given for existing rules for multiplying together two elements in the descent algebras of the Coxeter groups of type A and B. This framework is then used to derive a new multiplication rule for the descent algebra of the Coxeter groups of type D.
To address the second, a descent algebra over a field of prime characteristic, p, is defined. A homomorphism into the algebra of generalised p-modular characters is then described. This homomorphism is then used to obtain the radical, and allows the irreducible modules of the descent algebra to be determined.
Results from the two areas addressed are then exploited to give an explicit description of the radical of the descent algebra of the symmetric groups, over a finite field. In this instance, the nilpotency index of the radical and the irreducible representations are also described. Similarly, the descent algebra of the hyper-octahedral groups, over a finite field, has its radical, nilpotency index, and irreducible representations explicitly determined.
1997-01-01T00:00:00ZVan WIlligenburg, StephanieA descent algebra is a subalgebra of the group algebra of a Coxeter group. They were first defined over a field of characteristic zero. In this thesis, the main areas of research to be addressed are;
1. The formulation of a rule for multiplying two elements of descent algebra of the Coxeter groups of type D.
2. The identification of properties exhibited by descent algebras over a field of prime characteristic.
In addressing the first, a framework which exploits the specific properties of Coxeter groups is set up. With this framework, a new justification is given for existing rules for multiplying together two elements in the descent algebras of the Coxeter groups of type A and B. This framework is then used to derive a new multiplication rule for the descent algebra of the Coxeter groups of type D.
To address the second, a descent algebra over a field of prime characteristic, p, is defined. A homomorphism into the algebra of generalised p-modular characters is then described. This homomorphism is then used to obtain the radical, and allows the irreducible modules of the descent algebra to be determined.
Results from the two areas addressed are then exploited to give an explicit description of the radical of the descent algebra of the symmetric groups, over a finite field. In this instance, the nilpotency index of the radical and the irreducible representations are also described. Similarly, the descent algebra of the hyper-octahedral groups, over a finite field, has its radical, nilpotency index, and irreducible representations explicitly determined.Certain classes of group presentationsVatansever, Bilalhttp://hdl.handle.net/10023/137092019-04-01T08:38:32Z1993-01-01T00:00:00ZIn Chapter two we look at the class
F(n) = <R, S | Rⁿ = Sⁿ = (Rᵃ¹Sᵇ¹)ˣ¹(Rᶜ¹Sᵈ¹)ʸ¹(Rᵃ²Sᵇ²)ˣ² (Rᶜ²Sᵈ²)ʸ² …(RᵃᵐSᵇᵐ)ˣᵐ (RᶜᵐSᵈᵐ)ʸᵐ = 1 >.
For some values of n, a[sub]i , b[sub]i, d[sub]i, x[sub]i, y[sub]i we give results on these groups where we have been able to determine their order, either finite or infinite. In the last section in Chapter two we study two classes of groups generated by A and B and subject to the following relations:
Relations for class 1:
A⁴ = 1, B⁴ = 1, (B(AB)²)⁴ = 1, (B(BA)⁶)⁴ = 1, (B(BA)¹⁴)⁴ = 1, …,
B(BA)⁽²⁽ⁿ⁻¹⁾ᐟ²-2)⁴ = 1
A⁻¹B⁻¹)²⁽ⁿ⁻³⁾ᐟ²B(BA)⁽²⁽ⁿ⁻¹⁾ᐟ²-2)B(BA)⁽²⁽ⁿ⁻³⁾ᐟ²B⁻¹(A⁻¹B⁻¹)²⁽ⁿ⁻¹⁾ᐟ²-2) B⁻¹
A⁻¹B⁻¹)²⁽ⁿ⁺¹⁾ᐟ²-3) A(BA)⁽²⁽ⁿ⁻¹⁾ᐟ²-1)B⁻¹= 1
(BA)²⁽ⁿ⁻¹⁾ᐟ² B⁻¹(A⁻¹B⁻¹)²⁽ⁿ⁻¹⁾ᐟ²-2) B⁻¹(A⁻¹B⁻¹)²⁽ⁿ⁺¹⁾ᐟ²-3) A² =1
Relations for class 2:
A⁴ = 1, B⁴ = 1, (B(AB)²)⁴ = 1, (B(BA)⁶)⁴ = 1, (B(BA)¹⁴)⁴ = 1, …, B(BA)⁽²⁽ⁿᐟ²⁻²⁾)⁴ = 1 , B⁻¹(BA)² ⁽ⁿ⁻²⁾ᐟ²B(BA) ⁽²ⁿᐟ²⁻²⁾ B(A⁻¹B⁻¹)²⁽ⁿ⁻²⁾ᐟ²-1) = 1, (BA) ⁽²ⁿᐟ²+2⁽ⁿ⁻²⁾ᐟ²+2)B(BA) ⁽²ⁿᐟ²-2)B(A⁻¹B⁻¹)²⁽ⁿ⁻²⁾ᐟ²-1)A² =1.
The groups in the first class turn out to be the cyclic group of order 2 and the groups in the second class turn out to be metabelian groups of order 4. (2ⁿᐟ²-1)² . Moreover the derived group of the groups in the second class is the direct product of two copies of a cyclic group of order (2ⁿᐟ²-1)². In Chapter three we study the groups with a presentation of the form:
<A,B|A⁴ = 1, Bⁿ = 1, AⁱBʲAᵏBᵗ =1
and determine all possibilities with conditions: j+t = 0 and i,k ∊ { + 1, 2 }.
Also in the second section of Chapter three we study the groups with a presentation of the form:
<A,B | A⁴ = 1, Bⁿ =1, AⁱBʲAᵏBᵗA ᵐBᵖ =1>
and determine some of the possibilities with conditions: j = l,t = l,p = -2 and i,k,m ∊ ℤ. In Chapter four we give new efficient presentations for the groups PSL(2,p), where p is an odd prime, p ∊ { 5,7,11,13,17,19,23,29,31,37, 41,43,53,59,79,83,89,109,139,229 }. We give permutation generators for these groups which satisfy our efficient presentation. Also we give new efficient presentations for PSL(2,p), where p is a prime power and p ∊ { 9,25,27,49,169}. Also in Chapter four, permutation generators are given for these groups which satisfy our presentations. In Chapter five we give new efficient presentations for the groups SL(2,p), where p is an odd prime and p ∊ { 5,7,11,13,17,19,23,29,31,41, 43,53,79,89,109,139,229 }. Also we give new efficient presentations for the groups SL(2,p), where p is an prime power and p ∊ { 8,16,25,27,49,169 }. In Chapter six we study the class of groups with the presentation
<a,b |aᵖ =1, bᵐ⁺ᵖa⁻ᵐbᵐa⁻ᵐ =1, (ab)² = 1>
,p an odd number and m ∊ ℤ. For some values of p and m these groups have connections with the groups PSL(2,p). In Chapter 7 we attempt to show the efficiency of PSL(2, ℤ[sub]n ) x PSL(2, ℤ[sub]m). For some values of n and m we give efficient presentation for these groups. In the same chapter we also attempt to show the efficiency of PSL(2, ℤ [sub]p) x PSL(2,32). For some values of p we give an efficient presentation for these groups. In the last section of the thesis we give efficient presentations for the following direct products
(i) PSL(2,5) X PSL(2,3²)
(ii) PSL(2,7) X PSL(2,3²)
(iii) PSL(2,5) X PSL(2,3³)
Also in the last section of the thesis the structure of a perfect group of order 161280 is investigated.
1993-01-01T00:00:00ZVatansever, BilalIn Chapter two we look at the class
F(n) = <R, S | Rⁿ = Sⁿ = (Rᵃ¹Sᵇ¹)ˣ¹(Rᶜ¹Sᵈ¹)ʸ¹(Rᵃ²Sᵇ²)ˣ² (Rᶜ²Sᵈ²)ʸ² …(RᵃᵐSᵇᵐ)ˣᵐ (RᶜᵐSᵈᵐ)ʸᵐ = 1 >.
For some values of n, a[sub]i , b[sub]i, d[sub]i, x[sub]i, y[sub]i we give results on these groups where we have been able to determine their order, either finite or infinite. In the last section in Chapter two we study two classes of groups generated by A and B and subject to the following relations:
Relations for class 1:
A⁴ = 1, B⁴ = 1, (B(AB)²)⁴ = 1, (B(BA)⁶)⁴ = 1, (B(BA)¹⁴)⁴ = 1, …,
B(BA)⁽²⁽ⁿ⁻¹⁾ᐟ²-2)⁴ = 1
A⁻¹B⁻¹)²⁽ⁿ⁻³⁾ᐟ²B(BA)⁽²⁽ⁿ⁻¹⁾ᐟ²-2)B(BA)⁽²⁽ⁿ⁻³⁾ᐟ²B⁻¹(A⁻¹B⁻¹)²⁽ⁿ⁻¹⁾ᐟ²-2) B⁻¹
A⁻¹B⁻¹)²⁽ⁿ⁺¹⁾ᐟ²-3) A(BA)⁽²⁽ⁿ⁻¹⁾ᐟ²-1)B⁻¹= 1
(BA)²⁽ⁿ⁻¹⁾ᐟ² B⁻¹(A⁻¹B⁻¹)²⁽ⁿ⁻¹⁾ᐟ²-2) B⁻¹(A⁻¹B⁻¹)²⁽ⁿ⁺¹⁾ᐟ²-3) A² =1
Relations for class 2:
A⁴ = 1, B⁴ = 1, (B(AB)²)⁴ = 1, (B(BA)⁶)⁴ = 1, (B(BA)¹⁴)⁴ = 1, …, B(BA)⁽²⁽ⁿᐟ²⁻²⁾)⁴ = 1 , B⁻¹(BA)² ⁽ⁿ⁻²⁾ᐟ²B(BA) ⁽²ⁿᐟ²⁻²⁾ B(A⁻¹B⁻¹)²⁽ⁿ⁻²⁾ᐟ²-1) = 1, (BA) ⁽²ⁿᐟ²+2⁽ⁿ⁻²⁾ᐟ²+2)B(BA) ⁽²ⁿᐟ²-2)B(A⁻¹B⁻¹)²⁽ⁿ⁻²⁾ᐟ²-1)A² =1.
The groups in the first class turn out to be the cyclic group of order 2 and the groups in the second class turn out to be metabelian groups of order 4. (2ⁿᐟ²-1)² . Moreover the derived group of the groups in the second class is the direct product of two copies of a cyclic group of order (2ⁿᐟ²-1)². In Chapter three we study the groups with a presentation of the form:
<A,B|A⁴ = 1, Bⁿ = 1, AⁱBʲAᵏBᵗ =1
and determine all possibilities with conditions: j+t = 0 and i,k ∊ { + 1, 2 }.
Also in the second section of Chapter three we study the groups with a presentation of the form:
<A,B | A⁴ = 1, Bⁿ =1, AⁱBʲAᵏBᵗA ᵐBᵖ =1>
and determine some of the possibilities with conditions: j = l,t = l,p = -2 and i,k,m ∊ ℤ. In Chapter four we give new efficient presentations for the groups PSL(2,p), where p is an odd prime, p ∊ { 5,7,11,13,17,19,23,29,31,37, 41,43,53,59,79,83,89,109,139,229 }. We give permutation generators for these groups which satisfy our efficient presentation. Also we give new efficient presentations for PSL(2,p), where p is a prime power and p ∊ { 9,25,27,49,169}. Also in Chapter four, permutation generators are given for these groups which satisfy our presentations. In Chapter five we give new efficient presentations for the groups SL(2,p), where p is an odd prime and p ∊ { 5,7,11,13,17,19,23,29,31,41, 43,53,79,89,109,139,229 }. Also we give new efficient presentations for the groups SL(2,p), where p is an prime power and p ∊ { 8,16,25,27,49,169 }. In Chapter six we study the class of groups with the presentation
<a,b |aᵖ =1, bᵐ⁺ᵖa⁻ᵐbᵐa⁻ᵐ =1, (ab)² = 1>
,p an odd number and m ∊ ℤ. For some values of p and m these groups have connections with the groups PSL(2,p). In Chapter 7 we attempt to show the efficiency of PSL(2, ℤ[sub]n ) x PSL(2, ℤ[sub]m). For some values of n and m we give efficient presentation for these groups. In the same chapter we also attempt to show the efficiency of PSL(2, ℤ [sub]p) x PSL(2,32). For some values of p we give an efficient presentation for these groups. In the last section of the thesis we give efficient presentations for the following direct products
(i) PSL(2,5) X PSL(2,3²)
(ii) PSL(2,7) X PSL(2,3²)
(iii) PSL(2,5) X PSL(2,3³)
Also in the last section of the thesis the structure of a perfect group of order 161280 is investigated.Semigroups with length morphismsSaunders, Bryan Jameshttp://hdl.handle.net/10023/137062019-04-01T08:38:25Z1998-01-01T00:00:00ZThe class of metrical semigroups is defined as the set consisting of those semigroups which can be homomorphically mapped into the semigroup of natural numbers (without zero) under addition.
The finitely generated members of this class are characterised and the infinitely generated case is discussed. A semigroup is called locally metrical if every finitely generated subsemigroup is metrical.
The classical Green's relations are trivial on any metrical semigroup. Generalisations 𝓗+, 𝓛+ and 𝓡+ of the Green's relations are defined and it is shown that for any cancellative metrical semigroup, S, 𝓗 + is " as big as possible " if and only if S is isomorphic to a special type of semidirect product of 𝗡 and a group. Lyndon's characterisation of free groups by length functions is discussed andalink between length functions, metrical semigroups and semigroups embeddable into free semigroups is investigated. Next the maximal locally metrical ideal of a semigroup is discussed, and the class of t-compressible semigroups is defined as the set consisting of those semigroups that have no locally metrical ideal. The class of t-compressible semigroups is seen to contain the classes of regular and simple semigroups. Finally it is shown that a large class of semigroups can be decomposed into a chain of locally metrical ideals together with a t-compressible semigroup.
1998-01-01T00:00:00ZSaunders, Bryan JamesThe class of metrical semigroups is defined as the set consisting of those semigroups which can be homomorphically mapped into the semigroup of natural numbers (without zero) under addition.
The finitely generated members of this class are characterised and the infinitely generated case is discussed. A semigroup is called locally metrical if every finitely generated subsemigroup is metrical.
The classical Green's relations are trivial on any metrical semigroup. Generalisations 𝓗+, 𝓛+ and 𝓡+ of the Green's relations are defined and it is shown that for any cancellative metrical semigroup, S, 𝓗 + is " as big as possible " if and only if S is isomorphic to a special type of semidirect product of 𝗡 and a group. Lyndon's characterisation of free groups by length functions is discussed andalink between length functions, metrical semigroups and semigroups embeddable into free semigroups is investigated. Next the maximal locally metrical ideal of a semigroup is discussed, and the class of t-compressible semigroups is defined as the set consisting of those semigroups that have no locally metrical ideal. The class of t-compressible semigroups is seen to contain the classes of regular and simple semigroups. Finally it is shown that a large class of semigroups can be decomposed into a chain of locally metrical ideals together with a t-compressible semigroup.Infinite transformation semigroupsMarques, Maria Paulahttp://hdl.handle.net/10023/137052019-04-01T08:38:15Z1983-01-01T00:00:00ZIn this thesis some topics in the field of Infinite Transformation Semigroups are investigated.
In 1966 Howie considered the full transformation semigroup 𝓣 (x) on an infinite set x of cardinality m. For each 𝝰 in 𝓣 (x) he defined defect of 𝝰 = def 𝝰 and collapse of 𝝰= C(a) to be the sets X \ X 𝝰 and { 𝓍 ∊ x : (∃∊ x, y ≠ 𝓍) X𝝰 = Y𝝰 }, respectively. Later, in 1981 he introduced the set
S[sub]m̱ = {𝝰 ∊ 𝓣(x): |def 𝝰 | = | c(𝝰) | = | ran 𝝰 | = m, |y 𝝰 [super]-1 | <m,
(∀ y ∊ ran 𝝰) }
which is a subsemigroup of 𝓣 (x) provided the cardinal m is regular. Taking m to be a regular cardinal number, Howie proved that S[sub]m̱ is then a bisimple, idempotent-generated semigroup of depth 4. Next he considered the congruence defined in S[sub]m̱ by
△[sub]m̱ = {(𝝰, β) ∊ S[sub]m̱ x S[sub]m̱ : max (|D(𝝰, β) 𝝰| , | D((𝝰, β) β | ) < m̱ }
where D(𝝰, β) = { 𝓍 ∊ X : 𝓍 𝝰 ≠ 𝓍β } and showed that S[sub]m̱* = S[sub]m̱/ △[sub]m̱ is a bisimple, congruence-free and idempotent-generated semigroup of depth 4.
In this thesis comparable results are obtained for the semigroup P[sub]m̱ which is the top principal factor of the semigroup
𝓠[sub]m̱ = {𝝰 ∊ 𝓣(x): |def 𝝰 | = | c(𝝰) | = m̱}
Here it is no longer necessary to restrict to a regular cardinal m̱. The set S[sub]m̱ considered by Howie fails to be a subsemigroup of 𝓣 (𝓍) if m̱ is not regular. It is shown that in this case <S[sub]m̱ > = O[sub]m̱ . In the case where m̱ = 𝓍₀ (a regular cardinal) it is shown that △[sub]𝓍₀ is the only proper congruence on S[sub]m̱.
Within the symmetric inverse semigroup 𝓣(𝓍), the Baer-Levi semigroup B of type (m̱, m̱) on X is considered and a dual B* found. The products BB* and B*B are investigated and the semigroup Km̱ = <B*B> is described. The top principal factor of Km̱ is denoted by Lm̱ and it is shown that Lm̱ = B*B ⋃ {O}. On the set Lm̱ a congruence δ[sub]m̱, closely analogous to the congruence △[sub]m̱ defined above, is considered, and it is shown that Lm̱ / δ[sub]m̱ is a o-bisimple, inverse and nilpotent-generated semigroup.
Finally, two embedding theorems for inverse semigroups and semigroups in general are presented. The cardinalities of some of the semigroups introduced in this thesis are studied.
1983-01-01T00:00:00ZMarques, Maria PaulaIn this thesis some topics in the field of Infinite Transformation Semigroups are investigated.
In 1966 Howie considered the full transformation semigroup 𝓣 (x) on an infinite set x of cardinality m. For each 𝝰 in 𝓣 (x) he defined defect of 𝝰 = def 𝝰 and collapse of 𝝰= C(a) to be the sets X \ X 𝝰 and { 𝓍 ∊ x : (∃∊ x, y ≠ 𝓍) X𝝰 = Y𝝰 }, respectively. Later, in 1981 he introduced the set
S[sub]m̱ = {𝝰 ∊ 𝓣(x): |def 𝝰 | = | c(𝝰) | = | ran 𝝰 | = m, |y 𝝰 [super]-1 | <m,
(∀ y ∊ ran 𝝰) }
which is a subsemigroup of 𝓣 (x) provided the cardinal m is regular. Taking m to be a regular cardinal number, Howie proved that S[sub]m̱ is then a bisimple, idempotent-generated semigroup of depth 4. Next he considered the congruence defined in S[sub]m̱ by
△[sub]m̱ = {(𝝰, β) ∊ S[sub]m̱ x S[sub]m̱ : max (|D(𝝰, β) 𝝰| , | D((𝝰, β) β | ) < m̱ }
where D(𝝰, β) = { 𝓍 ∊ X : 𝓍 𝝰 ≠ 𝓍β } and showed that S[sub]m̱* = S[sub]m̱/ △[sub]m̱ is a bisimple, congruence-free and idempotent-generated semigroup of depth 4.
In this thesis comparable results are obtained for the semigroup P[sub]m̱ which is the top principal factor of the semigroup
𝓠[sub]m̱ = {𝝰 ∊ 𝓣(x): |def 𝝰 | = | c(𝝰) | = m̱}
Here it is no longer necessary to restrict to a regular cardinal m̱. The set S[sub]m̱ considered by Howie fails to be a subsemigroup of 𝓣 (𝓍) if m̱ is not regular. It is shown that in this case <S[sub]m̱ > = O[sub]m̱ . In the case where m̱ = 𝓍₀ (a regular cardinal) it is shown that △[sub]𝓍₀ is the only proper congruence on S[sub]m̱.
Within the symmetric inverse semigroup 𝓣(𝓍), the Baer-Levi semigroup B of type (m̱, m̱) on X is considered and a dual B* found. The products BB* and B*B are investigated and the semigroup Km̱ = <B*B> is described. The top principal factor of Km̱ is denoted by Lm̱ and it is shown that Lm̱ = B*B ⋃ {O}. On the set Lm̱ a congruence δ[sub]m̱, closely analogous to the congruence △[sub]m̱ defined above, is considered, and it is shown that Lm̱ / δ[sub]m̱ is a o-bisimple, inverse and nilpotent-generated semigroup.
Finally, two embedding theorems for inverse semigroups and semigroups in general are presented. The cardinalities of some of the semigroups introduced in this thesis are studied.Idempotents, nilpotents, rank and order in finite transformation semigroupsGarba, Goje Ubahttp://hdl.handle.net/10023/137032019-04-01T08:38:09Z1992-01-01T00:00:00ZLet E, E₁ denote, respectively, the set of singular idempotents in T[sub]n (the semigroup of all full transformations on a finite set X[sub]n = {1,..., n}) and the set of idempotents of defect 1. For a singular element 𝑎 in Tn, let k(𝑎),k₁ (𝑎) be defined by the properties
𝑎 ∈ Eᵏ⁽ᵃ⁾, 𝑎 ∉ Eᵏ⁽ᵃ⁾⁻¹,
𝑎 ∈ E₁ᵏ¹⁽ᵃ⁾, 𝑎 ∉ E₁ᵏ¹⁽ᵃ⁾⁻¹.
In this Thesis, we obtain results analogous to those of Iwahori (1977), Howie (1980), Saito (1989) and Howie, Lusk and McFadden (1990) concerning the values of k(𝑎) and k₁(𝑎) for the partial transformation semigroup P[sub]n. The analogue of Howie and McFadden's (1990) result on the rank of the semigroup K(n,r) = { 𝑎 ∈ T [sub]n: |im 𝑎 | ≤ r,2 ≤ r ≤ n-1} is also obtained.
The nilpotent-generated subsemigroup of P[sub]n was characterised by Sullivan in 1987. In this work, we have obtained its depth and rank.
Nilpotents in IO[sub]n and PO[sub]n (the semigroup of all partial one-one order-preserving maps, and all partial order-preserving maps) are studied. A characterisation of their nilpotent-generated subsemigroups is obtained. So also are their depth and rank. We have also characterised their nilpotent-generated subsemigroup for the infinite set X = {1,2,...}. The rank of the semigroup L(n,r) = {a ∈ S : |im 𝑎 | ≤r, 1 ≤ r ≤ n - 2} is investigated for S = O[sub]n,PO[sub]n,SPO[sub]n and I[sub]n (where O[sub]n is the semigroup of all order-preserving full transformations, SPO[sub]n the semigroup of all strictly partial order- preserving maps, and In the semigroup of one-one partial transformation).
1992-01-01T00:00:00ZGarba, Goje UbaLet E, E₁ denote, respectively, the set of singular idempotents in T[sub]n (the semigroup of all full transformations on a finite set X[sub]n = {1,..., n}) and the set of idempotents of defect 1. For a singular element 𝑎 in Tn, let k(𝑎),k₁ (𝑎) be defined by the properties
𝑎 ∈ Eᵏ⁽ᵃ⁾, 𝑎 ∉ Eᵏ⁽ᵃ⁾⁻¹,
𝑎 ∈ E₁ᵏ¹⁽ᵃ⁾, 𝑎 ∉ E₁ᵏ¹⁽ᵃ⁾⁻¹.
In this Thesis, we obtain results analogous to those of Iwahori (1977), Howie (1980), Saito (1989) and Howie, Lusk and McFadden (1990) concerning the values of k(𝑎) and k₁(𝑎) for the partial transformation semigroup P[sub]n. The analogue of Howie and McFadden's (1990) result on the rank of the semigroup K(n,r) = { 𝑎 ∈ T [sub]n: |im 𝑎 | ≤ r,2 ≤ r ≤ n-1} is also obtained.
The nilpotent-generated subsemigroup of P[sub]n was characterised by Sullivan in 1987. In this work, we have obtained its depth and rank.
Nilpotents in IO[sub]n and PO[sub]n (the semigroup of all partial one-one order-preserving maps, and all partial order-preserving maps) are studied. A characterisation of their nilpotent-generated subsemigroups is obtained. So also are their depth and rank. We have also characterised their nilpotent-generated subsemigroup for the infinite set X = {1,2,...}. The rank of the semigroup L(n,r) = {a ∈ S : |im 𝑎 | ≤r, 1 ≤ r ≤ n - 2} is investigated for S = O[sub]n,PO[sub]n,SPO[sub]n and I[sub]n (where O[sub]n is the semigroup of all order-preserving full transformations, SPO[sub]n the semigroup of all strictly partial order- preserving maps, and In the semigroup of one-one partial transformation).On a family of semigroup congruencesKopamu, Samuel Joseph Lyambianhttp://hdl.handle.net/10023/136992019-04-01T08:38:04Z1996-01-01T00:00:00ZWe introduce in this thesis a new family of semigroup congruences, and we set out to prove that it is worth studying them for the following very important reasons:
(a) that it provides an alternative way of studying algebraic structures of semigroups, thus shedding new light over semigroup structures already known, and it also provides new information about other structures not formerly understood;
(b) that it is useful for constructing new semigroups, hence producing new and interesting classes of semigroups from known classes; and
(c) that it is useful for classifying semigroups, particularly in describing lattices formed by semigroup species such as varieties, pseudovarieties, existence varieties etc.
This interesting family of congruences is described as follows: for any semigroup S, and any ordered pair (n,m) of non-negative integers, define ⦵(n,m) = {(a,b): uav = ubv, for all ⋿Sn and v ⋿Sm}, and we make the convention that S¹ = S and that S0 denotes the set containing only the empty word. The particular cases ⦵(0,1), ⦵(1,0) and ⦵(0,0) were considered by the author in his M.Sc. thesis (1991). In fact, one can recognise ⦵(1,0) to be the well known kernel of the right regular representation of S. It turns out that if S is reductive (for example, if S is a monoid), then ⦵(i,j) is equal to ⦵(0,0) - the identity relation on S, for every (i,j).
After developing the tools required for the latter part of the thesis in Chapters 0-2, in Chapter 3 we introduce a new class of semigroups - the class of all structurally regular semigroups. Making use of a new Mal'tsev-type product, in Chapters 4,5,6 and 7, we describe the lattices formed by certain varieties of structurally regular semigroups.
Many interesting open problems are posed throughout the thesis, and brief literature reviews are inserted in the text where appropriate.
1996-01-01T00:00:00ZKopamu, Samuel Joseph LyambianWe introduce in this thesis a new family of semigroup congruences, and we set out to prove that it is worth studying them for the following very important reasons:
(a) that it provides an alternative way of studying algebraic structures of semigroups, thus shedding new light over semigroup structures already known, and it also provides new information about other structures not formerly understood;
(b) that it is useful for constructing new semigroups, hence producing new and interesting classes of semigroups from known classes; and
(c) that it is useful for classifying semigroups, particularly in describing lattices formed by semigroup species such as varieties, pseudovarieties, existence varieties etc.
This interesting family of congruences is described as follows: for any semigroup S, and any ordered pair (n,m) of non-negative integers, define ⦵(n,m) = {(a,b): uav = ubv, for all ⋿Sn and v ⋿Sm}, and we make the convention that S¹ = S and that S0 denotes the set containing only the empty word. The particular cases ⦵(0,1), ⦵(1,0) and ⦵(0,0) were considered by the author in his M.Sc. thesis (1991). In fact, one can recognise ⦵(1,0) to be the well known kernel of the right regular representation of S. It turns out that if S is reductive (for example, if S is a monoid), then ⦵(i,j) is equal to ⦵(0,0) - the identity relation on S, for every (i,j).
After developing the tools required for the latter part of the thesis in Chapters 0-2, in Chapter 3 we introduce a new class of semigroups - the class of all structurally regular semigroups. Making use of a new Mal'tsev-type product, in Chapters 4,5,6 and 7, we describe the lattices formed by certain varieties of structurally regular semigroups.
Many interesting open problems are posed throughout the thesis, and brief literature reviews are inserted in the text where appropriate.Nonstandard quantum groups: twisting constructions and noncommutative differential geometryJacobs, Andrew D.http://hdl.handle.net/10023/136932019-04-01T08:37:32Z1998-01-01T00:00:00ZThe general subject of this thesis is quantum groups. The major original results are obtained in the particular areas of twisting constructions and noncommutative differential geometry.
Chapters 1 and 2 are intended to explain to the reader what are quantum groups. They are written in the form of a series of linked results and definitions. Chapter 1 reviews the theory of Lie algebras and Lie groups, focusing attention in particular on the classical Lie algebras and groups. Though none of the quoted results are due to the author, such a review, aimed specifically at setting up the paradigm which provides essential guidance in the theory of quantum groups, does not seem to have appeared already. In Chapter 2 the elements of the quantum group theory are recalled. Once again, almost none of the results are due to the author, though in Section 2.10, some results concerning the nonstandard Jordanian group are presented, by way of a worked example, which have not been published.
Chapter 3 concerns twisting constructions. We introduce a new class of 2-cocycles defined explicitly on the generators of certain multiparameter standard quantum groups. These allow us, through the process of twisting the familiar standard quantum groups, to generate new as well as previously known examples of non-standard quantum groups. In particular we are able to construct generalisations of both the Cremmer-Gervais deformation of SL(3) and the so called esoteric quantum groups of Fronsdal and Galindo in an explicit and straightforward manner.
In Chapter 4 we consider the differential calculus on Hopf algebras as introduced by Woronowicz. We classify all 4-dimensional first order bicovariant calculi on the Jordanian quantum group GL[sub]h,[sub]g(2) and all 3-dimensional first order bicovariant calculi on the Jordanian quantum group SL[sub]h(2). In both cases we assume that the bicovariant bimodules are generated as left modules by the differentials of the quantum group generators. It is found that there are 3 1-parameter families of 4-dimensional bicovariant first order calculi on GL[sub]h,[sub]g(2) and that there is a single, unique, 3-dimensional bicovariant calculus on SL[sub]h(2). This 3-dimensional calculus may be obtained through a classical-like reduction from any one of the three families of 4-dimensional calculi on GL[sub]h,[sub]g(2). Details of the higher order calculi and also the quantum Lie algebras are presented for all calculi. The quantum Lie algebra obtained from the bicovariant calculus on SL[sub]h(2) is shown to be isomorphic to the quantum Lie algebra we obtain as an ad-submodule within the Jordanian universal enveloping algebra U[sub]h(sl[sub]2(C)) and also through a consideration of the decomposition of the tensor product of two copies of the deformed adjoint module. We also obtain the quantum Killing form for this quantum Lie algebra.
1998-01-01T00:00:00ZJacobs, Andrew D.The general subject of this thesis is quantum groups. The major original results are obtained in the particular areas of twisting constructions and noncommutative differential geometry.
Chapters 1 and 2 are intended to explain to the reader what are quantum groups. They are written in the form of a series of linked results and definitions. Chapter 1 reviews the theory of Lie algebras and Lie groups, focusing attention in particular on the classical Lie algebras and groups. Though none of the quoted results are due to the author, such a review, aimed specifically at setting up the paradigm which provides essential guidance in the theory of quantum groups, does not seem to have appeared already. In Chapter 2 the elements of the quantum group theory are recalled. Once again, almost none of the results are due to the author, though in Section 2.10, some results concerning the nonstandard Jordanian group are presented, by way of a worked example, which have not been published.
Chapter 3 concerns twisting constructions. We introduce a new class of 2-cocycles defined explicitly on the generators of certain multiparameter standard quantum groups. These allow us, through the process of twisting the familiar standard quantum groups, to generate new as well as previously known examples of non-standard quantum groups. In particular we are able to construct generalisations of both the Cremmer-Gervais deformation of SL(3) and the so called esoteric quantum groups of Fronsdal and Galindo in an explicit and straightforward manner.
In Chapter 4 we consider the differential calculus on Hopf algebras as introduced by Woronowicz. We classify all 4-dimensional first order bicovariant calculi on the Jordanian quantum group GL[sub]h,[sub]g(2) and all 3-dimensional first order bicovariant calculi on the Jordanian quantum group SL[sub]h(2). In both cases we assume that the bicovariant bimodules are generated as left modules by the differentials of the quantum group generators. It is found that there are 3 1-parameter families of 4-dimensional bicovariant first order calculi on GL[sub]h,[sub]g(2) and that there is a single, unique, 3-dimensional bicovariant calculus on SL[sub]h(2). This 3-dimensional calculus may be obtained through a classical-like reduction from any one of the three families of 4-dimensional calculi on GL[sub]h,[sub]g(2). Details of the higher order calculi and also the quantum Lie algebras are presented for all calculi. The quantum Lie algebra obtained from the bicovariant calculus on SL[sub]h(2) is shown to be isomorphic to the quantum Lie algebra we obtain as an ad-submodule within the Jordanian universal enveloping algebra U[sub]h(sl[sub]2(C)) and also through a consideration of the decomposition of the tensor product of two copies of the deformed adjoint module. We also obtain the quantum Killing form for this quantum Lie algebra.Computing with simple groups: maximal subgroups and presentationsJamali, Ali-Rezahttp://hdl.handle.net/10023/136922019-04-01T08:37:08Z1989-01-01T00:00:00ZFor the non-abelian simple groups G of order up to 10⁶ , excluding the groups PSL(2,q), q > 9, the presentations in terms of an involution a and an element b of minimal order (with respect to a) such that G=<a,b> are well known. The presentations are complete in the sense that any pair (x,y) of generators of G satisfying x²=yᵐ=1, with m minimal, will satisfy the defining relations of just one presentation in the list. There are 106 such presentations.
Using a computer, we give generators for each maximal subgroup of the groups G. For each presentation of G, the generators of maximal subgroups are given as words in the group generators. Similarly generators for a Sylow p-subgroup of G, for each p, are given. For each group G, we give a representative for each conjugacy class of the group as a word in the group generators.
Minimal presentations for each Sylow p-subgroup of the groups G, and for most of the maximal subgroups of G are constructed. To obtain such presentations, the Schur multipliers of the underlying groups are calculated.
The same tasks are carried out for those groups PSL(2,q) of order less than 10⁶ which are included in the "ATLAS of finite groups". For these groups we consider a presentation on two generators x, y with x²=y³=1.
A finite group G is said to be efficient if it has a presentation on d generators and d+rank(M(G)) relations (for some d) where M(G) is the Schur multiplier of G. We show that the simple groups J₁, PSU(3,5) and M₂₂ are efficient. We also give efficient presentations for the direct products A₅xA₆, A₅xA₆,A₆xA₇ where Ĥ denotes the covering group of H.
1989-01-01T00:00:00ZJamali, Ali-RezaFor the non-abelian simple groups G of order up to 10⁶ , excluding the groups PSL(2,q), q > 9, the presentations in terms of an involution a and an element b of minimal order (with respect to a) such that G=<a,b> are well known. The presentations are complete in the sense that any pair (x,y) of generators of G satisfying x²=yᵐ=1, with m minimal, will satisfy the defining relations of just one presentation in the list. There are 106 such presentations.
Using a computer, we give generators for each maximal subgroup of the groups G. For each presentation of G, the generators of maximal subgroups are given as words in the group generators. Similarly generators for a Sylow p-subgroup of G, for each p, are given. For each group G, we give a representative for each conjugacy class of the group as a word in the group generators.
Minimal presentations for each Sylow p-subgroup of the groups G, and for most of the maximal subgroups of G are constructed. To obtain such presentations, the Schur multipliers of the underlying groups are calculated.
The same tasks are carried out for those groups PSL(2,q) of order less than 10⁶ which are included in the "ATLAS of finite groups". For these groups we consider a presentation on two generators x, y with x²=y³=1.
A finite group G is said to be efficient if it has a presentation on d generators and d+rank(M(G)) relations (for some d) where M(G) is the Schur multiplier of G. We show that the simple groups J₁, PSU(3,5) and M₂₂ are efficient. We also give efficient presentations for the direct products A₅xA₆, A₅xA₆,A₆xA₇ where Ĥ denotes the covering group of H.Semigroup presentationsIbrahim, Mohammed Ali Fayahttp://hdl.handle.net/10023/136892019-04-01T08:38:18Z1997-01-01T00:00:00ZIn this thesis we consider the following two fundamental problems for semigroup presentations:
1. Given a semigroup find a presentation defining it.
2. Given a presentation describe the semigroup defined by it.
We also establish other related results.
After an introduction in Chapter 1, we consider the first problem in Chapter 2, and establish a presentation for the commutative semigroup of integers Zpt. Dually, in Chapter 3 we consider the second problem and study presentations of semigroups related to the direct product of cyclic groups. In Chapter 4 we study presentations of semigroups related to dihedral groups and establish their V-classes structure in Chapter 5. In Chapter 6 we establish some results related to the Schutzenberger group which were suggested by our studies of the semigroup presentations in Chapters 3 and 4. Finally, in Chapter 7 we define and study new classes of semigroups which we call R, L-semi-commutative and semi-commutative semigroups and they were also suggested by our studies of the semigroup presentations in Chapters 3 and 4.
1997-01-01T00:00:00ZIbrahim, Mohammed Ali FayaIn this thesis we consider the following two fundamental problems for semigroup presentations:
1. Given a semigroup find a presentation defining it.
2. Given a presentation describe the semigroup defined by it.
We also establish other related results.
After an introduction in Chapter 1, we consider the first problem in Chapter 2, and establish a presentation for the commutative semigroup of integers Zpt. Dually, in Chapter 3 we consider the second problem and study presentations of semigroups related to the direct product of cyclic groups. In Chapter 4 we study presentations of semigroups related to dihedral groups and establish their V-classes structure in Chapter 5. In Chapter 6 we establish some results related to the Schutzenberger group which were suggested by our studies of the semigroup presentations in Chapters 3 and 4. Finally, in Chapter 7 we define and study new classes of semigroups which we call R, L-semi-commutative and semi-commutative semigroups and they were also suggested by our studies of the semigroup presentations in Chapters 3 and 4.Algorithms for subgroup presentations: computer implementation and applicationsHeggie, Patricia, M.http://hdl.handle.net/10023/136842019-04-01T08:39:30Z1991-01-01T00:00:00ZOne of the main algorithms of computational group theory is the Todd-Coxeter coset enumeration algorithm, which provides a systematic method for finding the index of a subgroup of a finitely presented group. This has been extended in various ways to provide not only the index of a subgroup, but also a presentation for the subgroup. These methods tie in with a technique introduced by Reidemeister in the 1920's and later improved by Schreier, now known as the Reidemeister-Schreier algorithm.
In this thesis we discuss some of these variants of the Todd-Coxeter algorithm and their inter-relation, and also look at existing computer implementations of these different techniques. We then go on to describe a new package for coset methods which incorporates various types of coset enumeration, including modified Todd- Coxeter methods and the Reidemeister-Schreier process. This also has the capability of carrying out Tietze transformation simplification. Statistics obtained from running the new package on a collection of test examples are given, and the various techniques compared.
Finally, we use these algorithms, both theoretically and as computer implementations, to investigate a particular class of finitely presented groups defined by the presentation: < a, b | aⁿ = b² = (ab-1) ß =1, ab² = ba²>. Some interesting results have been discovered about these groups for various values of β and n. For example, if n is odd, the groups turn out to be finite and metabelian, and if β= 3 or β= 4 the derived group has an order which is dependent on the values of n (mod 8) and n (mod 12) respectively.
1991-01-01T00:00:00ZHeggie, Patricia, M.One of the main algorithms of computational group theory is the Todd-Coxeter coset enumeration algorithm, which provides a systematic method for finding the index of a subgroup of a finitely presented group. This has been extended in various ways to provide not only the index of a subgroup, but also a presentation for the subgroup. These methods tie in with a technique introduced by Reidemeister in the 1920's and later improved by Schreier, now known as the Reidemeister-Schreier algorithm.
In this thesis we discuss some of these variants of the Todd-Coxeter algorithm and their inter-relation, and also look at existing computer implementations of these different techniques. We then go on to describe a new package for coset methods which incorporates various types of coset enumeration, including modified Todd- Coxeter methods and the Reidemeister-Schreier process. This also has the capability of carrying out Tietze transformation simplification. Statistics obtained from running the new package on a collection of test examples are given, and the various techniques compared.
Finally, we use these algorithms, both theoretically and as computer implementations, to investigate a particular class of finitely presented groups defined by the presentation: < a, b | aⁿ = b² = (ab-1) ß =1, ab² = ba²>. Some interesting results have been discovered about these groups for various values of β and n. For example, if n is odd, the groups turn out to be finite and metabelian, and if β= 3 or β= 4 the derived group has an order which is dependent on the values of n (mod 8) and n (mod 12) respectively.On the efficiency of finite groupsBrookes, Melaniehttp://hdl.handle.net/10023/136822019-04-01T08:37:29Z1996-01-01T00:00:00ZIn Chapter 2 of this thesis we look at methods for finding efficient presentations of the transitive permutation groups of degree ≤ 12. Chapter 3 gives efficient presentations for certain direct products of groups including PSL(2, P)2 SL(2, p) X SL(2, 8), PSL(2, p) x C2, for prime p ≥ 5 and PSL(2, 25)3. Chapter 4 introduces a new class of inefficient groups and Chapter 5 gives a brief survey of some of the open problems relating to the efficiency of finite groups.
1996-01-01T00:00:00ZBrookes, MelanieIn Chapter 2 of this thesis we look at methods for finding efficient presentations of the transitive permutation groups of degree ≤ 12. Chapter 3 gives efficient presentations for certain direct products of groups including PSL(2, P)2 SL(2, p) X SL(2, 8), PSL(2, p) x C2, for prime p ≥ 5 and PSL(2, 25)3. Chapter 4 introduces a new class of inefficient groups and Chapter 5 gives a brief survey of some of the open problems relating to the efficiency of finite groups.Application of the Todd-Coxeter coset enumeration algorithmCampbell, C. M. (Colin Matthew)http://hdl.handle.net/10023/135082019-04-01T08:39:35Z1975-01-01T00:00:00ZThis thesis is concerned with a topic in combinatorial group theory and, in particular, with a study of some groups with finite presentations. After preliminary definitions and theorems we describe the Todd-Coxeter coset enumeration algorithm and the modified Todd-Coxeter algorithm which shows that, given a finitely generated subgroup H of finite index in a finitely presented group G, we can find a presentation for H. We then give elementary examples illustrating the algorithms and include a discussion on the computer programmes that are to be used. In the main part of the thesis we investigate two classes of cyclically presented groups. Supposewhere w1 = w is a word in a1,a2,...,an, and wi+1 is obtained from wi by applying the permutation (1 2 ... n) to the suffices of the a's. The first class we investigate are the groups that is the groups G(l,m,n) are groups of type G2 (w). Secondly we investigate the Fibonacci-type groups H(r,n,k,s,h) obtained when, for some integers r,s,h > 1, k > O, the word w is given by Fibonacci groups being the special case given by k = s = h = 1. For both classes we begin by giving some homomorphisms and isomorphisms that may be obtained. We show, using the Todd-Coxeter algorithm when appropriate, that the six groups G(2,2,3), G(2,2,-3), G(-l,-l,4), G(2,3,-2), G(-2,2,-1) and G(-2,3,l) are finite non-metacyclic groups of deficiency zero, having orders 215.33, 28.33, 29.3.5, 23.33.7, 23.3.5.11 amd 23.36 respectively. We also show that the groups G(1-n, 6, n) where n = 1 mod 5 give an infinite series of non-metacyclic groups. We consider the structure of the non-metacyclic groups H(3,6,1,1,1) and H(3,6,5,l,2) both of order 1512, showing that neither is isomorphic to G(2, 3, -2) another non-metacyclic group of order 1512. In a paper on the Fibonacci groups D.L. Johnson, J.W. Wamsley and D. Wright pose two questions relating to the Fibonacci groups for the case r = 1 mod n, namely to find 2-generator 2-relation presentations for them and also their orders. We answer these questions and generalise the results to the class H(r,n,k,s,1) where it is shown that H(r,n,k,s,1) is metacyclic if (i) r = s mod n, (ii) (r,n) = 1, (iii) (r + k - 1, n) - 1, and a 2-generator 2-relation presentation is found for these groups. Further if (iv) (r,s) = 1, then we show that H(r,n,k,s,1) is a finite metacyclic group of order rn - sn. A possible generalisation to the groups H(r,n,k,s,h) is considered. Finally the metacyclic groups H(r,4,1,2,1), r odd are discussed.
1975-01-01T00:00:00ZCampbell, C. M. (Colin Matthew)This thesis is concerned with a topic in combinatorial group theory and, in particular, with a study of some groups with finite presentations. After preliminary definitions and theorems we describe the Todd-Coxeter coset enumeration algorithm and the modified Todd-Coxeter algorithm which shows that, given a finitely generated subgroup H of finite index in a finitely presented group G, we can find a presentation for H. We then give elementary examples illustrating the algorithms and include a discussion on the computer programmes that are to be used. In the main part of the thesis we investigate two classes of cyclically presented groups. Supposewhere w1 = w is a word in a1,a2,...,an, and wi+1 is obtained from wi by applying the permutation (1 2 ... n) to the suffices of the a's. The first class we investigate are the groups that is the groups G(l,m,n) are groups of type G2 (w). Secondly we investigate the Fibonacci-type groups H(r,n,k,s,h) obtained when, for some integers r,s,h > 1, k > O, the word w is given by Fibonacci groups being the special case given by k = s = h = 1. For both classes we begin by giving some homomorphisms and isomorphisms that may be obtained. We show, using the Todd-Coxeter algorithm when appropriate, that the six groups G(2,2,3), G(2,2,-3), G(-l,-l,4), G(2,3,-2), G(-2,2,-1) and G(-2,3,l) are finite non-metacyclic groups of deficiency zero, having orders 215.33, 28.33, 29.3.5, 23.33.7, 23.3.5.11 amd 23.36 respectively. We also show that the groups G(1-n, 6, n) where n = 1 mod 5 give an infinite series of non-metacyclic groups. We consider the structure of the non-metacyclic groups H(3,6,1,1,1) and H(3,6,5,l,2) both of order 1512, showing that neither is isomorphic to G(2, 3, -2) another non-metacyclic group of order 1512. In a paper on the Fibonacci groups D.L. Johnson, J.W. Wamsley and D. Wright pose two questions relating to the Fibonacci groups for the case r = 1 mod n, namely to find 2-generator 2-relation presentations for them and also their orders. We answer these questions and generalise the results to the class H(r,n,k,s,1) where it is shown that H(r,n,k,s,1) is metacyclic if (i) r = s mod n, (ii) (r,n) = 1, (iii) (r + k - 1, n) - 1, and a 2-generator 2-relation presentation is found for these groups. Further if (iv) (r,s) = 1, then we show that H(r,n,k,s,1) is a finite metacyclic group of order rn - sn. A possible generalisation to the groups H(r,n,k,s,h) is considered. Finally the metacyclic groups H(r,4,1,2,1), r odd are discussed.Centralisers and normalisers in symmetric and alternating groupsBilgiç, Huseyinhttp://hdl.handle.net/10023/135072019-04-01T08:37:24Z1998-01-01T00:00:00ZIn this thesis, we analyse the structure of the centraliser of an element and of the normaliser of a cyclic subgroup in both Sn and An. We show that the centraliser in Sn of a permutation can be written as a direct product of centralisers of regular permutations and that the centraliser of a regular permutation is a wreath product. In certain cases we prove that this wreath product splits as a direct product and we analyse the centre of the subgroup. We calculate the centraliser of a general permutation in An and show how this is related to the centralisers of regular permutations. We investigate the normaliser of the cyclic subgroup generated by an element of Sn and show how this is related to the centraliser of the permutation. We calculate the centre of the normaliser and investigate when the normaliser splits as a direct product. We carry out a similar investigation for normalisers of cyclic subgroups of An and investigate the relationship between normalisers in An and Sn. We give presentations for both centralisers and normalisers.
1998-01-01T00:00:00ZBilgiç, HuseyinIn this thesis, we analyse the structure of the centraliser of an element and of the normaliser of a cyclic subgroup in both Sn and An. We show that the centraliser in Sn of a permutation can be written as a direct product of centralisers of regular permutations and that the centraliser of a regular permutation is a wreath product. In certain cases we prove that this wreath product splits as a direct product and we analyse the centre of the subgroup. We calculate the centraliser of a general permutation in An and show how this is related to the centralisers of regular permutations. We investigate the normaliser of the cyclic subgroup generated by an element of Sn and show how this is related to the centraliser of the permutation. We calculate the centre of the normaliser and investigate when the normaliser splits as a direct product. We carry out a similar investigation for normalisers of cyclic subgroups of An and investigate the relationship between normalisers in An and Sn. We give presentations for both centralisers and normalisers.Some applications of computer algebra and interval mathematicsMonsi, Mansor Binhttp://hdl.handle.net/10023/135022019-04-01T08:37:12Z1988-01-01T00:00:00ZThis thesis contains some applications of Computer Algebra to unconstrained optimization and some applications of Interval Mathematics to the problem of simultaneously bounding the simple zeros of polynomials. Chapter 1 contains a brief introduction to Computer Algebra and Interval Mathematics, and several of the fundamental results from Interval Mathematics which are used in Chapters 4 and 5. Chapter 2 contains a survey of those features of the symbol manipulation package ALgLIB[Shew-85] which it is necessary to understand in order to use ALgLIB as explained in Chapter 3. Chapter 3 contains a description of Sisser's method [Sis-82a] for unconstrained minimization and several modifications thereof which are implemented using the pseudo-code of Dennis and Schnabel [DenS-83], and ALgLIB, Chapter 3 also contains numerical results corresponding to Sisser's method and its modifications for 7 examples. Chapter 4 contains a new algorithm PRSS for the simultaneous estimation of polynomial zeros and the corresponding interval form IRSS for simultaneously bounding real polynomial zeros. Comparisons are made with some related existing algorithms. Numerical results of the comparisons are also given in this chapter. Chapter 5 contains an application of an idea due to Neumaier [Neu-85] to the problem of constructing interval versions of point iterative procedures for the estimation of simple zeros of analytic functions. In particular, interval versions of some point iterative procedures for the simultaneous estimation of simple (complex) polynomial zeros are described. Finally, numerical results are given to show the efficiency of the new algorithm.
1988-01-01T00:00:00ZMonsi, Mansor BinThis thesis contains some applications of Computer Algebra to unconstrained optimization and some applications of Interval Mathematics to the problem of simultaneously bounding the simple zeros of polynomials. Chapter 1 contains a brief introduction to Computer Algebra and Interval Mathematics, and several of the fundamental results from Interval Mathematics which are used in Chapters 4 and 5. Chapter 2 contains a survey of those features of the symbol manipulation package ALgLIB[Shew-85] which it is necessary to understand in order to use ALgLIB as explained in Chapter 3. Chapter 3 contains a description of Sisser's method [Sis-82a] for unconstrained minimization and several modifications thereof which are implemented using the pseudo-code of Dennis and Schnabel [DenS-83], and ALgLIB, Chapter 3 also contains numerical results corresponding to Sisser's method and its modifications for 7 examples. Chapter 4 contains a new algorithm PRSS for the simultaneous estimation of polynomial zeros and the corresponding interval form IRSS for simultaneously bounding real polynomial zeros. Comparisons are made with some related existing algorithms. Numerical results of the comparisons are also given in this chapter. Chapter 5 contains an application of an idea due to Neumaier [Neu-85] to the problem of constructing interval versions of point iterative procedures for the estimation of simple zeros of analytic functions. In particular, interval versions of some point iterative procedures for the simultaneous estimation of simple (complex) polynomial zeros are described. Finally, numerical results are given to show the efficiency of the new algorithm.Proof diagrams and term rewriting with applications to computational algebraShand, Duncanhttp://hdl.handle.net/10023/134982019-04-01T08:37:52Z1997-01-01T00:00:00ZIn this thesis lessons learned from the use of computer algebra systems and machine assisted theorem provers are developed in order to give an insight into both the problems and their solutions. Many algorithms in computational algebra and automated deduction (for example Grobner basis computations and Knuth-Bendix completion) tend to produce redundant facts and can contain more than one proof of any particular fact. This thesis introduces proof diagrams in order to compare and contrast the proofs of facts which such procedures generate. Proof diagrams make it possible to analyse the effect of heuristics which can be used to guide implementations of such algorithms. An extended version of an inference system for Knuth-Bendix completion is introduced. It is possible to see that this extension characterises the applicability of critical pair criteria, which are heuristics used in completion. We investigate a number of executions of a completion procedure by analysing the associated proof diagrams. This leads to a better understanding of the heuristics used to control these examples. Derived rales of inference are also investigated in this thesis. This is done in the formalism of proof diagrams. Rewrite rules for proof diagrams are defined: this is motivated by the notion of a transformation tactic in the Nuprl proof development system. A method to automatically extract 'useful' derived inference rales is also discussed. 'Off the shelf' theorem provers, such as the Larch Prover and Otter, are compared to specialised programs from computational group theory. This analysis makes it possible to see where methods from automated deduction can improve on the tools which group theorists currently use. Problems which can be attacked with theorem provers but not with currently used specialised programs are also indicated. Tietze transformations, from group theory, are discussed. This makes it possible to link ideas used in Knuth-Bendix completion programs and group presentation simplification programs. Tietze transformations provide heuristics for more efficient and effective implementations of these programs.
1997-01-01T00:00:00ZShand, DuncanIn this thesis lessons learned from the use of computer algebra systems and machine assisted theorem provers are developed in order to give an insight into both the problems and their solutions. Many algorithms in computational algebra and automated deduction (for example Grobner basis computations and Knuth-Bendix completion) tend to produce redundant facts and can contain more than one proof of any particular fact. This thesis introduces proof diagrams in order to compare and contrast the proofs of facts which such procedures generate. Proof diagrams make it possible to analyse the effect of heuristics which can be used to guide implementations of such algorithms. An extended version of an inference system for Knuth-Bendix completion is introduced. It is possible to see that this extension characterises the applicability of critical pair criteria, which are heuristics used in completion. We investigate a number of executions of a completion procedure by analysing the associated proof diagrams. This leads to a better understanding of the heuristics used to control these examples. Derived rales of inference are also investigated in this thesis. This is done in the formalism of proof diagrams. Rewrite rules for proof diagrams are defined: this is motivated by the notion of a transformation tactic in the Nuprl proof development system. A method to automatically extract 'useful' derived inference rales is also discussed. 'Off the shelf' theorem provers, such as the Larch Prover and Otter, are compared to specialised programs from computational group theory. This analysis makes it possible to see where methods from automated deduction can improve on the tools which group theorists currently use. Problems which can be attacked with theorem provers but not with currently used specialised programs are also indicated. Tietze transformations, from group theory, are discussed. This makes it possible to link ideas used in Knuth-Bendix completion programs and group presentation simplification programs. Tietze transformations provide heuristics for more efficient and effective implementations of these programs.Tools and techniques for machine-assisted meta-theoryAdams, Andrew, 1969-http://hdl.handle.net/10023/133822019-04-01T08:37:15Z1997-01-01T00:00:00ZMachine-assisted formal proofs are becoming commonplace in certain fields of mathematics and theoretical computer science. New formal systems and variations on old ones are constantly invented. The meta-theory of such systems, i.e. proofs about the system as opposed to proofs within the system, are mostly done informally with a pen and paper. Yet the meta-theory of deductive systems is an area which would obviously benefit from machine support for formal proof. Is the software currently available sufficiently powerful yet easy enough to use to make machine assistance for formal meta-theory a viable proposition? This thesis presents work done by the author on formalizing proof theory from [DP97a] in various formal systems: SEQUEL [Tar93, Tar97], Isabelle [Pau94] and Coq [BB+96]. SEQUEL and Isabelle were found to be difficult to use for this type of work. In particular, the lack of automated production of induction principles in SEQUEL and Isabelle undermined confidence in the resulting formal proofs. Coq was found to be suitable for the formalisation methodology first chosen: the use of nameless dummy variables (de Bruijn indices) as pioneered in [dB72]. A second approach (inspired by the work of McKinna and Pollack [vBJMR94, MP97]) formalising named variables was also the subject of some initial work, and a comparison of these two approaches is presented. The formalisation was restricted to the implicational fragment of propositional logic. The informal theory has been extended to cover full propositional logic by Dyckhoff and Pinto, and extension of the formalisation using de Bruijn indices would appear to present few difficulties. An overview of other work in this area, in terms of both the tools and formalisation methods, is also presented. The theory formalised differs from other such work in that other formalisations have involved only one calculus. [DP97a] involves the relationships between three different calculi. There is consequently a much greater requirement for equality reasoning in the formalisation. It is concluded that a formalisation of any significance is still difficult, particularly one involving multiple calculi. No tools currently exist that allow for the easy representation of even quite simple systems in a way that fits human intuitions while still allowing for automatic derivation of induction principles. New work on integrating higher order abstract syntax and induction may be the way forward, although such work is still in the early stages.
1997-01-01T00:00:00ZAdams, Andrew, 1969-Machine-assisted formal proofs are becoming commonplace in certain fields of mathematics and theoretical computer science. New formal systems and variations on old ones are constantly invented. The meta-theory of such systems, i.e. proofs about the system as opposed to proofs within the system, are mostly done informally with a pen and paper. Yet the meta-theory of deductive systems is an area which would obviously benefit from machine support for formal proof. Is the software currently available sufficiently powerful yet easy enough to use to make machine assistance for formal meta-theory a viable proposition? This thesis presents work done by the author on formalizing proof theory from [DP97a] in various formal systems: SEQUEL [Tar93, Tar97], Isabelle [Pau94] and Coq [BB+96]. SEQUEL and Isabelle were found to be difficult to use for this type of work. In particular, the lack of automated production of induction principles in SEQUEL and Isabelle undermined confidence in the resulting formal proofs. Coq was found to be suitable for the formalisation methodology first chosen: the use of nameless dummy variables (de Bruijn indices) as pioneered in [dB72]. A second approach (inspired by the work of McKinna and Pollack [vBJMR94, MP97]) formalising named variables was also the subject of some initial work, and a comparison of these two approaches is presented. The formalisation was restricted to the implicational fragment of propositional logic. The informal theory has been extended to cover full propositional logic by Dyckhoff and Pinto, and extension of the formalisation using de Bruijn indices would appear to present few difficulties. An overview of other work in this area, in terms of both the tools and formalisation methods, is also presented. The theory formalised differs from other such work in that other formalisations have involved only one calculus. [DP97a] involves the relationships between three different calculi. There is consequently a much greater requirement for equality reasoning in the formalisation. It is concluded that a formalisation of any significance is still difficult, particularly one involving multiple calculi. No tools currently exist that allow for the easy representation of even quite simple systems in a way that fits human intuitions while still allowing for automatic derivation of induction principles. New work on integrating higher order abstract syntax and induction may be the way forward, although such work is still in the early stages.The Arabic translation of Theodosius's SphaericaMartin, Thomas J.http://hdl.handle.net/10023/133802019-04-01T08:37:49Z1975-01-01T00:00:00ZThe thesis "The Arabic Translation of Theodosius's Sphaerica" is an edition of the Istanbul manuscript Topkapi Seray Ahmet III 3464.2. Included is a comparative apparatus of the Greek and Arabic texts showing possible correspondence between the posited Greek exemplar of the translator and the various Greek manuscript traditions reported by J.L. Heiberg in his critical edition of the text. Further differences are pointed out in the English Trajislation. There is also a glossary of terminology- giving listings from Greek to Arabic and Arabic to Greek. An appendix discussing the execution of the drawings in the Arabic manuscript and their relation to the Greek drawings as reported by Heiberg is also given. Other appendices include a chart representing the convention seemingly adopted by the translator for lettering the drawings, a listing of inconsistent grammatical usage found in the manuscript, parallel passages from the Greek text, the text of the present edition, the versions of al-Maghribi and al-Tusi, and a privately owned manuscript, and finally a list of interlinear sigla found on the first few folios of the manuscript the purpose of which is unclear.
1975-01-01T00:00:00ZMartin, Thomas J.The thesis "The Arabic Translation of Theodosius's Sphaerica" is an edition of the Istanbul manuscript Topkapi Seray Ahmet III 3464.2. Included is a comparative apparatus of the Greek and Arabic texts showing possible correspondence between the posited Greek exemplar of the translator and the various Greek manuscript traditions reported by J.L. Heiberg in his critical edition of the text. Further differences are pointed out in the English Trajislation. There is also a glossary of terminology- giving listings from Greek to Arabic and Arabic to Greek. An appendix discussing the execution of the drawings in the Arabic manuscript and their relation to the Greek drawings as reported by Heiberg is also given. Other appendices include a chart representing the convention seemingly adopted by the translator for lettering the drawings, a listing of inconsistent grammatical usage found in the manuscript, parallel passages from the Greek text, the text of the present edition, the versions of al-Maghribi and al-Tusi, and a privately owned manuscript, and finally a list of interlinear sigla found on the first few folios of the manuscript the purpose of which is unclear.The life and work of Prof. George Chrystal (1851-1911)Yousuf, Mohammadhttp://hdl.handle.net/10023/133792019-04-01T08:37:04Z1990-01-01T00:00:00ZThis thesis is principally concerned with George Chrystal's life and his work, mainly in three directions viz., as an experimentalist, a mathematician, and an educationist. The main object is to bring to light the work of a personality who is representative of many more who are always forgotten. The majority of historians of science consider the works of the giants in science, ignoring totally the contributions made by the less prominent people like Prof. George Chrystal. In fact their contributions serve as one of the most important factors in propagation of scientific knowledge. His main contributions: verification of Ohm's Law experimentally; Non-Euclidean geometry; differential equations; text books on algebra; theory of seiches; institution of leaving certificate examination in Scottish education and many more have been discussed in detail. A survey of Chrystal's general thought is given in so far as it may be gathered from his scattered remarks. The references are mentioned by numerals in the superscript, details of which are given at the end of each chapter. The main text consists of six chapters. There are three appendices at the end,' Appendix 'A' consists of his correspondence with different scientists, most of which is still unpublished. Appendix 'B' contains a bibliography of his contributions in chronological order, and Appendix 'C contains his three Promoter's addresses. Tables and figures are attached at their proper places, including some rarely available photographs.
1990-01-01T00:00:00ZYousuf, MohammadThis thesis is principally concerned with George Chrystal's life and his work, mainly in three directions viz., as an experimentalist, a mathematician, and an educationist. The main object is to bring to light the work of a personality who is representative of many more who are always forgotten. The majority of historians of science consider the works of the giants in science, ignoring totally the contributions made by the less prominent people like Prof. George Chrystal. In fact their contributions serve as one of the most important factors in propagation of scientific knowledge. His main contributions: verification of Ohm's Law experimentally; Non-Euclidean geometry; differential equations; text books on algebra; theory of seiches; institution of leaving certificate examination in Scottish education and many more have been discussed in detail. A survey of Chrystal's general thought is given in so far as it may be gathered from his scattered remarks. The references are mentioned by numerals in the superscript, details of which are given at the end of each chapter. The main text consists of six chapters. There are three appendices at the end,' Appendix 'A' consists of his correspondence with different scientists, most of which is still unpublished. Appendix 'B' contains a bibliography of his contributions in chronological order, and Appendix 'C contains his three Promoter's addresses. Tables and figures are attached at their proper places, including some rarely available photographs.Normalisation techniques in proof theory and category theoryHamza, Taher Tawfik Ahmedhttp://hdl.handle.net/10023/133712019-04-01T08:37:20Z1986-01-01T00:00:00ZThe word problem for the free categories with some structure generated by a category X can be solved using proof-theoretical means. These free categories give a semantics in which derivations of GENTZEN's propositional sequent calculus can be interpreted by means of arrows of those categories. In this thesis we describe, implement and document the cut-elimination and the normalization techniques in proof theory as outlined in SZABO [1978]: we show how these are used in order to solve, mechanically, the word problem for the free categories with structure of : cartesian, bicartesian, distributive bicartesian, cartesian closed, and bicartesian closed. This implementation is extended by a procedure to interpret intuitionistic propositional sequent derivations as arrows of the above categories. Implementation of those techniques has forced us to modify the techniques in various inessential ways. The description and the representation in the syntax of our implementation of the above categories is contained in chapters 1 - 5, where each chapter describes one theory and concludes with examples of the system In use to represent concepts and solve simple word problems from category theory ( of various typos ). Appendix 1 contains some apparent printing errors we have observed in the work done by SZABO. The algorithms used in the proof of the cut-elimination theorems and normalization through chapters 1 - 5 are collected in appendices 2 - 4. Appendices 5 - 8 concern the implementation and its user manual.
1986-01-01T00:00:00ZHamza, Taher Tawfik AhmedThe word problem for the free categories with some structure generated by a category X can be solved using proof-theoretical means. These free categories give a semantics in which derivations of GENTZEN's propositional sequent calculus can be interpreted by means of arrows of those categories. In this thesis we describe, implement and document the cut-elimination and the normalization techniques in proof theory as outlined in SZABO [1978]: we show how these are used in order to solve, mechanically, the word problem for the free categories with structure of : cartesian, bicartesian, distributive bicartesian, cartesian closed, and bicartesian closed. This implementation is extended by a procedure to interpret intuitionistic propositional sequent derivations as arrows of the above categories. Implementation of those techniques has forced us to modify the techniques in various inessential ways. The description and the representation in the syntax of our implementation of the above categories is contained in chapters 1 - 5, where each chapter describes one theory and concludes with examples of the system In use to represent concepts and solve simple word problems from category theory ( of various typos ). Appendix 1 contains some apparent printing errors we have observed in the work done by SZABO. The algorithms used in the proof of the cut-elimination theorems and normalization through chapters 1 - 5 are collected in appendices 2 - 4. Appendices 5 - 8 concern the implementation and its user manual.Theory and observations of the magnetic field in the solar coronaCarcedo, Laurahttp://hdl.handle.net/10023/129482019-04-01T08:37:22Z2005-01-01T00:00:00ZAlthough the solar corona is one of the most studied areas in solar physics, its activity, such as flares, prominence eruptions and CMEs, is far from understood. Since the solar corona is a low-ß plasma, its structure and dynamics are driven by the magnetic field. The aim of this PhD thesis to study the magnetic field in the solar corona. Unfortunately, high quality direct measurements of the coronal magnetic field are not available and theoretical extrapolation using the observed photospheric magnetic field is required. The thesis is mainly divided in two parts. The first part deals with the comparison between theoretical models of magnetic fields and observed structures in the corona. For any theoretical model, a quantitative method to fit magnetic field lines to observed coronal loops is introduced. This method provides a quantity C that measures how closely a theoretical model can reproduce the observed coronal structures. Using linear force-free field extrapolation, the above field line fitting method is used to study the evolution of an active region. The method is also illustrated when the theoretical magnetic field depends on more than one parameter. The second part of the thesis focuses on the linear force-free field assumption using two different geometric configurations. Firstly a vertical rigid magnetic flux tube is considered. The analytical expression of the magnetic field is obtained as an expansion in terms of Bessel functions. The main properties of this system are discussed and compared with two cylindrically symmetric twist profiles. For the second system, the photosphere is assumed to be an infinite plane. Using translational geometry, the analytical expression of the linear force-free magnetic field that matches a prescribed line of sight magnetic field component is obtained. This solution is compared with the non-linear solution obtained by Roumeliotis (1993).
2005-01-01T00:00:00ZCarcedo, LauraAlthough the solar corona is one of the most studied areas in solar physics, its activity, such as flares, prominence eruptions and CMEs, is far from understood. Since the solar corona is a low-ß plasma, its structure and dynamics are driven by the magnetic field. The aim of this PhD thesis to study the magnetic field in the solar corona. Unfortunately, high quality direct measurements of the coronal magnetic field are not available and theoretical extrapolation using the observed photospheric magnetic field is required. The thesis is mainly divided in two parts. The first part deals with the comparison between theoretical models of magnetic fields and observed structures in the corona. For any theoretical model, a quantitative method to fit magnetic field lines to observed coronal loops is introduced. This method provides a quantity C that measures how closely a theoretical model can reproduce the observed coronal structures. Using linear force-free field extrapolation, the above field line fitting method is used to study the evolution of an active region. The method is also illustrated when the theoretical magnetic field depends on more than one parameter. The second part of the thesis focuses on the linear force-free field assumption using two different geometric configurations. Firstly a vertical rigid magnetic flux tube is considered. The analytical expression of the magnetic field is obtained as an expansion in terms of Bessel functions. The main properties of this system are discussed and compared with two cylindrically symmetric twist profiles. For the second system, the photosphere is assumed to be an infinite plane. Using translational geometry, the analytical expression of the linear force-free magnetic field that matches a prescribed line of sight magnetic field component is obtained. This solution is compared with the non-linear solution obtained by Roumeliotis (1993).Loop oscillations in the coronaJames, Lornahttp://hdl.handle.net/10023/129472019-04-01T08:37:26Z2004-01-01T00:00:00ZMagnetic loops in the Sun's corona have been discovered to oscillate in a variety of modes. The oscillations are observed to exhibit strong damping. A number of theories have been put forward to explain the damping, including resonant absorption and phase mixing. Here we consider the modelling of loop oscillations, paying particular attention to two effects: gravity, and the addition of a chromospheric layer below the corona. We develop an acoustic model of coronal loop oscillations and consider two ways of describing the effects of the gravitational stratification and the chromospheric layers, considering either two media separated by a discontinuous interface or a single medium with a sound speed that varies along the loop. A dispersion relation for the two-layer isothermal atmosphere case is obtained and investigated numerically using a bisection code. On comparison with roots obtained for a single isothermal atmosphere, it was found that the effect of chromospheric footpoints on the period of a mode is slight. However, the effect of gravity was found to be more notable, rising up to a twenty percent change in period when considering the longer observed loops. This result is of especial interest since gravity is often ignored by authors discussing loop oscillations. The case of a linear sound speed has been investigated analytically, obtaining a dispersion relation in terms of Bessel functions. Our results show that the Bessel equation is a possible solution for describing the wave modes.
2004-01-01T00:00:00ZJames, LornaMagnetic loops in the Sun's corona have been discovered to oscillate in a variety of modes. The oscillations are observed to exhibit strong damping. A number of theories have been put forward to explain the damping, including resonant absorption and phase mixing. Here we consider the modelling of loop oscillations, paying particular attention to two effects: gravity, and the addition of a chromospheric layer below the corona. We develop an acoustic model of coronal loop oscillations and consider two ways of describing the effects of the gravitational stratification and the chromospheric layers, considering either two media separated by a discontinuous interface or a single medium with a sound speed that varies along the loop. A dispersion relation for the two-layer isothermal atmosphere case is obtained and investigated numerically using a bisection code. On comparison with roots obtained for a single isothermal atmosphere, it was found that the effect of chromospheric footpoints on the period of a mode is slight. However, the effect of gravity was found to be more notable, rising up to a twenty percent change in period when considering the longer observed loops. This result is of especial interest since gravity is often ignored by authors discussing loop oscillations. The case of a linear sound speed has been investigated analytically, obtaining a dispersion relation in terms of Bessel functions. Our results show that the Bessel equation is a possible solution for describing the wave modes.Magnetic annihilation, null collapse and coronal heatingMellor, Christopherhttp://hdl.handle.net/10023/129462019-04-01T08:37:30Z2004-01-01T00:00:00ZThe problem of how the Sun's corona is heated is of central importance to solar physics research. In this thesis we model three main areas. The first, annihilation, is a feature of non-ideal MHD and focusses on how magnetic field of opposite polarity meets at a null point and annihilates, after having been advected with plasma toward a stagnation point in the plasma flow. Generally, the null point of the field and the stagnation point of the flow are coincident at the origin, but in chapter 2 a simple extension is considered where an asymmetry in the boundary conditions of the field moves the null point away from the origin. Chapter 3 presents a model of reconnective annihilation in three dimensions. It represents flux being advected through the fan plane of a 3D null, and diffusing through a thin diffusion region before being annihilated at the spine line, and uses the method of matched asymptotic expansions to find the solution for small values of the resistivity. The second area of the thesis covers null collapse. This is when the magnetic field in close proximity to a null point is disturbed, causing the field to fold up on itself and collapse. This is a feature of ideal MHD, and causes a strong current to build up, allowing non-ideal effects to become important. When using linearised equations for the collapse problem, we are in fact looking at a linear instability. If this instability initiates a collapse, this is only a valid model until non-linear effects become important. By talking about collapse in chapters 4 and 5 (as it is talked about in the literature), we mean that the linear instability initiates collapse, which in principle, non-linear effects could later stop. Chapter 4 introduces a two-dimensional model for collapse, using the ideal, compressible, linearised MHD equations. It is a general solution in which all spatially linear nulls and their supporting plasma flows and pressure gradients can be checked for susceptibility to collapse under open boundary conditions. Chapter 5 uses the model introduced in chapter 4 to investigate the collapse of three-dimensional, potential nulls (again, spatially linear) for all possible supporting plasma flows and pressure gradients. Using this model, all nulls under consideration are found to collapse and produce large currents, except for a group of 2D O-type nulls supported by highly super-Alfvenic plasma flows. The third area of this thesis involves numerically simulating a model of heating by coronal tectonics (Priest et al, 2002). A simple magnetic field is created and the boundary is driven, also in a simple manner. Current sheets which scale with grid resolution are seen to build up on the quasi-separatrix layers, and there is some evidence of magnetic reconnection.
2004-01-01T00:00:00ZMellor, ChristopherThe problem of how the Sun's corona is heated is of central importance to solar physics research. In this thesis we model three main areas. The first, annihilation, is a feature of non-ideal MHD and focusses on how magnetic field of opposite polarity meets at a null point and annihilates, after having been advected with plasma toward a stagnation point in the plasma flow. Generally, the null point of the field and the stagnation point of the flow are coincident at the origin, but in chapter 2 a simple extension is considered where an asymmetry in the boundary conditions of the field moves the null point away from the origin. Chapter 3 presents a model of reconnective annihilation in three dimensions. It represents flux being advected through the fan plane of a 3D null, and diffusing through a thin diffusion region before being annihilated at the spine line, and uses the method of matched asymptotic expansions to find the solution for small values of the resistivity. The second area of the thesis covers null collapse. This is when the magnetic field in close proximity to a null point is disturbed, causing the field to fold up on itself and collapse. This is a feature of ideal MHD, and causes a strong current to build up, allowing non-ideal effects to become important. When using linearised equations for the collapse problem, we are in fact looking at a linear instability. If this instability initiates a collapse, this is only a valid model until non-linear effects become important. By talking about collapse in chapters 4 and 5 (as it is talked about in the literature), we mean that the linear instability initiates collapse, which in principle, non-linear effects could later stop. Chapter 4 introduces a two-dimensional model for collapse, using the ideal, compressible, linearised MHD equations. It is a general solution in which all spatially linear nulls and their supporting plasma flows and pressure gradients can be checked for susceptibility to collapse under open boundary conditions. Chapter 5 uses the model introduced in chapter 4 to investigate the collapse of three-dimensional, potential nulls (again, spatially linear) for all possible supporting plasma flows and pressure gradients. Using this model, all nulls under consideration are found to collapse and produce large currents, except for a group of 2D O-type nulls supported by highly super-Alfvenic plasma flows. The third area of this thesis involves numerically simulating a model of heating by coronal tectonics (Priest et al, 2002). A simple magnetic field is created and the boundary is driven, also in a simple manner. Current sheets which scale with grid resolution are seen to build up on the quasi-separatrix layers, and there is some evidence of magnetic reconnection.Sir Arthur Eddington and the foundations of modern physicsDurham, Ian T.http://hdl.handle.net/10023/129332019-04-01T08:37:03Z2005-01-01T00:00:00ZIn this dissertation I analyze Sir Arthur Eddington's statistical theory as developed in the first six chapters of his posthumously published Fundamental Theory. In particular I look at the mathematical structure, philosophical implications, and relevancy to modern physics. This analysis is the only one of Fundamental Theory that compares it to modern quantum field theory and is the most comprehensive look at his statistical theory in four decades. Several major insights have been made in this analysis including the fact that he was able to derive Pauli's Exclusion Principle in part from Heisenberg's Uncertainty Principle. In addition the most profound general conclusion of this research is that Fundamental Theory is, in fact, an early quantum field theory, something that has never before been suggested. Contrary to the majority of historical reports and some comments by his contemporaries, this analysis shows that Eddington's later work is neither mystical nor was it that far from mainstream when it was published. My research reveals numerous profoundly deep ideas that were ahead of their time when Fundamental Theory was developed, but that have significant applicability at present. As such this analysis presents several important questions to be considered by modern philosophers of science, physicists, mathematicians, and historians. In addition it sheds new light on Eddington as a scientist and mathematician, in part indicating that his marginalization has been largely unwarranted.
2005-01-01T00:00:00ZDurham, Ian T.In this dissertation I analyze Sir Arthur Eddington's statistical theory as developed in the first six chapters of his posthumously published Fundamental Theory. In particular I look at the mathematical structure, philosophical implications, and relevancy to modern physics. This analysis is the only one of Fundamental Theory that compares it to modern quantum field theory and is the most comprehensive look at his statistical theory in four decades. Several major insights have been made in this analysis including the fact that he was able to derive Pauli's Exclusion Principle in part from Heisenberg's Uncertainty Principle. In addition the most profound general conclusion of this research is that Fundamental Theory is, in fact, an early quantum field theory, something that has never before been suggested. Contrary to the majority of historical reports and some comments by his contemporaries, this analysis shows that Eddington's later work is neither mystical nor was it that far from mainstream when it was published. My research reveals numerous profoundly deep ideas that were ahead of their time when Fundamental Theory was developed, but that have significant applicability at present. As such this analysis presents several important questions to be considered by modern philosophers of science, physicists, mathematicians, and historians. In addition it sheds new light on Eddington as a scientist and mathematician, in part indicating that his marginalization has been largely unwarranted.Extremal problems in combinatorial semigroup theoryMitchell, James Davidhttp://hdl.handle.net/10023/113222019-04-01T08:39:22Z2002-07-01T00:00:00ZIn this thesis we shall consider three types of extremal problems (i.e. problems involving maxima and minima) concerning semigroups. In the first chapter we show how to construct a minimal semigroup presentation that defines a group of non-negative deficiency given a minimal group presentation for that group. This demonstrates that the semigroup deficiency of a group of non-negative deficiency is equal to the group deficiency of that group. Given a finite monoid we find a necessary and sufficient condition for the monoid deficiency to equal the semigroup deficiency. We give a class of infinite monoids for which this equality also holds. The second type of problem we consider concerns infinite semigroups of relations and transformations. We find the relative rank of the full transformation semigroup, over an infinite set, modulo some standard subsets and subsemigroups, including the set of contraction maps and the set of order preserving maps (for some infinite ordered sets). We also find the relative rank of the semigroup of all binary relations (over an infinite set) modulo the partial transformation semigroup, the full transformation semigroup, the symmetric inverse semigroup, the symmetric group and the set of idempotent relations. Analogous results are also proven for the symmetric inverse semigroup. The third, and final, type of problem studied concerns generalising notions of independence from linear algebra to semigroups and groups. We determine the maximum cardinality of an independent set in finite abelian groups, Brandt semigroups, free nilpotent semigroups, and some examples of infinite groups.
2002-07-01T00:00:00ZMitchell, James DavidIn this thesis we shall consider three types of extremal problems (i.e. problems involving maxima and minima) concerning semigroups. In the first chapter we show how to construct a minimal semigroup presentation that defines a group of non-negative deficiency given a minimal group presentation for that group. This demonstrates that the semigroup deficiency of a group of non-negative deficiency is equal to the group deficiency of that group. Given a finite monoid we find a necessary and sufficient condition for the monoid deficiency to equal the semigroup deficiency. We give a class of infinite monoids for which this equality also holds. The second type of problem we consider concerns infinite semigroups of relations and transformations. We find the relative rank of the full transformation semigroup, over an infinite set, modulo some standard subsets and subsemigroups, including the set of contraction maps and the set of order preserving maps (for some infinite ordered sets). We also find the relative rank of the semigroup of all binary relations (over an infinite set) modulo the partial transformation semigroup, the full transformation semigroup, the symmetric inverse semigroup, the symmetric group and the set of idempotent relations. Analogous results are also proven for the symmetric inverse semigroup. The third, and final, type of problem studied concerns generalising notions of independence from linear algebra to semigroups and groups. We determine the maximum cardinality of an independent set in finite abelian groups, Brandt semigroups, free nilpotent semigroups, and some examples of infinite groups.Optimized automated survey design in wildlife population assessmentStrindberg, Samanthahttp://hdl.handle.net/10023/113182020-02-20T11:54:50Z2001-05-01T00:00:00ZIncreased pressure on the environment has placed numerous ecological populations under threat of extinction. Management schemes dedicated to the future conservation of wildlife populations rely on effective monitoring of the size of those populations. This requires that accurate and precise abundance estimates are obtained for the purposes of wildlife population assessment. The accuracy and precision of estimates are determined to a large extent by the survey design used to obtain population samples. Methods for optimizing the survey design process are detailed, with a particular- focus on automating the sui-vey designs using computer software. The technique of automated survey design is a simulation-based tool, which provides the means to assess the properties of any type of survey design, permits the evaluation of abundance estimates over sui-vey regions with assumed population densities, and from a practical standpoint facilitates the creation of a survey plan that can be implemented in the field. Survey design properties include the probability of a particular location being included in the sample, the spatial distribution of the sampling locations within the survey region, and the distances covered by observers to obtain the sample data. The design properties are directly linked to the accuracy and precision of estimates, as well as the efficiency, achieved by a type of design. A comparative study of a number of different survey designs that can be broadly classified as systematic or non-systematic is presented. The simulation results show their performance with regard to the above-mentioned properties and the abundance estimates obtained if the designs are applied to some known population densities. Due to the more even spatial distribution of the systematic designs the estimates they produce are potentially more precise and the distances covered by observers less variable as well. It is also shown how biased estimates can result if the probability of a particular location being included in the sample is assumed to be even over the entire survey region when it is not. The problems associated with surveying along the boundary of a survey region and within non-convex regions are addressed. The methods are illustrated with a number of survey design examples.
2001-05-01T00:00:00ZStrindberg, SamanthaIncreased pressure on the environment has placed numerous ecological populations under threat of extinction. Management schemes dedicated to the future conservation of wildlife populations rely on effective monitoring of the size of those populations. This requires that accurate and precise abundance estimates are obtained for the purposes of wildlife population assessment. The accuracy and precision of estimates are determined to a large extent by the survey design used to obtain population samples. Methods for optimizing the survey design process are detailed, with a particular- focus on automating the sui-vey designs using computer software. The technique of automated survey design is a simulation-based tool, which provides the means to assess the properties of any type of survey design, permits the evaluation of abundance estimates over sui-vey regions with assumed population densities, and from a practical standpoint facilitates the creation of a survey plan that can be implemented in the field. Survey design properties include the probability of a particular location being included in the sample, the spatial distribution of the sampling locations within the survey region, and the distances covered by observers to obtain the sample data. The design properties are directly linked to the accuracy and precision of estimates, as well as the efficiency, achieved by a type of design. A comparative study of a number of different survey designs that can be broadly classified as systematic or non-systematic is presented. The simulation results show their performance with regard to the above-mentioned properties and the abundance estimates obtained if the designs are applied to some known population densities. Due to the more even spatial distribution of the systematic designs the estimates they produce are potentially more precise and the distances covered by observers less variable as well. It is also shown how biased estimates can result if the probability of a particular location being included in the sample is assumed to be even over the entire survey region when it is not. The problems associated with surveying along the boundary of a survey region and within non-convex regions are addressed. The methods are illustrated with a number of survey design examples.Helioseismology and diagnostics of internal magnetic layersFoullon, Claire-Uriel Armelle Marie Alinehttp://hdl.handle.net/10023/113152020-02-20T11:54:44Z2002-03-01T00:00:00ZSolar magnetic fields, as well as temperature changes, introduce pressure deviations that play a significant role in modulating the resonant frequencies of p-mode oscillations. Those pressure deviations occurring in the atmosphere or sub-surface of the Sun can explain the frequency shifts observed on the timescale of the solar activity cycle. A separate study of the contribution of internal magnetic layers can clarify the relative importance of surface effects. Results from helioseismology provide realistic constraints for choosing parameters suitable to represent the magnetic layers buried in the solar interior and available for modelling, i.e. at the base of the convection zone and in the sunspots’ anchoring zone. Diagnostics of the internal magnetic layers are obtained through a schematic model in which the Sun is plane-stratified. The influence of a buried magnetic field on p-modes is explored, and the nature of various waves and instabilities that can arise on such a buried magnetic field is assessed. By treating the effects of internal magnetic layers, this thesis contributes to the building of a bridge between theories and observations. On the one hand, the theoretical analysis is explored carefully in the course of its formulation, which generates new hypotheses that were not obvious so far. On the other hand, observations help to understand which explanations of the solar cycle frequency shifts may apply.
2002-03-01T00:00:00ZFoullon, Claire-Uriel Armelle Marie AlineSolar magnetic fields, as well as temperature changes, introduce pressure deviations that play a significant role in modulating the resonant frequencies of p-mode oscillations. Those pressure deviations occurring in the atmosphere or sub-surface of the Sun can explain the frequency shifts observed on the timescale of the solar activity cycle. A separate study of the contribution of internal magnetic layers can clarify the relative importance of surface effects. Results from helioseismology provide realistic constraints for choosing parameters suitable to represent the magnetic layers buried in the solar interior and available for modelling, i.e. at the base of the convection zone and in the sunspots’ anchoring zone. Diagnostics of the internal magnetic layers are obtained through a schematic model in which the Sun is plane-stratified. The influence of a buried magnetic field on p-modes is explored, and the nature of various waves and instabilities that can arise on such a buried magnetic field is assessed. By treating the effects of internal magnetic layers, this thesis contributes to the building of a bridge between theories and observations. On the one hand, the theoretical analysis is explored carefully in the course of its formulation, which generates new hypotheses that were not obvious so far. On the other hand, observations help to understand which explanations of the solar cycle frequency shifts may apply.Magnetohydrodynamic waves and instabilities in solar magnetic structuresBoddie, Davidhttp://hdl.handle.net/10023/113082019-04-01T08:37:54Z2001-06-01T00:00:00ZMotions of plasma in magnetic structures in the solar atmosphere may be successfully modelled using the theory of magnetohydrodynamics (MHD) describing oscillatory motion, in the form of standing and propagating waves, and unstable behaviour. In this thesis we consider two forms of magnetic structuring, the current sheet and the thin magnetic flux tube. The current sheet finds particular application in the solar corona and solar wind; the thin flux tube is of particular importance in solar photospheric magnetism. A model of a current sheet with a continuous magnetic field profile is studied as a waveguide. The equation of motion for small perturbations to a current sheet equilibrium is obtained from the equations of ideal linear MHD and solved numerically to determine the nature of magnetoacoustic waves propagating parallel to the applied magnetic field. A number of approximation methods are used to shed light on the significance of the numerical results. We consider a variation of this model, applicable to the solar corona, and examine the possibility of impulsively generated magnetohydro dynamic waves in the sheet. Such waves exhibit wavepacket properties, similar to those found in slab models of magnetic structures. The process of convective collapse in a vertical magnetic flux tube located in the solar photospheric network is treated using the thin flux tube equations of ideal linear MED. We consider the critical stability of a thin flux tube embedded in convection zone models of varying complexity, taking into account the effects of an overlying chromospheric atmosphere and temperature imbalance between the flux tube and its environment. The dependence of the instability on various sets of boundary conditions is discussed; the choice of boundary conditions is a subject of some debate in the current literature. Possible future directions for work which extends the description of dynamic phenomena in both the current sheet and thin flux tube structure is discussed and ideas for linking these areas of research are presented.
2001-06-01T00:00:00ZBoddie, DavidMotions of plasma in magnetic structures in the solar atmosphere may be successfully modelled using the theory of magnetohydrodynamics (MHD) describing oscillatory motion, in the form of standing and propagating waves, and unstable behaviour. In this thesis we consider two forms of magnetic structuring, the current sheet and the thin magnetic flux tube. The current sheet finds particular application in the solar corona and solar wind; the thin flux tube is of particular importance in solar photospheric magnetism. A model of a current sheet with a continuous magnetic field profile is studied as a waveguide. The equation of motion for small perturbations to a current sheet equilibrium is obtained from the equations of ideal linear MHD and solved numerically to determine the nature of magnetoacoustic waves propagating parallel to the applied magnetic field. A number of approximation methods are used to shed light on the significance of the numerical results. We consider a variation of this model, applicable to the solar corona, and examine the possibility of impulsively generated magnetohydro dynamic waves in the sheet. Such waves exhibit wavepacket properties, similar to those found in slab models of magnetic structures. The process of convective collapse in a vertical magnetic flux tube located in the solar photospheric network is treated using the thin flux tube equations of ideal linear MED. We consider the critical stability of a thin flux tube embedded in convection zone models of varying complexity, taking into account the effects of an overlying chromospheric atmosphere and temperature imbalance between the flux tube and its environment. The dependence of the instability on various sets of boundary conditions is discussed; the choice of boundary conditions is a subject of some debate in the current literature. Possible future directions for work which extends the description of dynamic phenomena in both the current sheet and thin flux tube structure is discussed and ideas for linking these areas of research are presented.Field line resonances in the earth's magnetosphere driven by convectively unstable magnetospheric waveguide modesMcRobbie, Mairi Catrionahttp://hdl.handle.net/10023/113032019-04-01T08:39:32Z2002-07-01T00:00:00ZShear flow instabilities, such as Kelvin-Helmholtz instabilities, occurring on the Earth’s magnetospheric flanks may cause fast magnetosonic wave modes to propagate through the non-homogeneous environment of the Earth’s magnetospheric cavity. The non-uniformity in this plasma environment means the fast wave mode couples to a standing Alfvén wave mode along a closed field line in the magnetosphere with natural frequency equal to the fast wave frequency. The one-dimensional hydromagnetic box model of Southwood (1974), which treats the Earth’s magnetic field as a set of straight field lines between two ionospheric boundaries which are not perfectly reflecting, is used to model the resonance. There is a finite height-integrated Pedersen conductivity, Σp, at the boundaries of the one-dimensional box which is responsible for the damping of the field line resonance. The coupling process between the fast and Alfvén modes is represented by a simple harmonic oscillator equation driven by a time-dependent function representing the fast mode azimuthal pressure gradient, Wright (1992a,b). A fourth-order Runge-Kutta numerical integration technique is used to obtain the solution to the simple harmonic oscillation. These numerical routines are verified using analytically derived solutions for a test case of a simple driving function d{t) = Dsin(wdt). Following this test of the numerical routines, realistic driving functions from Wright et al (2002), which represent convectively unstable fast wave modes propagating through the magnetospheric cavity as a result of a Kelvin-Helmholtz instability occurring on the flanks of the magnetosphere, are used to drive the simple harmonic system. Four different unstable drivers are used, these being the fundamental and the second harmonic mode for two different values of azimuthal coordinate. For all four drivers clear resonance characteristics emerged, suggesting these may drive field line resonances in the Earth’s magnetosphere.
2002-07-01T00:00:00ZMcRobbie, Mairi CatrionaShear flow instabilities, such as Kelvin-Helmholtz instabilities, occurring on the Earth’s magnetospheric flanks may cause fast magnetosonic wave modes to propagate through the non-homogeneous environment of the Earth’s magnetospheric cavity. The non-uniformity in this plasma environment means the fast wave mode couples to a standing Alfvén wave mode along a closed field line in the magnetosphere with natural frequency equal to the fast wave frequency. The one-dimensional hydromagnetic box model of Southwood (1974), which treats the Earth’s magnetic field as a set of straight field lines between two ionospheric boundaries which are not perfectly reflecting, is used to model the resonance. There is a finite height-integrated Pedersen conductivity, Σp, at the boundaries of the one-dimensional box which is responsible for the damping of the field line resonance. The coupling process between the fast and Alfvén modes is represented by a simple harmonic oscillator equation driven by a time-dependent function representing the fast mode azimuthal pressure gradient, Wright (1992a,b). A fourth-order Runge-Kutta numerical integration technique is used to obtain the solution to the simple harmonic oscillation. These numerical routines are verified using analytically derived solutions for a test case of a simple driving function d{t) = Dsin(wdt). Following this test of the numerical routines, realistic driving functions from Wright et al (2002), which represent convectively unstable fast wave modes propagating through the magnetospheric cavity as a result of a Kelvin-Helmholtz instability occurring on the flanks of the magnetosphere, are used to drive the simple harmonic system. Four different unstable drivers are used, these being the fundamental and the second harmonic mode for two different values of azimuthal coordinate. For all four drivers clear resonance characteristics emerged, suggesting these may drive field line resonances in the Earth’s magnetosphere.Three dimensional numerical simulations of non-linear MHD instabilities in the solar coronaGerrard, Catherine Louisehttp://hdl.handle.net/10023/112972019-04-01T08:39:38Z2002-01-01T00:00:00ZThe aim of this thesis has been to carry out 3D MHD simulations to investigate nonlinear MHD instabilities and the behaviour of solar coronal loops. The simulations have been carried out on a parallel computer using a new shock-capturing Lagrangian-remap code, LareSd. As part of the PhD this code has been extended to include resistivity allowing the study of the non-linear resistive evolution of the instability. In particular the kink instability in line-tied coronal loops has been studied. This was suggested as a possible explanation of compact loop flares, sudden brightenings of a coronal loop due to a release of energy which does not destroy the loop. For the kink instability to explain such flares it must drive reconnection. This requires high current densities, i.e. current sheets. The results presented in this thesis suggest that the formation of current sheets during the non-linear evolution of the kink instability is more complicated than was previously believed. Indeed, if the loop is allowed to evolve slowly until the instability is triggered than the current appears to saturate at a finite value. This suggests that the kink instability cannot explain a compact loop flare. LareSd has also been used to model space observations from NASA’s SoHO (a joint NASA/ESA satellite) and TRACE satellites. These observations showed a group of rotating sunspots and their overlying system of loops. The simulations will allow further investigations of this behaviour to be carried out.
2002-01-01T00:00:00ZGerrard, Catherine LouiseThe aim of this thesis has been to carry out 3D MHD simulations to investigate nonlinear MHD instabilities and the behaviour of solar coronal loops. The simulations have been carried out on a parallel computer using a new shock-capturing Lagrangian-remap code, LareSd. As part of the PhD this code has been extended to include resistivity allowing the study of the non-linear resistive evolution of the instability. In particular the kink instability in line-tied coronal loops has been studied. This was suggested as a possible explanation of compact loop flares, sudden brightenings of a coronal loop due to a release of energy which does not destroy the loop. For the kink instability to explain such flares it must drive reconnection. This requires high current densities, i.e. current sheets. The results presented in this thesis suggest that the formation of current sheets during the non-linear evolution of the kink instability is more complicated than was previously believed. Indeed, if the loop is allowed to evolve slowly until the instability is triggered than the current appears to saturate at a finite value. This suggests that the kink instability cannot explain a compact loop flare. LareSd has also been used to model space observations from NASA’s SoHO (a joint NASA/ESA satellite) and TRACE satellites. These observations showed a group of rotating sunspots and their overlying system of loops. The simulations will allow further investigations of this behaviour to be carried out.An investigation of rotating magnetospheresRyan, Richard Danielhttp://hdl.handle.net/10023/112942019-04-01T08:37:36Z2002-11-01T00:00:00ZIn this thesis we will construct simple models of rotating stellar and planetary magnetospheres within the framework of ideal MHD. These models will take the basic outline of a stellar magnetosphere that we have outlined above as a starting point from which to proceed further. In summary, this simple magnetosphere will be that of a single, rapidly rotating star' with an axisymmetric dipole magnetic field at the base of its corona and with an axis that is in alignment with that of the rotation axis. It is the isothermal plasma associated with this field that will give rise to the magnetospheric emission and which is held in strict corotation with the stellar surface. Equatorial and rotational symmetry reduce the domain to one quarter of a two dimensional quadrant. We will consider timescales that are much longer than the typical time scales of the system, which will allow us to model the evolution of the system quasi-statically by calculating sequences of MHS equilibria. This is achieved by numerical solution of the Grad-Shafranov equation (in terms of the flux function. A) Which requires us to specify a suitable surface pressure distribution and specify the toroidal component of the magnetic field as a function of A. The second chapter will outline the numerical procedure that will be employed to calculate these equilibrium sequences, and the practical realisation of this procedure. The third chapter will discuss different models which will be characterised by different surface pressure distributions but all of which will lack a toroidal magnetic field component. The fourth chapter will discuss results from a model which includes a toroidal magnetic field component. The models successfully reproduce the observed saturation and supersaturation of stellar emission with rotation. The fifth chapter will address the question of analytically constructing three dimensional equilibria that may be of use in the modelling of magnetospheres with magnetic field geometries that are not in alignment with their rotation axes or which are displaced from the centre of the rotating body, such as the giant gas planets Uranus and Neptune. The last section of the thesis will be a brief discussion of our conclusions, a review of the work of the thesis and will consider the outlook for further development, extension and refinement of our models.
2002-11-01T00:00:00ZRyan, Richard DanielIn this thesis we will construct simple models of rotating stellar and planetary magnetospheres within the framework of ideal MHD. These models will take the basic outline of a stellar magnetosphere that we have outlined above as a starting point from which to proceed further. In summary, this simple magnetosphere will be that of a single, rapidly rotating star' with an axisymmetric dipole magnetic field at the base of its corona and with an axis that is in alignment with that of the rotation axis. It is the isothermal plasma associated with this field that will give rise to the magnetospheric emission and which is held in strict corotation with the stellar surface. Equatorial and rotational symmetry reduce the domain to one quarter of a two dimensional quadrant. We will consider timescales that are much longer than the typical time scales of the system, which will allow us to model the evolution of the system quasi-statically by calculating sequences of MHS equilibria. This is achieved by numerical solution of the Grad-Shafranov equation (in terms of the flux function. A) Which requires us to specify a suitable surface pressure distribution and specify the toroidal component of the magnetic field as a function of A. The second chapter will outline the numerical procedure that will be employed to calculate these equilibrium sequences, and the practical realisation of this procedure. The third chapter will discuss different models which will be characterised by different surface pressure distributions but all of which will lack a toroidal magnetic field component. The fourth chapter will discuss results from a model which includes a toroidal magnetic field component. The models successfully reproduce the observed saturation and supersaturation of stellar emission with rotation. The fifth chapter will address the question of analytically constructing three dimensional equilibria that may be of use in the modelling of magnetospheres with magnetic field geometries that are not in alignment with their rotation axes or which are displaced from the centre of the rotating body, such as the giant gas planets Uranus and Neptune. The last section of the thesis will be a brief discussion of our conclusions, a review of the work of the thesis and will consider the outlook for further development, extension and refinement of our models.On the application of numerical continuation methods to two- and three-dimensional solar and astrophysical problemsRomeou, Zahareniahttp://hdl.handle.net/10023/112932019-04-01T08:37:56Z2002-06-01T00:00:00ZIn this thesis, applications of a numerical continuation method to two- and three-dimensional bifurcation problems are presented. The 2D problems are motivated by solar applications. In particular, it is shown that the bifurcation properties of a previously studied model for magnetic arcades depend strongly on the pressure function used in the model. The bifurcation properties of a straight flux model for coronal loops are investigated and compared with the results of linear ideal MHD stability analysis. It is shown that for line-tied boundary conditions, the method for the calculation of the equilibrium sequence determines whether the first or the second bifurcation point coincides with the linear stability threshold. Also, in this thesis, the 3D version of the continuation code is applied for the first time. The problems treated with the 3D code are therefore chosen with the intention to demonstrate the general capabilities of the code and to see where its limitations are. Whereas the code performs as expected for relatively simple albeit nonlinear bifurcation problems, a clear need for further development is shown by more involved problems.
2002-06-01T00:00:00ZRomeou, ZahareniaIn this thesis, applications of a numerical continuation method to two- and three-dimensional bifurcation problems are presented. The 2D problems are motivated by solar applications. In particular, it is shown that the bifurcation properties of a previously studied model for magnetic arcades depend strongly on the pressure function used in the model. The bifurcation properties of a straight flux model for coronal loops are investigated and compared with the results of linear ideal MHD stability analysis. It is shown that for line-tied boundary conditions, the method for the calculation of the equilibrium sequence determines whether the first or the second bifurcation point coincides with the linear stability threshold. Also, in this thesis, the 3D version of the continuation code is applied for the first time. The problems treated with the 3D code are therefore chosen with the intention to demonstrate the general capabilities of the code and to see where its limitations are. Whereas the code performs as expected for relatively simple albeit nonlinear bifurcation problems, a clear need for further development is shown by more involved problems.Investigations of current build up in topologically simple magnetic fieldsBocquet, Francois-Xavierhttp://hdl.handle.net/10023/112912019-04-01T08:38:29Z2005-06-01T00:00:00ZThe solar corona is a highly conductive plasma which is dominated by the coronal magnetic field. Observations show that important solar phenomena like flares or the heating of the corona are driven by magnetic energy, probably through the process of magnetic reconnection. The release of magnetic energy by reconnection requires that non-ideal processes take place in contradiction to the high conductivity of the corona. One possibility to overcome this problem is to generate strong electrical currents in strongly localised regions. In this thesis we investigate how such localised currents can be formed by slow ideal evolution of topologically simple magnetic fields. To this purpose numerical simulations are carried out using an Eulerian and a Lagrangian MHD relaxation code. We first use a simple example (twisting of a uniform field) to investigate the advantages and disadvantages of both codes and to discover possible limitations for their application. We show that for the problems addressed in this thesis the Lagrangian code is more suited because it can resolve the localised current densities much better than the Eulerian code. We then focus in particular on magnetic fields containing a so-called Hyperbolic Flux Tube (HPT). A recently proposed analytical theory predicts that HFT’s are sites where under certain conditions strong current build-up can be expected. We use our code to carry out a systematic parametric study of the dependence of current growth for a typical HFT configuration. We have also developed a completely new version of the analytical theory which is directly based on the set-up of our numerical simulations. We find that the simulations agree with the analytical prediction in a quantitative way but that the analytical theory underestimates the current growth quite substantially, probably by not taking into account the non-linear character of the full problem.
2005-06-01T00:00:00ZBocquet, Francois-XavierThe solar corona is a highly conductive plasma which is dominated by the coronal magnetic field. Observations show that important solar phenomena like flares or the heating of the corona are driven by magnetic energy, probably through the process of magnetic reconnection. The release of magnetic energy by reconnection requires that non-ideal processes take place in contradiction to the high conductivity of the corona. One possibility to overcome this problem is to generate strong electrical currents in strongly localised regions. In this thesis we investigate how such localised currents can be formed by slow ideal evolution of topologically simple magnetic fields. To this purpose numerical simulations are carried out using an Eulerian and a Lagrangian MHD relaxation code. We first use a simple example (twisting of a uniform field) to investigate the advantages and disadvantages of both codes and to discover possible limitations for their application. We show that for the problems addressed in this thesis the Lagrangian code is more suited because it can resolve the localised current densities much better than the Eulerian code. We then focus in particular on magnetic fields containing a so-called Hyperbolic Flux Tube (HPT). A recently proposed analytical theory predicts that HFT’s are sites where under certain conditions strong current build-up can be expected. We use our code to carry out a systematic parametric study of the dependence of current growth for a typical HFT configuration. We have also developed a completely new version of the analytical theory which is directly based on the set-up of our numerical simulations. We find that the simulations agree with the analytical prediction in a quantitative way but that the analytical theory underestimates the current growth quite substantially, probably by not taking into account the non-linear character of the full problem.The theory of rational integral functions of several sets of variables and associated linear transformationsWallace, Andrew Hughhttp://hdl.handle.net/10023/112122019-04-01T08:38:05Z1949-04-01T00:00:00ZThe theme of this paper is the unification of two theories which arose and were developed independently of one another in the latter part of the 19th century and the beginning of the 20th, namely the theory of series expansion of rational integral functions of several sets of variables, homogeneous in the variables of each set, that is the series expansion of algebraic forms in several sets of variables, and the theory of induces linear transformations, or invariant matrices. I have divided the work into five chapters of which the first and third are purely historical; Chapter I is an account of various methods, devised before the introduction of the ideas of standard order and standard tableaux, of forming series expansions of algebraic forms, while Chapter III is mainly occupied by an account of Schnur’s work on invariant matrices. Chapters II, IV and V establish the link between the two theories and, at the expense of one or two points of repetition of definitions, are self-contained and may be read consecutively, more or less without reference to the other two chapters.
1949-04-01T00:00:00ZWallace, Andrew HughThe theme of this paper is the unification of two theories which arose and were developed independently of one another in the latter part of the 19th century and the beginning of the 20th, namely the theory of series expansion of rational integral functions of several sets of variables, homogeneous in the variables of each set, that is the series expansion of algebraic forms in several sets of variables, and the theory of induces linear transformations, or invariant matrices. I have divided the work into five chapters of which the first and third are purely historical; Chapter I is an account of various methods, devised before the introduction of the ideas of standard order and standard tableaux, of forming series expansions of algebraic forms, while Chapter III is mainly occupied by an account of Schnur’s work on invariant matrices. Chapters II, IV and V establish the link between the two theories and, at the expense of one or two points of repetition of definitions, are self-contained and may be read consecutively, more or less without reference to the other two chapters.James Gregory : a survey of his work in mathematical analysisInglis, Alexanderhttp://hdl.handle.net/10023/112112019-04-01T08:37:35Z1933-05-01T00:00:00Z1933-05-01T00:00:00ZInglis, AlexanderThe statistical analysis of point events associated with a fixed pointLawson, Andrew B.http://hdl.handle.net/10023/72942019-04-01T08:37:28Z1991-01-01T00:00:00ZThis work concerns the analysis of point events which are distributed on a planar region and are thought to be related to a fixed point. Data examples are considered from Epidemiology, where morbidity events are thought to be related to a pollution source, and Ecology and Geology where events associated with a central point are to be modelled. We have developed a variety of Heterogeneous Poisson Process (HEPP) models for the above examples. In particular, I have developed interaction and 8-dependence models for angular-linear correlation, with their ML estimation and associated score/W aId tests. In the Epidemiological case we have developed case-control models and tests. The possibility of second-order effects being important has also led to the development of Bayesian Spatial Prior (BSP) models. In addition, we have developed a new deviance residual for HEPP models and explored the use of GLIM for modelling purposes. A variety of results were found in data analysis. In some cases HEPP models provide adequate descriptions of the process. In others, BSP models yield better fits. In general, the discrete case admits a simple spatial Poisson model for counts and does not require BSP model extensions.
1991-01-01T00:00:00ZLawson, Andrew B.This work concerns the analysis of point events which are distributed on a planar region and are thought to be related to a fixed point. Data examples are considered from Epidemiology, where morbidity events are thought to be related to a pollution source, and Ecology and Geology where events associated with a central point are to be modelled. We have developed a variety of Heterogeneous Poisson Process (HEPP) models for the above examples. In particular, I have developed interaction and 8-dependence models for angular-linear correlation, with their ML estimation and associated score/W aId tests. In the Epidemiological case we have developed case-control models and tests. The possibility of second-order effects being important has also led to the development of Bayesian Spatial Prior (BSP) models. In addition, we have developed a new deviance residual for HEPP models and explored the use of GLIM for modelling purposes. A variety of results were found in data analysis. In some cases HEPP models provide adequate descriptions of the process. In others, BSP models yield better fits. In general, the discrete case admits a simple spatial Poisson model for counts and does not require BSP model extensions.Carl Friedrich Geiser and Ferdinand Rudio : the men behind the first International Congress of MathematiciansEminger, Stefanie Ursulahttp://hdl.handle.net/10023/65362019-04-01T08:38:56Z2015-06-26T00:00:00ZThe first International Congress of Mathematicians (ICM) was held in Zurich in 1897, setting the standards for all future ICMs. Whilst giving an overview of the congress itself, this thesis focuses on the Swiss organisers, who were predominantly university professors and secondary school teachers. As this thesis aims to offer some insight into their lives, it includes their biographies, highlighting their individual contributions to the congress. Furthermore, it explains why Zurich was chosen as the first host city and how the committee proceeded with the congress organisation.
Two of the main organisers were the Swiss geometers Carl Friedrich Geiser (1843-1934) and Ferdinand Rudio (1856-1929). In addition to the congress, they also made valuable contributions to mathematical education, and in Rudio’s case, the history of mathematics. Therefore, this thesis focuses primarily on these two mathematicians.
As for Geiser, the relationship to his great-uncle Jakob Steiner is explained in more detail. Furthermore, his contributions to the administration of the Swiss Federal Institute of Technology are summarised. Due to the overarching theme of mathematical education and collaborations in this thesis, Geiser’s schoolbook "Einleitung in die synthetische Geometrie" is considered in more detail and Geiser’s methods are highlighted.
A selection of Rudio’s contributions to the history of mathematics is studied as well. His book "Archimedes, Huygens, Lambert, Legendre" is analysed and compared to E W Hobson’s treatise "Squaring the Circle". Furthermore, Rudio’s papers relating to the commentary of Simplicius on quadratures by Antiphon and Hippocrates are considered, focusing on Rudio’s translation of the commentary and on "Die Möndchen des Hippokrates". The thesis concludes with an analysis of Rudio’s popular lectures "Leonhard Euler" and "Über den Antheil der mathematischen Wissenschaften an der Kultur der Renaissance", which are prime examples of his approach to the history of mathematics.
2015-06-26T00:00:00ZEminger, Stefanie UrsulaThe first International Congress of Mathematicians (ICM) was held in Zurich in 1897, setting the standards for all future ICMs. Whilst giving an overview of the congress itself, this thesis focuses on the Swiss organisers, who were predominantly university professors and secondary school teachers. As this thesis aims to offer some insight into their lives, it includes their biographies, highlighting their individual contributions to the congress. Furthermore, it explains why Zurich was chosen as the first host city and how the committee proceeded with the congress organisation.
Two of the main organisers were the Swiss geometers Carl Friedrich Geiser (1843-1934) and Ferdinand Rudio (1856-1929). In addition to the congress, they also made valuable contributions to mathematical education, and in Rudio’s case, the history of mathematics. Therefore, this thesis focuses primarily on these two mathematicians.
As for Geiser, the relationship to his great-uncle Jakob Steiner is explained in more detail. Furthermore, his contributions to the administration of the Swiss Federal Institute of Technology are summarised. Due to the overarching theme of mathematical education and collaborations in this thesis, Geiser’s schoolbook "Einleitung in die synthetische Geometrie" is considered in more detail and Geiser’s methods are highlighted.
A selection of Rudio’s contributions to the history of mathematics is studied as well. His book "Archimedes, Huygens, Lambert, Legendre" is analysed and compared to E W Hobson’s treatise "Squaring the Circle". Furthermore, Rudio’s papers relating to the commentary of Simplicius on quadratures by Antiphon and Hippocrates are considered, focusing on Rudio’s translation of the commentary and on "Die Möndchen des Hippokrates". The thesis concludes with an analysis of Rudio’s popular lectures "Leonhard Euler" and "Über den Antheil der mathematischen Wissenschaften an der Kultur der Renaissance", which are prime examples of his approach to the history of mathematics.Peter Guthrie Tait : new insights into aspects of his life and work : and associated topics in the history of mathematicsLewis, Elizabeth Faithhttp://hdl.handle.net/10023/63302019-04-01T08:38:25Z2015-06-26T00:00:00ZIn this thesis I present new insights into aspects of Peter Guthrie Tait’s life and work, derived principally from largely-unexplored primary source material: Tait’s scrapbook, the Tait–Maxwell school-book and Tait’s pocket notebook. By way of associated historical insights, I also come to discuss the innovative and far-reaching mathematics of the elusive Frenchman, C.-V. Mourey.
P. G. Tait (1831–1901) F.R.S.E., Professor of Mathematics at the Queen’s College, Belfast (1854–1860) and of Natural Philosophy at the University of Edinburgh (1860–1901), was one of the leading physicists and mathematicians in Europe in the nineteenth century. His expertise encompassed the breadth of physical science and mathematics. However, since the nineteenth century he has been unfortunately overlooked—overshadowed, perhaps, by the brilliance of his personal friends, James Clerk Maxwell (1831–1879), Sir William Rowan Hamilton (1805–1865) and William Thomson (1824–1907), later Lord Kelvin.
Here I present the results of extensive research into the Tait family history. I explore the spiritual aspect of Tait’s life in connection with The Unseen Universe (1875) which Tait co-authored with Balfour Stewart (1828–1887). I also reveal Tait’s surprising involvement in statistics and give an account of his introduction to complex numbers, as a schoolboy at the Edinburgh Academy. A highlight of the thesis is a re-evaluation of C.-V. Mourey’s 1828 work, La Vraie Théorie des quantités négatives et des quantités prétendues imaginaires, which I consider from the perspective of algebraic reform. The thesis also contains: (i) a transcription of an unpublished paper by Hamilton on the fundamental theorem of algebra which was inspired by Mourey and (ii) new biographical information on Mourey.
2015-06-26T00:00:00ZLewis, Elizabeth FaithIn this thesis I present new insights into aspects of Peter Guthrie Tait’s life and work, derived principally from largely-unexplored primary source material: Tait’s scrapbook, the Tait–Maxwell school-book and Tait’s pocket notebook. By way of associated historical insights, I also come to discuss the innovative and far-reaching mathematics of the elusive Frenchman, C.-V. Mourey.
P. G. Tait (1831–1901) F.R.S.E., Professor of Mathematics at the Queen’s College, Belfast (1854–1860) and of Natural Philosophy at the University of Edinburgh (1860–1901), was one of the leading physicists and mathematicians in Europe in the nineteenth century. His expertise encompassed the breadth of physical science and mathematics. However, since the nineteenth century he has been unfortunately overlooked—overshadowed, perhaps, by the brilliance of his personal friends, James Clerk Maxwell (1831–1879), Sir William Rowan Hamilton (1805–1865) and William Thomson (1824–1907), later Lord Kelvin.
Here I present the results of extensive research into the Tait family history. I explore the spiritual aspect of Tait’s life in connection with The Unseen Universe (1875) which Tait co-authored with Balfour Stewart (1828–1887). I also reveal Tait’s surprising involvement in statistics and give an account of his introduction to complex numbers, as a schoolboy at the Edinburgh Academy. A highlight of the thesis is a re-evaluation of C.-V. Mourey’s 1828 work, La Vraie Théorie des quantités négatives et des quantités prétendues imaginaires, which I consider from the perspective of algebraic reform. The thesis also contains: (i) a transcription of an unpublished paper by Hamilton on the fundamental theorem of algebra which was inspired by Mourey and (ii) new biographical information on Mourey.Mathematics for history's sake : a new approach to Ptolemy's GeographyMintz, Daniel V.http://hdl.handle.net/10023/21522019-04-01T08:38:50Z2011-06-22T00:00:00ZAlmost two thousand years ago, Claudius Ptolemy created a guide to drawing maps of the world, identifying the names and coordinates of over 8,000 settlements and geographical features. Using the coordinates of those cities and landmarks which have been identified with modern locations, a series of best-fit transformations has been applied to several of Ptolemy’s regional maps, those of Britain, Spain, and Italy. The transformations relate Ptolemy’s coordinates to their modern equivalents by rotation and skewed scaling. These reflect the types of error that appear in Ptolemy’s data, namely those of distance and orientation.
The mathematical techniques involved in this process are all modern. However, these techniques have been altered in order to deal with the historical difficulties of Ptolemy’s maps. To think of Ptolemy’s data as similar to that collected from a modern random sampling of a population and to apply unbiased statistical methods to it would be erroneous. Ptolemy’s data is biased, and the nature of that bias is going to be informed by the history of the data. Using such methods as cluster analysis, Procrustes analysis, and multidimensional scaling, we aimed to assess numerically the accuracy of Ptolemy’s maps. We also investigated the nature of the errors in the data and whether or not these could be linked to historical developments in the areas mapped.
2011-06-22T00:00:00ZMintz, Daniel V.Almost two thousand years ago, Claudius Ptolemy created a guide to drawing maps of the world, identifying the names and coordinates of over 8,000 settlements and geographical features. Using the coordinates of those cities and landmarks which have been identified with modern locations, a series of best-fit transformations has been applied to several of Ptolemy’s regional maps, those of Britain, Spain, and Italy. The transformations relate Ptolemy’s coordinates to their modern equivalents by rotation and skewed scaling. These reflect the types of error that appear in Ptolemy’s data, namely those of distance and orientation.
The mathematical techniques involved in this process are all modern. However, these techniques have been altered in order to deal with the historical difficulties of Ptolemy’s maps. To think of Ptolemy’s data as similar to that collected from a modern random sampling of a population and to apply unbiased statistical methods to it would be erroneous. Ptolemy’s data is biased, and the nature of that bias is going to be informed by the history of the data. Using such methods as cluster analysis, Procrustes analysis, and multidimensional scaling, we aimed to assess numerically the accuracy of Ptolemy’s maps. We also investigated the nature of the errors in the data and whether or not these could be linked to historical developments in the areas mapped.