Topics in perturbation theory

In providing a means of progressively improving an initial estimate, perturbation series have become a ubiquitous tool in modern physics. However, and mainly because this stepwise process of improvement rapidly becomes increasingly involved, surprisingly little is known about the formal properties of the series obtained. This thesis therefore investigates some aspects of these properties and how they effect the application of these techniques, with an emphasis on quantum field theory and the phenomenology of e+e(^-) colliders. One of the better understood examples of a perturbative series is the WKB one which is widely used to approximate the energy levels of quantum mechanical systems. Recently much interest has centred on a modification of this, the SWKB series. Apart from (possibly) offering an improvement on the original, this is intrinsically interesting in being related to the supersymmetry of field theory. Furthermore, as Chapter 1 explains, there is a close connection between the cases where the initial estimate requires no correction and an important set of quantum mechanical problems (the "shape invariant" ones) which can be solved elegantly and completely. The situation in field theory is more complicated, not least because the series for any particular problem is no longer unique. While this presents few theoretical difficulties, it has serious consequences when attempts are made to compare predictions with experiment. This obstacle is particularly severe in Quantum Chromodynamics and its fundamental constant (A(_QCD)) is therefore only roughly known at present. It will be argued that current responses to this are all imperfect, but that tests of the theory can be envisaged that circumvent the problem. This leads into questions concerning the origin of the divergences in the perturbation series - for although it may initially provide usefully improved estimates, the series probably breaks down eventually. Existing arguments about this topic are critically reviewed - and in one case substantially simplified - before an alternative one is proposed in some detail. By concentrating on a particularly restricted situation, the Common Effective Charge Approach simplifies matters to the extent that issues such as non-analyticity of functions and the potential accuracy of perturbative techniques in realistic applications can be conveniently investigated

A study of Borsuk's hyperspace 2 .

Abstract Not Availabl

Restricted permutations, antichains, atomic classes and stack sorting

Involvement is a partial order on all finite permutations, of infinite dimension and having subsets isomorphic to every countable partial order with finite descending chains. It has attracted the attention of some celebrated mathematicians including Paul Erdős and, due to its close links with sorting devices, Donald Knuth. We compare and contrast two presentations of closed classes that depend on the partial order of involvement: Basis or Avoidance Set, and Union of Atomic Classes. We examine how the basis is affected by a comprehensive list of closed class constructions and decompositions. The partial order of involvement contains infinite antichains. We develop the concept of a fundamental antichain. We compare the concept of 'fundamental' with other definitions of minimality for antichains, and compare fundamental permutation antichains with fundamental antichains in graph theory. The justification for investigating fundamental antichains is the nice patterns they produce. We forward the case for classifying the fundamental permutation antichains. Sorting devices have close links with closed classes. We consider two sorting devices, constructed from stacks in series, in detail. We give a comment on an enumerative conjecture by Ira Gessel. We demonstrate, with a remarkable example, that there exist two closed classes, equinumerous, one of which has a single basis element, the other infinitely many basis elements. We present this paper as a comprehensive analysis of the partial order of permutation involvement. We regard the main research contributions offered here to be the examples that demonstrate what is, and what is not, possible; although there are numerous structure results that do not fall under this category. We propose the classification of fundamental permutation antichains as one of the principal problems for closed classes today, and consider this as a problem whose solution will have wide significance for the study of partial orders, and mathematics as a whole

Dual processes in mathematics: reasoning about conditionals

This thesis studies the reasoning behaviour of successful mathematicians. It is based on the philosophy that, if the goal of an advanced education in mathematics is to develop talented mathematicians, it is important to have a thorough understanding of their reasoning behaviour. In particular, one needs to know the processes which mathematicians use to accomplish mathematical tasks. However, Rav (1999) has noted that there is currently no adequate theory of the role that logic plays in informal mathematical reasoning. The goal of this thesis is to begin to answer this specific criticism of the literature by developing a model of how conditional “if…then” statements are evaluated by successful mathematics students. Two stages of empirical work are reported. In the first the various theories of reasoning are empirically evaluated to see how they account for mathematicians’ responses to the Wason Selection Task, an apparently straightforward logic problem (Wason, 1968). Mathematics undergraduates are shown to have a different range of responses to the task than the general well-educated population. This finding is followed up by an eve-tracker inspection time experiment which measured which parts of the task participants attended to. It is argued that Evans’s (1984, 1989, 1996, 2006) heuristic-analytic theory provides the best account of these data. In the second stage of empirical work an in-depth qualitative interview study is reported. Mathematics research students were asked to evaluate and prove (or disprove) a series of conjectures in a realistic mathematical context. It is argued that preconscious heuristics play an important role in determining where participants allocate their attention whilst working with mathematical conditionals. Participants’ arguments are modelled using Toulmin’s (1958) argumentation scheme, and it is suggested that to accurately account for their reasoning it is necessary to use Toulmin’s full scheme, contrary to the practice of earlier researchers. The importance of recognising that arguments may sometimes only reduce uncertainty in the conditional statement’s truth/falsity, rather than remove uncertainty, is emphasised. In the final section of the thesis, these two stages are brought together. A model is developed which attempts to account for how mathematicians evaluate conditional statements. The model proposes that when encountering a mathematical conditional “if P then Q”, the mathematician hypothetically adds P to their stock of knowledge and looks for a warrant with which to conclude Q. The level of belief that the reasoner has in the conditional statement is given by a modal qualifier which they are prepared to pair with their warrant. It is argued that this level of belief is fixed by conducting a modified version of the so-called Ramsey Test (Evans & Over, 2004). Finally the differences between the proposed model and both formal logic and everyday reasoning are discussed

Dual processes in mathematics : reasoning about conditionals

This thesis studies the reasoning behaviour of successful mathematicians. It is based on the philosophy that, if the goal of an advanced education in mathematics is to develop talented mathematicians, it is important to have a thorough understanding of their reasoning behaviour. In particular, one needs to know the processes which mathematicians use to accomplish mathematical tasks. However, Rav (1999) has noted that there is currently no adequate theory of the role that logic plays in informal mathematical reasoning. The goal of this thesis is to begin to answer this specific criticism of the literature by developing a model of how conditional “if…then” statements are evaluated by successful mathematics students. Two stages of empirical work are reported. In the first the various theories of reasoning are empirically evaluated to see how they account for mathematicians’ responses to the Wason Selection Task, an apparently straightforward logic problem (Wason, 1968). Mathematics undergraduates are shown to have a different range of responses to the task than the general well-educated population. This finding is followed up by an eve-tracker inspection time experiment which measured which parts of the task participants attended to. It is argued that Evans’s (1984, 1989, 1996, 2006) heuristic-analytic theory provides the best account of these data. In the second stage of empirical work an in-depth qualitative interview study is reported. Mathematics research students were asked to evaluate and prove (or disprove) a series of conjectures in a realistic mathematical context. It is argued that preconscious heuristics play an important role in determining where participants allocate their attention whilst working with mathematical conditionals. Participants’ arguments are modelled using Toulmin’s (1958) argumentation scheme, and it is suggested that to accurately account for their reasoning it is necessary to use Toulmin’s full scheme, contrary to the practice of earlier researchers. The importance of recognising that arguments may sometimes only reduce uncertainty in the conditional statement’s truth/falsity, rather than remove uncertainty, is emphasised. In the final section of the thesis, these two stages are brought together. A model is developed which attempts to account for how mathematicians evaluate conditional statements. The model proposes that when encountering a mathematical conditional “if P then Q”, the mathematician hypothetically adds P to their stock of knowledge and looks for a warrant with which to conclude Q. The level of belief that the reasoner has in the conditional statement is given by a modal qualifier which they are prepared to pair with their warrant. It is argued that this level of belief is fixed by conducting a modified version of the so-called Ramsey Test (Evans ;Over, 2004). Finally the differences between the proposed model and both formal logic and everyday reasoning are discussed.EThOS - Electronic Theses Online ServiceEconomic and Social Research Council (Great Britain) (ESRC)GBUnited Kingdo

The role of the crucial experiment in student modelling

As the range of models which tutoring systems can capture is extended, efficient diagnosis becomes more difficult. This thesis describes a solution to this problem based on the generation of 'Critical Problems'; their role in student modelling is analogous to that of the 'Crucial Experiment' in science. We argue that great diagnostic power can be obtained by generating discriminatory problem examples. In general, efficient diagnosis is just not possible without such an hypothesis-testing capability. We describe a program, PO, which given a pair of production rule models and a description of the class of problems which the student must solve, generates an abstract specification of the problems which discriminate between those two hypotheses. Through a process termed 'Abstract Interpretation', PO tips the balance in favour of diagnostic measurement. The key to this problem lies in the realisation that we are only interested in the abstract mapping between a model's inputs and outputs; from the point of view of generating a Critical Problem, the intermediate processing of the model is irrelevant