This thesis investigates nondeterminism in functional programming languages. To establish a precise understanding of nondeterministic language properties, Sondergaard and Sestoft's analysis and definitions of functional language properties are adopted as are the characterizations of weak and strong nondeterminism. This groundwork is followed by a denotational semantic description of a nondeterministic language (suggested by Sondergaard and Sestoft). In this manner, a precise characterization of the effects of strong nondeterminism is developed. Methods used to hide nondeterminism to in order to overcome or sidestep the problem of strong nondeterminism in pure functional languages are defined. These different techniques ensure that functional languages remain pure but also include some of the advantages of nondeterminism. Lastly, this discussion of nondeterminism is applied to the area of functional parallel language implementation to indicate that the related problem and the possible solutions are not purely academic. This application gives rise to an interesting discussion on optimization of list parallelism. This technique relies on the ability to decide when a bag may be used instead of a list.

Static analysis is the name given to a number of compile time analysis techniques used to automatically generate information which can lead to improvements in the execution performance of function languages. This thesis provides an introduction to these techniques and their implementation. The abstract interpretation framework is an example of a technique used to extract information from a program by providing the program with an alternate semantics and evaluating this program over a non-standard domain. The elements of this domain represent certain properties of interest. This framework is examined in detail, as well as various extensions and variants of it. The use of binary logical relations and program logics as alternative formulations of the framework , and partial equivalence relations as an extension to it, are also looked at. The projection analysis framework determines how much of a sub-expression can be evaluated by examining the context in which the expression is to be evaluated, and provides an elegant method for finding particular types of information from data structures. This is also examined. The most costly operation in implementing an analysis is the computation of fixed points. Methods developed to make this process more efficient are looked at. This leads to the final chapter which highlights the dependencies and relationships between the different frameworks and their mathematical disciplines. / KMBT_223

Monitoring the functional connectivity between brain networks is becoming increasingly important in elucidating brain functionality in normal and disease states. Current methods of detecting networks in the recorded EEG such as correlation and coherence are limited by the fact that they assume stationarity of the relationship between channels, and rely on linear dependencies. Here we utilize mutual information (MI) as the metric for determining nonlinear statistical dependencies between electroencephalographic (EEG) channels. Previous work investigating MI between EEG channels in subjects with widespread diseases of the cerebral cortex had subjects simply rest quietly with their eyes closed. In motor disorders such as Parkinson’s disease (PD), abnormalities are only expected during performance of motor tasks, but this makes the assumption of stationarity of relationships between EEG channels untenable. We therefore propose a novel EEG segmentation method based on the temporal dynamics of the cross-spectrogram of the computed Independent Components (ICs). After suitable thresholding of the MI values between channels in the temporally segmented EEG, graphical theoretical analysis approaches are applied to the derived networks. The method was applied to EEG data recorded from six normal subjects and seven PD subjects on and off medication performing a motor task involving either their right hand only or both hands simultaneously. One-way analysis of variance (ANOVA) tests demonstrated statistically significant difference between subject groups. This proposed segmentation/MI network method appears to be a promising approach for EEG analysis. / Applied Science, Faculty of / Electrical and Computer Engineering, Department of / Graduate

Regression models with a scalar response and a functional predictor have been extensively studied. One approach is to approximate the functional predictor using basis function or eigenfunction expansions. In the expansion, the coefficient vector can either be fixed or random. The random coefficient vector is also known as random effects and thus the regression models are in a mixed effects framework. The random effects provide a model for the within individual covariance of the observations. But it also introduces an additional parameter into the model, the covariance matrix of the random effects. This additional parameter complicates the covariance matrix of the observations. Possibly, the covariance parameters of the model are not identifiable. We study identifiability in normal linear mixed effects models. We derive necessary and sufficient conditions of identifiability, particularly, conditions of identifiability for the regression models with a scalar response and a functional predictor using random effects. We study the regression model using the eigenfunction expansion approach with random effects. We assume the random effects have a general covariance matrix and the observed values of the predictor are contaminated with measurement error. We propose methods of inference for the regression model's functional coefficient. As an application of the model, we analyze a biological data set to investigate the dependence of a mouse's wheel running distance on its body mass trajectory. / Science, Faculty of / Statistics, Department of / Graduate

Many functional programming languages rely on the elimination of 'impure' features: assignment to variables, exceptions and even input/output. But some of these are genuinely useful, and it is of real interest to establish how they can be reintroducted in a controlled way. This dissertation looks in detail at one example of this: the addition to a functional language of dynamically generated names. Names are created fresh, they can be compared with each other and passed around, but that is all. As a very basic example of state, they capture the graduation between private and public, local and global, by their interaction with higher-order functions. The vehicle for this study is the nu-calculus, an extension of the simply-typed lambdacalculus. The nu-calculus is equivalent to a certain fragment of Standard ML, omitting side-effects, exceptions, datatypes and recursion. Even without all these features, the interaction of name creation with higher-order functions can be complex and subtle. Various operational and denotational methods for reasoning about the nu-calculus are developed. These include a computational metalanguage in the style of Moggi, which distinguishes in the type system between values and computations. This leads to categorical models that use a strong monad, and examples are devised based on functor categories. The idea of logical relations is used to derive powerful reasoning methods that capture some of the distinction between private and public names. These techniques are shown to be complete for establishing contextual equivalence between first-order expressions; they are also used to construct a correspondingly abstract categorical model. All the work with the nu-calculus extends cleanly to Reduced ML, a larger language that introduces integer references: mutable storage cells that are dynamically allocated. It turns out that the step up is quite simple, and both the computational metalanguage and the sample categorical models can be reused.

Certain properties of an unknown element u in a Hilbert space are investigated. For u satisfying certain linear constraints, it is shown that approximations to u and error bounds for the approximations may be obtained in terms of functional representers. The general approximation method is applied to homogeneous systems of ordinary linear differential equations and various formulae are derived. An Alwac III-E digital computer was used to compute optimal approximations and error bounds with the aid of these formulae. Numerous applications to particular systems are mentioned. On the basis of the numerical results, certain remarks are given as a guide for the numerical application of the method, at least in the framework of ordinary differential equations. From the cases studied it is seen that this can be a practicable method for the numerical solution of differential equations. / Science, Faculty of / Mathematics, Department of / Graduate

M.Sc.

M.Sc. (Mathematics) / Any sensible piece of writing has an intended readership. Conversely, any piece of writing that has no intended readership has no sense. These are axioms of authorship and necessary directions to any prospective author. The aim of this dissertation was to serve as an experimental exposition of the analysis of the resolvent operator. Its intended readership is therefore graduate-level students in operator theory and Banach algebras. The analysis included in this dissertation is of a specific kind: it includes and occasionally extends beyond the analysis of a function at certain of its singularities of finite order. The exposition is experimental in the sense that it does not even aim at a comprehensive review of analysis of the resolvent operator, but it is concerned with that part of it which seems to have interesting and useful results and which appears to be the most suggestive of further research. In order to obtain an exhaustive exposition, we still lack a study of the properties of the resolvent operator where it is differentiable (which seemingly entails little more than undergraduate-level complex analysis), and a study of essential singularities of the resolvent operator (which seems too difficult for the expository style). A brief overview of the contents of this dissertation is in order: a chapter introducing some analytic concepts used throughout this dissertation; a chapter on poles of order 1 follows (so-called simple poles), where the Gelfand theorem (2.1.1) is the most important result; a chapter on poles of higher order, where the Hille theorem is the most prominent; and lastly some topics that have arisen out of the study of poles of the resolvent, collected in chapter 4. I should make it abundantly clear to the reader that although this dissertation is my work, it does not for the most part follow that the result are my own. What is my own is the arrangement, but as it is a literature study, the results are mainly those of other authors. My own addition has been mostly notes, usually in italics. The literature study has benefited very much from Zemanek's paper (Zemanek,[54]), and I am deeply indebted to him for it. Incidentally, this has also been a chance to exhibit my style of citation; the number corresponds to the number of the citation in the bibliography. There are numerous instances where I have indicated possible extensions and recumbent studies that could be roused effectively, but which have swelled this volume unnecessarily. For instance, the last subsection is little more than such indications.

Human activities are altering species' environments, consequently driving many to extinction and changing biotic communities worldwide. Functional traits are species’ intrinsic characteristics that shape their roles in an environment. Loss of functional diversity compromises ecosystem processes and potentially the resilience of communities facing further change. Biological homogenization reflects the non-random loss of species and possibly also of trait distributions within community, leading to the increasing ubiquity of some traits and growing rarity of others. By changing the composition and configuration of species’ local habitat and their regional surroundings, land use can alter community dynamics. The extent to which land uses within habitats and across the surrounding landscape matrix alter the distribution of functional traits in biotic communities remains highly uncertain but could determine how to design management strategies intended to aid conservation. Here, I investigate the relative contributions of compositional and configurational landscape characteristics at local and regional scales on the diversity and composition of functional traits within butterfly communities. I constructed models to identify landscape predictors of functional trait diversity but found no significant associations with individual traits. Managing habitat quality by fostering compositional and configurational heterogeneity in the local landscape can improve functional diversity. However, efforts to facilitate conservation of species with rare traits necessitates maintaining compositional and configurational variety within habitats and in the surrounding matrix.

A cone K in a vector space X is a subset which is closed under addition, positive scalar multiplication and the only element with additive inverse is zero. The pair (X, K) is called an ordered vector space. In this study, we consider the characterizations of reflexive Banach spaces. This is done by considering cones with bounded and unbounded bases and the second characterization is by reflexive cones. The relationship between cones with bounded and unbounded bases and reflexive cones is also considered. We provide an example to show distinction between such cones. / Dissertation (MSc (Mathematics))--University of Pretoria 2020. / UCDP - 523 / Mathematics and Applied Mathematics / MSc (Mathematics) / Unrestricted