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The TARA digital repository system captures, stores, indexes, preserves, and distributes digital research material.Fri, 28 Oct 2016 10:31:42 GMT2016-10-28T10:31:42ZSolutions to dilation equations
http://hdl.handle.net/2262/77576
Solutions to dilation equations
Malone, David
This thesis aims to explore part of the wonderful world of dilation equations. Dilation equations have a convoluted history, having reared their heads in various mathematical ﬁelds. One of the early appearances was in the construction of continuous but nowhere differentiable functions. More recently dilation equations have played a signiﬁcant role in the study of subdivision schemes and in the construction of wavelets. The intention here is to study dilatione quations as entities of interest in their own right, just as the similar subjects of differential and difference equations are often studied. It will often be Lp(R) properties we are interested in and we will often use Fourier Analysis as a tool. This is probably due to the author’s original introduction to dilation equations through wavelets. A short introduction to the subject of dilation equations is given in Chapter 1. The introduction is ﬂeeting, but references to further material are given in the conclusion. Chapter 2 considers the problem of ﬁnding all solutions of the equation which arises when the Fourier transform is applied to a dilation equation. Applying this result to the Haar dilation equation allows us ﬁrst to catalogue the L2(R) solutions of this equation and then to produce some nice operator results regarding shift and dilation operators. We then consider the same problem in Rn where, unfortunately, techniques using dilation equations are not as easy to apply. However, the operator results are retrieved using traditional multiplier techniques. In Chapter 3 we attempt to do some hands-on calculations using the results of Chapter 2. We discover a simple ‘factorisation’ of the solutions of the Haar dilation equation. Using this factorisation we produce many solutions of the Haar dilation equation. We then examine how all these results might be applied to the solutions of other dilation equations. A technique which I have not seen exploited elsewhere is developed in Chapter 4. This technique examines a left-hand or right-hand ‘end’ of a dilation equation. It is initially developed to search for reﬁnable characteristic functions and leads to a characterisation of reﬁnable functions which are constant on intervals of the form [n, n +1). This left-hand end method is then applied successfully to the problem of 2- and 3- reﬁnable functions and used to obtain bounds on smoothness and boundedness. Chapter 5 is a collection of smaller results regarding dilation equations. The relatively simple problem of polynomial solutions of dilation equations is covered, as are some methods for producing new solutions and equations from known solutions and equations. Results regarding when self-similar tiles can be of a simple form are also presented.
Mon, 01 Jan 2001 00:00:00 GMThttp://hdl.handle.net/2262/775762001-01-01T00:00:00ZInstitutional and empirical essays on regulatory impact assessment
http://hdl.handle.net/2262/77574
Institutional and empirical essays on regulatory impact assessment
Lyons, Seán
This thesis is concerned with institutional arrangements and empirical tools that may be used to assess the impact of regulation on the economy, and thereby improve regulatory policymaking. We start in Chapter 2 with a qualitative introduction to regulatory impact assessment, highlighting some of the challenges facing small regional governments wishing to improve the quality of regulation they enact. The remainder of the thesis is divided into two parts. Part 1 considers and evaluates the optimal design of tests aimed at discovering specific classes of regulatory impact. Our particular example is a negative clearance test for the effect of regulations on competition. One such tool, the UK competition filter, has been in operation in the UK since 2002. Applying it to case studies from Ireland (in Chapter 3), we show that particular design features may easily lead to excessive false negative results. We then propose a more appropriate structure for tests of this kind. A paper based on this chapter is forthcoming in the Journal of the Statistical and Social Inquiry Society of Ireland. The analysis is extended in Chapter 4 to show that the UK filter is also prone to give excessive false negatives in its own jurisdiction; a paper drawing on these results and co-authored with my supervisor, Dr. Francis O’Toole, is under review by the Journal of Competition Law and Economics. Part 2 focuses on empirical studies of regulatory impact. Two of the three chapters in this part are concerned with aspects of mobile telephony regulation. The first of these, Chapter 5, notes that an increasing number of countries require mobile telephone networks to offer mobile number portability (MNP), which allows customers who wish to switch mobile operator to keep their mobile numbers (including the prefix) and avoid the costs of switching to new numbers. Ex ante assessments suggest that MNP should reduce switching costs and strengthen competition. We construct a new international dataset that allows us to estimate econometric models of the benefits of MNP and we test MNP’s impact on market outcomes using this international time-series cross-section data. The results show that MNP reduces average prices and encourages churn (a proxy for switching) when the switching process is rapid (e.g. less than 5 days) but not when it is slower. A paper based on the research in Chapter 5 is under review by Information Economics and Policy. Chapter 6 also employs the mobile telephony dataset introduced in Chapter 5, focusing this time on a metric popular among regulators and market analysts: average revenue per user, or ARPU. This chapter, which draws on a co-authored paper forthcoming in Telecommunications Policy, develops an econometric model of the determinants of ARPU and shows that this statistic may not be a good indicator of competitive conditions. ARPU proves to be better explained by subscribers’ incomes than by market concentration measures. In Chapter 7, we ask how far the constraints perceived and reported in surveys of small businesses (e.g. excessive regulation, access to capital) are reflected in such firms’ subsequent survival rates. There appears to be a relationship, but it is neither as strong nor as straightforward as the popularity of this sort of evidence among policymakers might lead us to expect. Indeed, the perceived barriers expected to be most significant on the basis of previous research – excessive regulation and constrained access to finance – show no significant relationship to survival rates. Until more convincing evidence is found showing that this type of survey evidence helps explain market outcomes, using it in the assessment or design of public policy may be problematical. The chapter suggests ways of enhancing surveys so that the real effects of perceived constraints can be more clearly identified. A paper drawing on it is currently under review by Small Business Economics.
Mon, 01 Jan 2007 00:00:00 GMThttp://hdl.handle.net/2262/775742007-01-01T00:00:00ZFourier analysis, multiresolution analysis and dilation equations
http://hdl.handle.net/2262/77575
Fourier analysis, multiresolution analysis and dilation equations
Malone, David
This thesis has essentially two parts. The ﬁrst two chapters are an introduction to the related areas of Fourier analysis, multiresolution analysis and wavelets. Dilation equations arise in the context of multi resolution analysis. The mathematics of these two chapters is informal, and is intended to provide a feeling for the general subject. This work is loosely based on two talks which I gave, one during the 1997 Inter-varsity Mathematics competition and the other at the 1997 Dublin Institute for Advanced Studies Easter Symposium. The second part, Chapters 3 and 4, contain original work. Chapter 3 provides a new formal construction of the Fourier transform on Lp(Rn) (1 ≤ p ≤ 2) based on the ideas introduced in Chapter 2. The idea is to take some basic properties of the Fourier transform and show we can construct a bounded operator on L2(R) with these properties. I do this by constructing an operator on each level of the Haar multiresolution analysis, which I then show is well enough behaved to be extended by a limiting process to all of L2(R). Some of the important properties of the Fourier transform are also derived in terms of this construction, and the generalisations to Lp(Rn) are explored. Chapter 4 builds on the work of Chapters 3 and provides a uniqueness result for the Fourier transform. While searching for this result I also establish a related result for dilation equations (a subject also introduced in Chapter 2). Here the exact set of properties which were used to deﬁne the Fourier transform are varied in an effort to discover which are merely consistent with the Fourier transform and which strong enough to deﬁne it. I end up examining sets of dilation equations and determining when these will have a unique solution.
Thu, 01 Jan 1998 00:00:00 GMThttp://hdl.handle.net/2262/775751998-01-01T00:00:00ZPlant and snail communities in three habitat types in a limestone landscape in the west of Ireland, and the effects of exclusion of large grazing animals
http://hdl.handle.net/2262/77573
Plant and snail communities in three habitat types in a limestone landscape in the west of Ireland, and the effects of exclusion of large grazing animals
Long, Maria P.
This thesis documents the plant and snail communities found in woodland, scrub and grassland in the Burren region in the west of Ireland. The flora of the Burren is renowned and has been well studied, but the vegetation communities are less well understood. In relation to molluscs, their distribution in Ireland is quite well documented, but studies on molluscan ecology and community structure are lacking, particularly for land snails. Additional work with the molluscan data includes an examination of the population structure and an assessment of methodological issues. This thesis also investigates experimentally the short term changes in plant and snail communities following cessation of grazing by large herbivores. This study is exceptional in assessing the effects of grazing by looking at multiple habitats using a replicated, balanced multisite design. The project design enables the study of ecological change on a number of scales – quadrat, site, habitat and landscape. Scrub encroachment is a big issue for many land owners and managers in the Burren, with hazel, Corylus avellana, being the most significant species involved. The woodland and scrub habitats selected for study were hazel-dominated, and all of the grasslands had hazel scrub nearby. The findings of the vegetation study indicate that, unsurprisingly, the vegetation of woodlands and grasslands differ substantially, with soil fertility as well as light penetration being important in the separation. Interestingly, the scrub vegetation differed floristically from both woodland and grassland. Further, it could be split into two distinct subsets – ‘woody’ and ‘grassy’. These elements formed reasonably distinct entities which were related to the woodland and grassland vegetation communities respectively, but were distinct from either. A total of 30 species of snail was recorded, which is approximately 45% of the total number of land snails in Ireland. This included a number of species from the ‘Red List’ for Irish non-marine molluscs. The woodlands and scrub had higher abundances of snails and were more species rich than the grasslands. The amount of litter in a quadrat was found to be an important factor correlated with species richness of snails, while plant species richness was not found to be correlated with snail richness. With regard to population structure of snails, the populations at the study sites were shown to be composed mainly of juveniles, with only 28% adults. The inclusion of dead and immature individuals in the results added six to the species list, but these species occurred in very low numbers. The relative abundances of species was shifted, however, if only adults were included. The advantages of using a 0.5mm sieve mesh size for processing samples were shown by the large numbers of snails found in this smaller size fraction and by the demonstration that a number of species are underestimated when sampling using a 1mm sieve mesh. However, these advantages need to be weighed against the benefits gained in terms of decreased lab work time. The changes in the vegetation brought about by the cessation of grazing were rapid and dramatic in the grasslands. Many plant species declined in abundance and several flowering plant species disappeared. There was a major build-up of litter, and cover of grasses increased significantly. Both diversity and species richness of plants decreased. The woodlands presented contrasting findings to the grasslands, with plant diversity increasing significantly. Species richness increased also, although the change was not statistically significant. The amount of bare earth decreased sharply, and the cover of field layer plants increased in parallel. There was little detectable pattern of change in the scrub vegetation; this can be ascribed to the heterogeneity and variability of the habitat and the restricted timeframe of the 24-month study period. A large change was seen in the snail communities in the grasslands in this case, abundance and species richness increased. The changes were linked with the litter build-up, and the denser, taller vegetation within the fenced plots. Few individual species showed strong trends, with the pattern instead being a small and variable, but relatively consistent, increase across all species. The snail communities showed little appreciable changes in the woodlands and scrub during the timespan of this survey. Again, a period of 24 months may not have been long enough for measurable changes to manifest themselves. Land abandonment is a major threat in many ecosystems and the cessation of existing management regimes (e.g. grazing) is likely to have a large impact on plant and animal communities. Changes have been seen in the Burren in recent decades, with perhaps the most dramatic example being the expansion of hazel scrub. This has been attributed mainly (though not exclusively) to changes in grazing practices. The network of fenced exclosures, and their associated control plots, set up during this project are an important resource for the study and documentation, now and in the future, of how changing management practices are affecting plant and animal communities. Already, the loss of some plant species has been documented from grasslands in the absence of grazing, indicating how essential grazing is for the maintenance of semi-natural grasslands in the region. However, this loss of diversity is offset by the success of the snails in the ungrazed plots, reminding us that solutions are rarely straightforward in conservation management, and that a variety of structural elements (e.g. grazed and ungrazed patches) is probably optimal for biodiversity.
Sat, 01 Jan 2011 00:00:00 GMThttp://hdl.handle.net/2262/775732011-01-01T00:00:00Z