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Subject: search.filters.subject.Mathematical analysis

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    Convexity and linear distortion : a thesis presented in partial fulfilment of the requirements for the degree of Master of Science in Mathematics, Institute of Natural and Mathematical Science, Massey University of Albany, New Zealand
    (Massey University, 2017) Hashemi, Seyed Mohsen
    This thesis is primarily concerned with the convexity properties of distortion functionals (particularly the linear distortion) defined on quasiconformal homeomorphisms of domains in Euclidean n-spaces, though we will mainly stick to three-dimensions. The principal applica-tion is in identifying the lower semi-continuity of distortion on uniformly convergent limits of sequences of quasiconformal mappings. For example, given the curve family or analytic definitions of quasiconformality - discussed in this thesis - it is known that if {fn}n=1 is a sequence of K-quasiconformal mappings (and here K depends on the particular distortion but is the same for every element of the sequence) which converges to a function f, then the limit function is also K-quasiconformal. Despite a widespread belief that this was also true for the geometric definition of quasi-conformality (via the linear distortion H(f) defined below) Tadeusz Iwaniec gave a specific surprising example to show that the linear distortion function is not lower semicontinuous. The main aim of this thesis is to show that this failure of lower semicontinuity is actually far more common, perhaps even generic in the sense that it might be true that under mild restrictions on a quasiconformal f, there may be a sequence {fn}n=1 with fn → f uniformly and with lim supn→∞ H(fn) < H(f). The main result of this thesis is to show this is true for a wide class of linear mappings.
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    Simulating the RNA-world and computational ribonomics : a thesis presented for the degree of Doctor of Philosophy in Biomathematics at Massey University, Palmerston North, New Zealand
    (Massey University, 2003) Gardner, Paul Phillip
    Project 1: Experiments by Piccirilli et al (Nature, Lond. 343, 33-37 (1990)) have shown that the canonical RNA genetic alphabet, AUCG (or ATCG in DNA), is not the only possible nucleotide alphabet. In this work we address the question "Is the canonical alphabet optimal?" Computational tools are used to infer RNA secondary structures (shapes) from RNA sequences of various possible alphabets, and measures of RNA shape are gathered with respect to alphabet size. Then, simulations based upon replication and selection of fixed sized RNA populations are used to investigate the effect of alternative alphabets upon RNAs ability to evolve through a fitness landscape. Those results imply that for low copy fidelity the canonical alphabet is fitter than two, six and eight letter alphabets. Under high copy fidelity conditions, a six letter alphabet out-performed the four letter alphabets, which suggests that the canonical alphabet is indeed a relic of the RNA-world. Project 2: Non-coding RNA genes produce functional RNA molecules rather than proteins. One such family is the H/ACA snoRNAs. Unlike the related C/D snoRNAs, these have resisted automated detection until recently. We develop an algorithm for screening the Saccharomyces cerevisiae genome for novel H/ACA snoRNAs. To achieve this, we introduce some new methods to facilitate the search for non-coding RNAs in genomic sequences which are based on properties of predicted minimum free energy (MFE) secondary structures. The algorithm has been implemented and can be generalised to enable screening of other eukaryote genomes. We find that use of primary sequence data alone is insufficient for identifying novel H/ACA snoRNAs. The use of secondary structure filters reduces the number of candidates to a manageable size. On the basis of genomic location data, we identify three strong H/ACA snoRNA candidates. These together with a further 47 candidates obtained by our analysis are being screened experimentally and investigated (along with known H/ACA snoRNAs) using comparative genomic analysis.
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    Evolutionary analyses of large data sets : trees and beyond : a thesis presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Mathematics at Massey University
    (Massey University, 2001) Holland, Barbara Ruth
    The increasing amount of molecular data available for phylogenetic studies means that larger, often intra-species, data sets are being analysed. Treating such data sets with methods designed for small interspecies data may not be useful. This thesis comprises four projects within the field of phylogenetics that focus on cases where the application of current tree estimation methods is not sufficient to answer the biological questions of interest. A simulation study contrasts the accuracy of several tree estimation methods for a particular class of five-taxon, equal-rate, trees. This study highlights several difficulties with tree estimation, including the fact that some tree topologies produce “misleading" patterns that are incorrectly interpreted; that correction for multiple changes does not always increase accuracy, because of increased variance; and the difficulty of correctly placing outgroup taxa. A mitochondrial DNA data set, containing over 400 modern and ancient Adélie penguin samples, is used to estimate the rate of evolution. Straightforward tree-estimation is unhelpful because the amount of homoplasy in the data makes the construction of a single reliable tree impossible. Instead the data is represented by a network. A method, that extends statistical geometry, assesses whether or not a data set can be well-represented by a tree. The "tree-likeness" of each quartet in the data is evaluated and displayed visually, either for the entire data set or by taxon. This aids in identifying reticulate (or simply noisy) data sets, and also particular taxa that confound tree-like signal. Novel methods are developed that use pairwise dissimilarities between isolates in intra-species microbial data sets, to identify strains that are good representatives of their species or subspecies.