A history of vegetation and climate change during the last two millennia is elucidated from ombrogenous blanket peat sequences from the central and eastern North York Moors. The evidence is derived from five mires Harwood Dale Bog, May Moss, Fen Bogs, Yarlsey Moss and Bluewath Beck. May Moss received particular attention because it is the only remaining unmodified blanket mire on the North York Moors. All the sites were cored, with May Moss yielding seven cores, four of which were extruded along a five metres transect. The cores were selectively analysed for plant macrofossil, testate amoebae, humification and pollen. Chronologies were constructed using 14C dating and the judicious use of biostratigraphic marker horizons. Comparison of 14C dates obtained on bulk peat samples and on pure Sphagnum remains encountered substantial differences, which raises anxieties about 14C dating of a material as heterogeneous as peat. The regional vegetation history elucidated from the pollen evidence reflects changes in the demography, culture, economy and climate of the North York Moors. Evidence of woodland decline and abundant agricultural taxa are attributed to phases of increased agricultural exploitation of the uplands in response to a commercial approach to farming during the Romano-British period, population expansion during the Anglo-Scandinavian period, and attempts to exploit the moorlands during the boom periods of the 12th-13th and 15th-16th centuries. Conversely, phases of woodland expansion and agricultural decline are associated with the Roman withdrawal from England, the 'harrying of the north' in AD 1069-70 and demographic collapse during the 14th century. T estate amoebae, plant macrofossil and humification stratigraphies provide a record of mire palaeohydrology, which is used to infer a history of effective precipitation. There is a broad consistency within the palaeohydrological indications from a single core, which indicates that the techniques support each other. Furthermore, similar testate amoebae, plant macrofossil and humification stratigraphies were encountered in adjacent cores at May Moss. There is evidence of pronounced shifts to wetter/cooler conditions circa 500 BC, AD 450, 850, 1400, 1625 and 1825 separated by unambiguously drier/warmer phases circa AD 200-450, 700-800, 1100-1200, 1550-1600 and 1750-1800. The palaeoclimate time series displays a strong correlation with the record of solar variability; however, biosphere, atmosphere and oceanic interactions in the North Atlantic region and global volcanism also affect regional climate.

The concept of a tree has been used in various areas of mathematics for over a century. In particular, trees appear to be one of the most fundamental notions in computer science. Sequential algorithms for trees are generally well studied. Unfortunately many of these sequential algorithms use methods which seem to be inherently sequential. One of the contributions of this thesis is the introduction of several parallel techniques for the construction of various types of trees and the presentation of new parallel tree construction algorithms using these methods. Along with the parallel tree construction techniques presented here, we develop techniques which have broader applications. We use the Parallel Random Access Machine as our model of computation. We consider two basic methods of constructing trees:tree expansion and tree synthesis. In the tree expansion method, we start with a single vertex and construct a tree by adding nodes of degree one and/or by subdividing edges. We use the parallel tree expansion technique to construct the tree representation for graphs in the family of graphs known as cographs. In the tree synthesis method, we start with a forest of single node subtrees and construct a tree by adding edges or (for rooted trees) by creating parent nodes for some roots of the trees in the forest. We present a family of parallel and sequential algorithms to construct various approximations to the Huffman tree. All these algorithms apply the tree synthesis method by constructing a tree in a level-by-level fashion. To support one of the algorithms in the family we develop a technique which we call the cascading sampling technique. One might suspect that the parallel tree synthesis method can be applied only to trees of polylogarithmic height, but this is not the case.We present a technique which we call the valley filling technique and develop its accelerated version called the accelerated valley filling technique. We present an application of this technique to an optimal parallel algorithm for construction of minimax trees. / Science, Faculty of / Computer Science, Department of / Graduate

M.Sc. (Mathematics) / Please refer to full text to view abstract

In this paper we determine when there exists a matrix M in PGL2(F), and its form, such that L_k = D^M_k where D^M_k is a higher-order Dickson polynomial. We first examine the cases where M has projective orders 3, 4, and 6. For the order 3 case, we find that M has entries in, at worst, a quadratic extension of F. This is also true for the orders 4 and 6, but requires a restriction on the coefficients of h(x), the characteristic polynomial of L. In all cases, an explicit formula for M is given, and in the order 4 case the meaning of the extension is interpreted in terms of the Galois group of h. Lastly, we examine the case where F is finite, and offer a formula for M of order 5.

In this dissertation, I discuss bounds for the set of possible number of zeros of a homogeneous linear recurring sequence over a finite field of q elements, based on an irreducible minimal polynomials of degree d and order m as the characteristic polynomial. I prove upper and lower bounds on the cardinality of the set of number of zeros. The set is determined when t= (qd-1)/m has the form qa+1 or q2a-qa+1 where a is a positive integer. The connection with coding theory is a key ingredient. Also it is proved that the upper bound defined here is the best bound for the cardinality of the set of zeros, in the sense that it is reached infinitely often.

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Large scale genomic sequencing efforts have resulted in a massive inflow of raw sequence data. This raw data, when appropriately processed and analyzed, can provide insight to a trained biologist and aid in hypothesis-driven research. Given the time and resource requirements necessary for biological experiments, computational predictions of gene functions can aid in reducing a large list of candidate genes to a few promising targets. Various computational solutions have been proposed and developed for gene function prediction. These solutions utilize various forms of data, such as DNA/RNA/protein sequences, protein structures, interaction networks, literature mining, and a combination of these data sources. However, these methods do not always produce precise results as the underlying data sets used for training or modeling are quite sparse. We developed and used a massive sequence similarity network build over 108 million known protein sequences to aid in protein function prediction. Predictions are made through the alignment of query sequences to representative sequences for a given cluster derived from the massive sequence similarity network. Derived clusters aggregate information (particularly that from the Gene Ontology) from respective members, which we then consolidate through a novel weighted path method. We evaluate our method on four holdout datasets using CAFA evaluation metrics. Our results suggest that clustering significantly reduces the time and memory requirements, with a marginal impact on predictive power. At lower sequence similarity thresholds, our method outperforms other gold standard methods. / Master of Science / We often think of a protein as a nutritional requirement. However, proteins are far more than just food, they play countless and unappreciated roles in facilitating life. From transporting nutrients in the body, synthesis of hormones, functioning as enzymes to expediting chemical reactions, serving as the scaffold for cells and tissues, to protecting the body against foreign pathogens. On a molecular level, each protein is made up of chains of 20 different amino acids, just like a chain of beads, that are then folded to create a 3-dimensional structure. The variations in the ordering of amino acids result in different types of proteins. There are millions of genes across known life, and they perform different functions when translated into proteins. Nature has given us many proteins with interesting properties, and the low cost of sequencing their precursors (DNA) has resulted in large amounts of sequence data that is not yet associated with a function. Biological experiments to determine the function of a protein can be time consuming and expensive. We built a massive network encompassing 108 million protein sequences based on sequence similarity. This ensures that we make use of as much data as possible to make better predictions. Specifically, our work focuses on utilizing this information of similar proteins to aid in predicting the functions of a protein given its sequences. It is based on the idea of guilt by association, such that if two proteins are similar in sequences, they perform similar functions. We show that using computationally efficient methods and large datasets, one can achieve fast and highly precise predictions.

Piaget and Freudenthal advocated thinking levels. In the 1950's the van Hieles developed a five level model of geometric thought. Judith Land adapted the model in 1990, utilising four levels to teach the concept of functions. These four levels have been considered here in the formation of concepts of sequences and series. The origin and relevance of sequences and series have been studied and the importance of visualisation, patterning and generalisation in the instructional process investigated. A series of lessons on these topics was taught to a group of six higher grade matriculation students of mixed ability and gender. Questionnaires related to student progress through the various levels were answered, categorised, graphed and analysed. Despite the small number of students, results seem to indicate that emphasising visualisation, exploring patterns and generalisation and teaching the topics as a reinvention had made a positive contribution towards progress through the various thought levels. / Mathematics Education / M.A. (Mathematics Education)