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Copy number Variants (CNVs), which comprise deletions, insertions and inversions of genomic sequence, are a main form of genetic variation between individual genomes. CNVs are commonly present in the genomes of human and other species. However, they have not been extensively characterized as their ascertainment is challenging. I reviewed current CNV studies and CNV discovery methods, especially the algorithms which infer CNVs from whole genome Single Nucleotide Polymorphism (SNP) arrays and compared the performance of three analytical tools in order to identify the best method of CNV identification. Then I applied this method to identify CNV events in three European population isolates—the island of Vis in Croatia, the islands of Orkney in Scotland and villages in the South Tyrol in Italy - from Illumina genome-wide array data with more than 300,000 SNPs. I analyzed and compared CNV features across these three populations, including CNV frequencies, genome distribution, gene content, segmental duplication overlap and GC content. With the pedigree information for each population, I investigated the inheritance and segregation of CNVs in families. I also looked at association between CNVs and quantitative traits measured in the study samples. CNVs were widely found in study samples and reference genomes. Discrepancies were found between sets of CNVs called by different analytical tools. I detected 4016 CNVs in 1964 individuals, out of a total of 2789 participants from the three population isolates, which clustered into 743 copy number variable regions (CNVRs). Features of these CVNRs, including frequency and distribution, were compared and were shown to differ significantly between the Orcadian, South Tyrolean and Dalmatian population samples. Consistent with the inference that this indicated population-specific CNVR identity and origin, it was also demonstrated that CNV variation within each population can be used to measure genetic relatedness. Finally, I discovered that individuals who had extreme values of some metabolic traits possessed rare CNVs which overlapped with known genes more often than in individuals with moderate trait values.

A number of exciting recent developments in the field of sieve theory have been done concerning bounded gaps between prime numbers. One of the main techniques used in these papers is a modified version of Selberg's Sieve from the 1940's. While there are a number of sources that explain the original sieve, most, if not all, are quite inaccessible to those without significant experience in analytic number theory. The goal of this exposition is to change that. The statement and proof of the general form of Selberg's sieve is, by itself, difficult to understand and appreciate. For this reason, the inital exposition herein will be about one particular application: to recover Chebysheff's upper bound on the order of magnitude of the number of primes less than a given number. As Selberg's sieve follows some of the same initial steps as the more elementary sieve of Eratosthenes, this latter sieve will be worked through as well. To help the reader get a better sense of Selberg's sieve, a few particular applications are worked through, including an upper bound on the number of twin primes less than a number. This will then be used to show the convergence of the reciprocals of the twin primes.

This thesis considers the importance of the inferior parietal lobe for calculation and Arabic number comparison. The first experiment demonstrates that repetitive Transcranial Magnetic Stimulation (rTMS) can be used on normal subjects to replicate findings from studies of patients whose ability to calculate after brain injury was impaired. While subjects were solving addition tasks, rTMS was applied over anterior and posterior areas of the inferior parietal lobule and the adjoining intraparietal sulcus (aIPL+S, pIPL+S). In line with results from patient studies, magnetic stimulation showed a disruptive effect only over left IPL+S. It had no disruptive effect when delivered over right inferior parietal lobule and the adjoining intraparietal sulcus. To investigate the representation of number magnitude in the human brain rTMS was subsequently applied to the same inferior parietal regions while subjects performed a number comparison task. With numbers between 31 and 99, repetitive TMS over the pIPL+S disrupted organisation of the putative "number line". rTMS had no disruptive effect when delivered over aIPL+S, in either the left or right hemisphere. With numbers between 1 and 9, however, TMS over the pIPL+S did not impair task performance. Here, TMS had a disruptive effect when delivered over aIPL+S, in either the left or right hemisphere, thus suggesting that areas in the inferior parietal lobes might be specialised for certain number sizes. The idea of a spatial mental number line was further investigated in a detailed single-case description of a person with an automatic mental number line. In the last experiment, functional Magnetic Resonance Imaging (fMRI) was used to investigate number comparison. The fMRI study gave some indication that small numbers might be represented in the aIPL+S region. In general, the fMRI results suggest that parietal cortical contribution to number magnitude representation is intimately related to its role in basic sensorimotor processes.

The purpose of this thesis is to investigate the properties of ideals in quadratic number fields, A field F is said to be an algebraic number field if F is a finite extension of R, the field of rational numbers. A field F is said to be a quadratic number field if F is an extension of degree 2 over R. The set 1 of integers of R will be called the rational integers.

Random numbers are important for gaming, simulation and cryptography. Random numbers have been generated using analog circuitry. Two problems exist with using analog circuits in a digital design: (1) analog components require an analog circuit designer to insure proper structure and functionality and (2) analog components are not easily transmigrated into a different fabrication technology. This paper proposes a class of random number generators that are constructed using only digital components and typical digital design methodology. The proposed classification is called divergent path since the path of generated numbers through the range of possible values diverges at every sampling. One integrated circuit was fabricated and several models were synthesized into a FPGA. Test results are given.

Thesis (Ph.D.)--University of the Witwatersrand, Faculty of Science, 1998. / A thesis submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in fulfilment of the requirements for the degree of Doctor of Philosophy. Johannesburg 1998 / Let A = (aij) be an m x n matrix. There is a natural way to associate a poset PA with A. A jump in a linear extension of PA is a pair of consecutive elements which are incomparable in Pa. The jump number of A is the minimum number of jumps in any linear extension of PA. The maximum jump number over a class of n x n matrices of zeros and ones with constant row and column sum k, M (n, k), has been investigated in Chapter 2 and 3. Chapter 2 deals with extremization problems concerning M (n ,k). In Chapter 3, we obtain the exact values for M (11,k). M(n,Q), M (n,n-3) and M(n,n-4). The concept of frequency hyperrectangle generalizes the concept of latin square. In Chapter 4 we derive a bound for the maximum number of mutually orthogonal frequency hyperrectangles. Chapter 5 gives two algorithms to construct mutually orthogonal frequency hyperrectangles. Chapter 6 is devoted to some enumerative results about Carlitz compositions (compositions with different adjacent parts).

by Li King Hung. / Thesis submitted in: July, 1998. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1999. / Includes bibliographical references (leaves 54-56). / Abstract also in Chinese. / Chapter 1 --- Description of the Number Field Sieve --- p.6 / Chapter 1.1 --- Introduction --- p.6 / Chapter 1.2 --- Outline of the algorithm --- p.8 / Chapter 2 --- Algebraic knowledge --- p.17 / Chapter 2.1 --- Factorization of an ideal over the class of ideals --- p.17 / Chapter 2.2 --- Existence of the square roots of an element in the ring of algebraic integers --- p.26 / Chapter 3 --- Run time analysis and Practical result --- p.31 / Chapter 3.1 --- Relation between sieving over X and finding the linear dependencies --- p.32 / Chapter 3.2 --- Relation between the size of a factor base and finding the linear dependencies --- p.34 / Chapter 3.3 --- Practical consideration --- p.36 / Chapter 4 --- Improvement of the algorithm --- p.38 / Chapter 4.1 --- Quadratic characters --- p.38 / Chapter 4.2 --- Finding the square root --- p.40 / Chapter 4.3 --- Solving the linear system of equation --- p.42 / Chapter 4.4 --- Reusing the computation --- p.47 / Chapter 4.5 --- Using more general purpose data --- p.50 / Chapter 4.6 --- Examples --- p.51 / Chapter 4.6.1 --- "A 18-digit example,761260375069630873" --- p.52 / Chapter 4.6.2 --- "A 23-digit example, 16504377514594481520559" --- p.52 / Bibliography

We give an explicit formula for the arithmetic intersection number of CM cycles on Lubin-Tate spaces for all levels. We prove our formula by formulating the intersection number on the infinite level. Our CM cycles are constructed by choosing two separable quadratic extensions K1, K2/F of non-Archimedean local fields F . Our formula works for all cases, K1 and K2 can be either the same or different, ramify or unramified. As applications, this formula translate the linear Arithmetic Fundamental Lemma (linear AFL) into a comparison of integrals. This formula can also be used to recover Gross and Keating’s result on lifting endomorphism of formal modules.

Song Hao = 湍流熱對流中的熱傳導測量 / 宋浩. / Thesis submitted in: December 2004. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2005. / Includes bibliographical references (leaves 65-70). / Text in English; abstracts in English and Chinese. / Song Hao = Tuan liu re dui liu zhong de re zhuan dao ce liang / Song Hao. / Abstract (in English) --- p.i / Abstract (in Chinese) --- p.ii / Acknowledgements --- p.iii / Table of Contents --- p.iv / List of Figures --- p.vi / List of Tables --- p.ix / Chapters / Chapter 1. --- Introduction --- p.1 / Chapter 2. --- Experimental Setup and Methods --- p.7 / Chapter 2.1 --- The rough boundary cell / Chapter 2.1.1 --- Convection Cell --- p.7 / Chapter 2.1.2 --- Temperature Probes --- p.9 / Chapter 2.1.3 --- Working Fluids --- p.10 / Chapter 2.1.4 --- Temperature-stabilized Box --- p.12 / Chapter 2.2 --- Rectangular and Square Cell --- p.13 / Chapter 2.3 --- The Big Cell --- p.13 / Chapter 2.4 --- The Thermal Measurements --- p.16 / Chapter 3. --- Nusselt Number Measurement in the Rough Boundary Cell --- p.19 / Chapter 3.1 --- Nu correction --- p.19 / Chapter 3.2 --- Non-Boussinesq Effect --- p.25 / Chapter 3.3 --- Experimental Results --- p.28 / Chapter 3.3.1 --- Water --- p.28 / Chapter 3.3.2 --- 1-Pentonal --- p.29 / Chapter 3.3.3 --- Dipropylene Glycol --- p.30 / Chapter 3.3.4 --- Triethylene Glycol --- p.32 / Chapter 3.4 --- Discussion on the Results --- p.34 / Chapter 3.4.1 --- Nusselt number --- p.34 / Chapter 3.4.2 --- Comparison with the smooth cell --- p.35 / Chapter 3.4.3 --- Normalized Nusselt number enhancement --- p.37 / Chapter 3.4.4 --- Nu~Pr-Relation --- p.40 / Chapter 3.4.5 --- Effects of Roughness Size --- p.43 / Chapter 3.4.6 --- Temperature Fluctuation Measurement --- p.44 / Chapter 4. --- Geometry Dependence of Nusselt Number and Temperature Fluctuation --- p.47 / Chapter 4.1 --- Nusselt Number Measurement --- p.48 / Chapter 4.2 --- Temperature Fluctuation's Dependence on Geometry --- p.50 / Chapter 5. --- Nusselt Number in the Big Cell --- p.53 / Chapter 5.1 --- The Big Cell --- p.53 / Chapter 5.2 --- Correction for Big Cell --- p.57 / Chapter 5.3 --- Results and Discussion --- p.61 / Chapter 6. --- Conclusions --- p.63 / References --- p.65