In this thesis we study low correlation zone (LCZ) sequence sets and a class of quadratic forms. In the first part we obtain two new classes of optimal LCZ sequence sets. In our first construction using a suitable orthogonal transformation we extend some results of [21]. We give new classes of LCZ sequence sets defined over Z4 in our second construction. We show that our LCZ sequence sets are optimal with respect to the Tang, Fan and Matsufiji bound [37]. In the second part we consider some special linearized polynomials and corresponding quadratic forms. We compute the number of solutions of certain equations related to these quadratic forms and we apply these result to obtain curves with many rational points.

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The linear feedback shift-register is a widely used tool for generating cryptographic sequences. The properties of the Galois model discussed here offer many opportunities to improve the implementations that already exist. We explore the overall properties of the phases of the Galois model and conjecture a relation with modular Golomb rulers. This conjecture points to an efficient method for constructing non-linear filtering generators which fulfil Golic s design criteria in order to maximise protection against his inversion attack. We also produce a number of methods which can improve the rate of output of sequences by combining particular distinct phases of smaller elementary sequences.

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Although the eDNA sequences for the 10 k:Da (hsp 10, hsp 1 0) and the 60 k:Da (hsp60, cpn60) heat shock proteins have been obtained for a number of mammalian species, until very recently information was not available on the functional genes encoding these proteins. The primary objective of this work was to clone and sequence the functional genes for these proteins from CHO, Chinese hamster ovary cells. Screening of a lambda EMBL3 CHO genomic library with the CHO hsp 10 eDNA identified a clone containing the putative hsp 10 functional gene. A -5.5 kb fragment was isolated from one of these clones by enzymatic digestion and -3.3 kb was sequenced. The clone was found to contain consensus regulatory sequences upstream of the putative transcription initiation site, + 1, including two Sp 1 binding sites, a CAAT box, and a single heat shock element, HSE, but lacked a TATA box. The coding region consists of four exons, identical to the hsp10 CHO eDNA sequence, separated by three introns, of 200 bp, 600 bp and 1600 bp in size, containing conserved splice sites. Screening of the same EMBL3 CHO genomic library with the CHO hsp 10 eDNA also resulted in isolation of a full length processed pseudogene with -90 % identity to the eDNA. This pseudogene lacked introns, contained a poly(A) tract, as well as various single bp changes, additions and deletions. The upstream region of this pseudo gene was found to contain similarity to the human LINE sequence, a DNA repetitive element. PCR amplification ofCHO-WT genomic DNA resulted in isolation offive additional processed pseudogenes, corresponding to the central -270 bp of the CHO hsplO eDNA. All the pseudogenes displayed a high degree of similarity to the CHO hsp 10 eDNA sequence despite the presence of numerous mutations. Prior to this report, pseudogenes had not been found associated with hsp 10. The identification of these pseudogenes suggests the presence of a multi gene family for this heat shock protein in the CHO genome. Previously, a semi-processed pseudogene, Gel, was identified for hsp60 from CHO cells which contained a single -87 bp intron near its 3' end (Venner eta/., 1990). From this pseudo gene, a fragment containing the -87 bp intron was isolated for use as a probe to screen a lambda EMBL3 CHO genomic library. This resulted in isolation of several positive clones, two of which were purified, a -1.0 kb fragment amplified by PCR and then sequenced revealing two additional semi-processed pseudogenes, designated .A4 and .AS. These pseudo genes were found to be homologous to the GC 1 clone, containing many similar mutations as well as the -87 bp intron. Utilizing CHO-WT genomic DNA, a separate PCR amplification resulted in isolation of a -2.5 kb fragment which was partially sequenced and found to correspond to the putative hsp60 functional gene. The fragment contained one exon, which was identical to the CHO hsp60 eDNA in the region sequenced, and two introns of800 bp and 1500 bp. This fragment can now provide an ideal probe for isolation ofthe CHO hsp60 functional gene. / Thesis / Master of Science (MSc)

Sequences of matrices arise in many applications in science and engineering. In this thesis we consider matrices that are closely related (or closely related in groups), and we take advantage of the small differences between them to efficiently solve sequences of linear systems and eigenproblems. Recycling techniques, such as recycling preconditioners or subspaces, are popular approaches for reducing computational cost. In this thesis, we introduce two novel approaches for recycling previously computed information for a subsequent system or eigenproblem, and demonstrate good results for sequences arising in several applications. Preconditioners are often essential for fast convergence of iterative methods. However, computing a good preconditioner can be very expensive, and when solving a sequence of linear systems, we want to avoid computing a new preconditioner too often. Instead, we can recycle a previously computed preconditioner, for which we have good convergence behavior of the preconditioned system. We propose an update technique we call the sparse approximate map, or SAM update, that approximately maps one matrix to another matrix in our sequence. SAM updates are very cheap to compute and apply, preserve good convergence properties of a previously computed preconditioner, and help to amortize the cost of that preconditioner over many linear solves. When solving a sequence of eigenproblems, we can reduce the computational cost of constructing the Krylov space starting with a single vector by warm-starting the eigensolver with a subspace instead. We propose an algorithm to warm-start the Krylov-Schur method using a previously computed approximate invariant subspace. We first compute the approximate Krylov decomposition for a matrix with minimal residual, and use this space to warm-start the eigensolver. We account for the residual matrix when expanding, truncating, and deflating the decomposition and show that the norm of the residual monotonically decreases. This method is effective in reducing the total number of matrix-vector products, and computes an approximate invariant subspace that is as accurate as the one computed with standard Krylov-Schur. In applications where the matrix-vector products require an implicit linear solve, we incorporate Krylov subspace recycling. Finally, in many applications, sequences of matrices take the special form of the sum of the identity matrix, a very low-rank matrix, and a small-in-norm matrix. We consider convergence rates for GMRES applied to these matrices by identifying the sources of sensitivity. / Doctor of Philosophy / Problems in science and engineering often require the solution to many linear systems, or a sequence of systems, that model the behavior of physical phenomena. In order to construct highly accurate mathematical models to describe this behavior, the resulting matrices can be very large, and therefore the linear system can be very expensive to solve. To efficiently solve a sequence of large linear systems, we often use iterative methods, which can require preconditioning techniques to achieve fast convergence. The preconditioners themselves can be very expensive to compute. So, we propose a cheap update technique that approximately maps one matrix to another in the sequence for which we already have a good preconditioner. We then combine the preconditioner and the map and use the updated preconditioner for the current system. Sequences of eigenvalue problems also arise in many scientific applications, such as those modeling disk brake squeal in a motor vehicle. To accurately represent this physical system, large eigenvalue problems must be solved. The behavior of certain eigenvalues can reveal instability in the physical system but to identify these eigenvalues, we must solve a sequence of very large eigenproblems. The eigensolvers used to solve eigenproblems generally begin with a single vector, and instead, we propose starting the method with several vectors, or a subspace. This allows us to reduce the total number of iterations required by the eigensolver while still producing an accurate solution. We demonstrate good results for both of these approaches using sequences of linear systems and eigenvalue problems arising in several real-world applications. Finally, in many applications, sequences of matrices take the special form of the sum of the identity matrix, a very low-rank matrix, and a small-in-norm matrix. We examine the convergence behavior of the iterative method GMRES when solving such a sequence of matrices.

In this thesis we study the zeros and asymptotics of sequences that satisfy linear recurrence relations with generally nonconstant coefficients. By the theorem of Skolem-Mahler-Lech, the set of zero terms of a sequence that satisfies a linear recurrence relation with constant coefficients taken from a field of characteristic zero is comprised of the union of finitely many arithmetic progressions together with a finite exceptional set. Further, in the nondegenerate case, we can eliminate the possibility of arithmetic progressions and conclude that there are only finitely many zero terms. For generally nonconstant coefficients, there are generalizations of this theorem due to Bézivin and to Methfessel that imply, under fairly general conditions, that we obtain a finite union of arithmetic progressions together with an exceptional set of density zero. Further, a condition is given under which one can exclude the possibility of arithmetic progressions and obtain a set of zero terms of density zero. In this thesis, it is shown that this condition reduces to the nondegeneracy condition in the case of constant coefficients. This allows for a consistent definition of nondegeneracy valid for generally nonconstant coefficients and a unified result is obtained. The asymptotic theory of sequences that satisfy linear recurrence relations with generally nonconstant coefficients begins with the basic theorems of Poincaré and Perron. There are some generalizations of these theorems that hold in greater generality, but if we restrict the coefficient sequences of our linear recurrences to be polynomials in the index, we obtain full asymptotic expansions of a predictable form for the solution sequences. These expansions can be obtained by applying a transfer method of Flajolet and Sedgewick or, in some cases, by applying a bivariate method of Pemantle and Wilson. In this thesis, these methods are applied to a family of binomial sums and full asymptotic expansions are obtained. The leading terms of the expansions are obtained explicitly in all cases, while in some cases a field containing the asymptotic coefficients is obtained and some divisibility properties for the asymptotic coefficients are obtained using a generalization of a method of Stoll and Haible.

Contemporary research in digital signal processing (DSP) is focused on issues of computational complexity, very high data rate and large quantum of data. Thus, the success in newer applications and areas hinge on handling these issues. Conventional ways to address these challenges are to develop newer structures like Multirate signal processing, Multiple Input Multiple Output(MIMO), bandpass sampling, compressed domain sensing etc. In the implementation domain, the approach is to look at floating point over fixed point representation and / or longer wordlength etc., related to number representations and computations. Of these, a simple approach is to look at number representation, perhaps with a simple integer. This automatically guarantees accuracy and zero quantization error as well as longer wordlength. Thus, it is necessary and interesting to explore viable DSP alternatives that can reduce complexity and yet match the required performance. The main aim of this work is to highlight the importance, use and analysis of integer sequences. Firstly, the thesis explores the use of integer sequences as windowing functions. The results of these investigations show that integer sequences and their convolution, indeed, outperform many of the classical real valued window functions in terms of mainlobe width, sidelobe attenuation etc. Secondly, the thesis proposes techniques to approximate discrete Gaussian distribution using integer sequences. The key idea is to convolve symmetrized integer sequences and examine the resulting profiles. These profiles are found to approximate discrete Gaussian distribution with a mean square error of the order of 10−8 or less. While looking at integer sequences to approximate discrete Gaussian, Fibonacci sequence was found to exhibit some interesting properties. The third part of the thesis proves certain fascinating optimal probabilistic limit properties (mean and variance) of Fibonacci sequence. The thesis also provides complete generalization of these properties to probability distributions generated by second order linear recurrence relation with integer coefficients and any kth order linear recurrence relation with unit coefficients. In addition to the above, the thesis also throws light on possible architectural implications of using integer sequences in DSP applications and ideas for further exploration.