published_or_final_version / Mathematics / Master / Master of Philosophy

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Thesis (M.Phil.)--University of Hong Kong, 2003. / Includes bibliographical references (leaves 52) Also available in print.

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Thesis (Ph.D.). / Written for the Dept. of Mathematics and Statistics. Title from title page of PDF (viewed 2009/06/08). Includes bibliographical references.

Doctor of Philosophy / Department of Mathematics / Christopher G. Pinner

For the exponential response η_u = α + βe^γZ_u (u = 1,2,…,N) where α and β lie on the real line (-∞,∞), and γ is a positive integer; the designs are given which minimize the bias due to the inherent inability of the approximation function ŷ_u = Σ_j=0<sup>d</sub>b_je^jZ_u to fit such a model. Transformation to η_u = α + βx_u^γ and ŷ_u = Σ_j=0<sup>d</sub>b_jx_u^j facilitates the solution for minimum bias designs. The requirements for minimum bias designs follow along lines similar to those given by Box and Draper (J. Amer. Stat. Assoc., 54, 1959, p. 622). The minimum bias designs are obtained for specific values of N with a maximum protection level, γ_d*(N), for the parameter γ and an approximation function of degree d. These designs obtained possess several degrees of freedom in the choice of the design levels of the x_u or the Z_uu , which may be used to satisfy additional design requirements. It is shown that for a given N, the same designs which minimize bias for approximation functions of degree one also minimize bias for general degree d, with a decrease in γ_d*(N) as d increases. In fact γ_d*(N) = γ₁*(N) - d + 1, but with the decrease in γ_d*(N) is a compensating decrease in the actual level of the minimum bias. Furthermore, γ_d*(N) increases monotonically with N, thereby allowing the maximum protection level on 1 to be increased as desired by increasing N. In the course of obtaining solutions, some interesting techniques are developed for determining the nature of the roots of a polynomial equation which has several known coefficients and several variable coefficients. / Ph. D.

Cette thèse concerne quelques problèmes liés à la répartition des entiers sans facteur carré dansles progressions arithmétiques. Ces problèmes s’expriment en termes de majorations du terme d’erreurassocié à cette répartition.Les premier, deuxième et quatrième chapitres sont concentrés sur l’étude statistique des termesd’erreur quand on fait varier la progression arithmétique modulo q. En particulier on obtient une formuleasymptotique pour la variance et des majorations non triviales pour les moments d’ordre supérieur. Onfait appel à plusieurs techniques de théorie analytique des nombres comme les méthodes de crible et lessommes d’exponentielles, notamment une majoration récente pour les sommes d’exponentielles courtesdue à Bourgain dans le deuxième chapitre.Dans le troisième chapitre on s’intéresse à estimer le terme d’erreur pour une progression fixée. Onaméliore un résultat de Hooley de 1975 dans deux directions différentes. On utilise ici des majorationsrécentes de sommes d’exponentielles courtes de Bourgain-Garaev et de sommes d’exponentielles torduespar la fonction de Möbius dues à Bourgain et Fouvry-Kowalski-Michel. / This thesis concerns a few problems linked with the distribution of squarefree integers in arithmeticprogressions. Such problems are usually phrased in terms of upper bounds for the error term relatedto this distribution.The first, second and fourth chapter focus on the satistical study of the error terms as the progres-sions varies modulo q. In particular we obtain an asymptotic formula for the variance and non-trivialupper bounds for the higher moments. We make use of many technics from analytic number theorysuch as sieve methods and exponential sums. In particular, in the second chapter we make use of arecent upper bound for short exponential sums by Bourgain.In the third chapter we give estimates for the error term for a fixed arithmetic progression. Weimprove on a result of Hooley from 1975 in two different directions. Here we use recent upper boundsfor short exponential sums by Bourgain-Garaev and exponential sums twisted by the Möbius functionby Bourgain et Fouvry-Kowalski-Michel.

by Wai Ho Mow. / Thesis (Ph.D.)--Chinese University of Hong Kong, 1993. / Includes bibliographical references (leaves 116-124). / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Spread Spectrum Technique --- p.2 / Chapter 1.1.1 --- Pulse Compression Radars --- p.3 / Chapter 1.1.2 --- Spread Spectrum Multiple Access Systems --- p.6 / Chapter 1.2 --- Definitions and Notations --- p.8 / Chapter 1.3 --- Organization of this Thesis --- p.12 / Chapter 2 --- Lower Bounds on Correlation of Sequences --- p.15 / Chapter 2.1 --- Welch's Lower Bounds and Sarwate's Generalization --- p.16 / Chapter 2.2 --- A New Construction and Bounds on Odd Correlation --- p.23 / Chapter 2.3 --- Known Sequence Sets Touching the Correlation Bounds --- p.26 / Chapter 2.4 --- Remarks on Other Bounds --- p.27 / Chapter 3 --- Perfect Polyphase Sequences: A Unified Approach --- p.29 / Chapter 3.1 --- Generalized Bent Functions and Perfect Polyphase Sequences --- p.30 / Chapter 3.2 --- The General Construction of Chung and Kumar --- p.32 / Chapter 3.3 --- Classification of Known Constructions ...........； --- p.34 / Chapter 3.4 --- A Unified Construction --- p.39 / Chapter 3.5 --- Desired Properties of Sequences --- p.41 / Chapter 3.6 --- Proof of the Main Theorem --- p.45 / Chapter 3.7 --- Counting the Number of Perfect Polyphase Sequences --- p.49 / Chapter 3.8 --- Results of Exhaustive Searches --- p.53 / Chapter 3.9 --- A New Conjecture and Its Implications --- p.55 / Chapter 3.10 --- Sets of Perfect Polyphase Sequences --- p.58 / Chapter 4 --- Aperiodic Autocorrelation of Generalized P3/P4 Codes --- p.61 / Chapter 4.1 --- Some Famous Polyphase Pulse Compression Codes --- p.62 / Chapter 4.2 --- Generalized P3/P4 Codes --- p.65 / Chapter 4.3 --- Asymptotic Peak-to-Side-Peak Ratio --- p.66 / Chapter 4.4 --- Lower Bounds on Peak-to-Side-Peak Ratio --- p.67 / Chapter 4.5 --- Even-Odd Transformation and Phase Alphabet --- p.70 / Chapter 5 --- Upper Bounds on Partial Exponential Sums --- p.77 / Chapter 5.1 --- Gauss-like Exponential Sums --- p.77 / Chapter 5.1.1 --- Background --- p.79 / Chapter 5.1.2 --- Symmetry of gL(m) and hL(m) --- p.80 / Chapter 5.1.3 --- Characterization on the First Quarter of gL(m) --- p.83 / Chapter 5.1.4 --- Characterization on the First Quarter of hL(m) --- p.90 / Chapter 5.1.5 --- Bounds on the Diameters of GL(m) and HL(m) --- p.94 / Chapter 5.2 --- More General Exponential Sums --- p.98 / Chapter 5.2.1 --- A Result of van der Corput --- p.99 / Chapter 6 --- McEliece's Open Problem on Minimax Aperiodic Correlation --- p.102 / Chapter 6.1 --- Statement of the Problem --- p.102 / Chapter 6.2 --- A Set of Two Sequences --- p.105 / Chapter 6.3 --- A Set of K Sequences --- p.110 / Chapter 7 --- Conclusion --- p.113 / Bibliography --- p.124

Consider either (i) f = f1 ⊠ f2 for two Maass cusp forms for SLm(ℤ) and SLm′(ℤ), respectively, with 2 ≤ m ≤ m′, or (ii) f= f1 ⊠ f2 ⊠ f3 for three weight 2k holomorphic cusp forms for SL2(ℤ). Let λf(n) be the normalized coefficients of the associated L-function L(s, f), which is either (i) the Rankin-Selberg L-function L(s, f1 ×f2), or (ii) the Rankin triple product L-function L(s, f1 ×f2 ×f3). First, we derive a Voronoi-type summation formula for λf (n) involving the Meijer G-function. As an application we obtain the asymptotics for the smoothly weighted average of λf (n) against e(αnβ), i.e. the asymptotics for the associated resonance sums. Let ℓ be the degree of L(s, f). When β = 1/ℓ and α is close or equal to ±ℓq 1/ℓ for a positive integer q, the average has a main term of size |λf (q)|X 1/2ℓ+1/2 . Otherwise, when α is fixed and 0 < β < 1/ℓ it is shown that this average decays rapidly. Similar results have been established for individual SLm(ℤ) automorphic cusp forms and are due to the oscillatory nature of the coefficients λf (n).