It is well known that the n-th eigenfunction to one-dimensional Sturm-Liouville eigenvalue problems has exactly n-1 nodes, i.e. non-degenerate zeros. For higher dimensions, it is much more complicated to obtain general statements on the zeros of eigenfunctions. The author states a new conjecture on the number of nodal domains of spherical harmonics, i.e. of connected components of S^2 \ N(u) with the nodal set N(u) = (x in S^2 : u(x) = 0) of the eigenfunction u, and proves it for the first six eigenvalues. It is a sharp upper bound, thus improving known bounds as the Courant nodal domain theorem, see S. Y. Cheng, Comment. Math. Helv. 51, 43-55 (1976; Zbl 334.35022). The proof uses facts on real projective plane algebraic curves (see D. A. Gudkov, Usp. Mat. Nauk 29(4), 3-79, Russian Math. Surveys 29(4), 1-79 (1979; Zbl 316.14018)), because they are the zero sets of homogeneous polynomials, and the spherical harmonics are the restrictions of spherical harmonic homogeneous polynomials in the space to the plane. / Series: Preprint Series / Department of Applied Statistics and Data Processing

Este trabalho foi realizado com o objetivo de se estudar: - a concordância de um mesmo pesquisador, ao classificar, em ocasiões diferentes, as curvas de mortalidade proporcional, - a concordância entre diferentes pesquisadores na interpretação das curvas de mortalidade proporcional e ou - a concordância entre a curva de mortalidade proporcional e cada um dos indicadores, coeficiente de mortalidade infantil e razão de mortalidade proporcional, na indicação de alteração do nível de saúde de coletividades. Empreqou-se a estatística do tipo Kappa e encontrou-se: (GRÁFICO) / This study was realized with the purpose of examining: - the agreement of the same researcher, in classifying, on different occasions, the proportional mortality curves, - the agreement among different researchers in the interpretation of the proportional mortality curves, - the agreement among the proportional mortality curve and each one of the health indicators, infant mortality rate and proportional mortality ratio, in the indication of alteration of the level of health for communities. The statistics, type Kapna, was utilized and these were the findings: (GRAFICO)

by Leung Tak. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1992. / Includes bibliographical references (leaves 76-77). / Chapter 1 --- Introduction --- p.1 / Chapter 2 --- Elliptic curve --- p.4 / Chapter 2.1 --- Elliptic Curve in Normal Form --- p.4 / Chapter 2.2 --- Geometry and Group Law --- p.7 / Chapter 2.3 --- Special Class of Elliptic Curves --- p.10 / Chapter 2.4 --- Mordell's Conjecture --- p.12 / Chapter 2.5 --- Torsion Group --- p.14 / Chapter 2.6 --- Selmer Group and Tate-Shafarevitch. Group --- p.16 / Chapter 2.7 --- Endomorphism of Elliptic Curves --- p.19 / Chapter 2.8 --- Formal Group over Elliptic Curves --- p.23 / Chapter 2.9 --- The Finite Field Case --- p.26 / Chapter 2.10 --- The Local Field Case --- p.27 / Chapter 2.11 --- The Global Field Case --- p.29 / Chapter 3 --- Class Field Theory --- p.31 / Chapter 3.1 --- Valuation and Local Field --- p.31 / Chapter 3.2 --- Unramified and Totally Ramified Extensions and Their Norm Groups --- p.35 / Chapter 3.3 --- Formal Group and Abelian Extension of Local Field --- p.36 / Chapter 3.4 --- Abelian Extenion and Norm Residue Map --- p.41 / Chapter 3.5 --- Finite Extension and Ramification Group --- p.43 / Chapter 3.6 --- "Hilbert Symbols [α, β］w and (α, β)f" --- p.46 / Chapter 3.7 --- Adele and Idele --- p.48 / Chapter 3.8 --- Galois Extension and Kummer Extension --- p.50 / Chapter 3.9 --- Global Reciprocity Law and Global Class Field --- p.52 / Chapter 3.10 --- Ideal-Theoretic Formulation of Class Field Theory --- p.57 / Chapter 4 --- Hasse-Weil L-function of elliptic curves --- p.60 / Chapter 4.1 --- Classical Zeta Functions and L-functions --- p.60 / Chapter 4.2 --- Congruence Zeta Function --- p.63 / Chapter 4.3 --- Hasse-Weil L-function and Birch-Swinnerton-Dyer Conjecture --- p.64 / Chapter 4.4 --- A Sketch of the Proof from the Joint Paper of Coates and Wiles --- p.67 / Chapter 4.5 --- The works of other mathematicians --- p.73

Tang Ko Cheung, Simon. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2001. / Includes bibliographical references (leaves 89-91). / Abstracts in English and Chinese. / Chapter 1 --- Theory of Optimal Normal Bases --- p.3 / Chapter 1.1 --- Introduction --- p.3 / Chapter 1.2 --- The minimum number of terms --- p.6 / Chapter 1.3 --- Constructions for optimal normal bases --- p.7 / Chapter 1.4 --- Existence of optimal normal bases --- p.10 / Chapter 2 --- Implementing Multiplication in GF(2m) --- p.13 / Chapter 2.1 --- Defining the Galois fields GF(2m) --- p.13 / Chapter 2.2 --- Adding and squaring normal basis numbers in GF(2m) --- p.14 / Chapter 2.3 --- Multiplication formula --- p.15 / Chapter 2.4 --- Construction of Lambda table for Type I ONB in GF(2m) --- p.16 / Chapter 2.5 --- Constructing Lambda table for Type II ONB in GF(2m) --- p.21 / Chapter 2.5.1 --- Equations of the Lambda matrix --- p.21 / Chapter 2.5.2 --- An example of Type IIa ONB --- p.23 / Chapter 2.5.3 --- An example of Type IIb ONB --- p.24 / Chapter 2.5.4 --- Creating the Lambda vectors for Type II ONB --- p.26 / Chapter 2.6 --- Multiplication in practice --- p.28 / Chapter 3 --- Inversion over optimal normal basis --- p.33 / Chapter 3.1 --- A straightforward method --- p.33 / Chapter 3.2 --- High-speed inversion for optimal normal basis --- p.34 / Chapter 3.2.1 --- Using the almost inverse algorithm --- p.34 / Chapter 3.2.2 --- "Faster inversion, preliminary subroutines" --- p.37 / Chapter 3.2.3 --- "Faster inversion, the code" --- p.41 / Chapter 4 --- Elliptic Curve Cryptography over GF(2m) --- p.49 / Chapter 4.1 --- Mathematics of elliptic curves --- p.49 / Chapter 4.2 --- Elliptic Curve Cryptography --- p.52 / Chapter 4.3 --- Elliptic curve discrete log problem --- p.56 / Chapter 4.4 --- Finding good and secure curves --- p.58 / Chapter 4.4.1 --- Avoiding weak curves --- p.58 / Chapter 4.4.2 --- Finding curves of appropriate order --- p.59 / Chapter 5 --- The performance of 17x bit Elliptic Curve Scalar Multiplication --- p.63 / Chapter 5.1 --- Choosing finite fields --- p.63 / Chapter 5.2 --- 17x bit test vectors for onb --- p.65 / Chapter 5.3 --- Testing methodology and sample runs --- p.68 / Chapter 5.4 --- Proposing an elliptic curve discrete log problem for an 178bit curve --- p.72 / Chapter 5.5 --- Results and further explorations --- p.74 / Chapter 6 --- On matrix RSA --- p.77 / Chapter 6.1 --- Introduction --- p.77 / Chapter 6.2 --- 2 by 2 matrix RSA scheme 1 --- p.80 / Chapter 6.3 --- Theorems on matrix powers --- p.80 / Chapter 6.4 --- 2 by 2 matrix RSA scheme 2 --- p.83 / Chapter 6.5 --- 2 by 2 matrix RSA scheme 3 --- p.84 / Chapter 6.6 --- An example and conclusion --- p.85 / Bibliography --- p.91

by Yuen Wai Ching. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1998. / Includes bibliographical references (leaves 54-55). / Abstract also in Chinese. / Chapter 1 --- Introduction --- p.5 / Chapter 2 --- Group Analysis --- p.9 / Chapter 2.1 --- Groups and Differential Equations --- p.9 / Chapter 2.2 --- Prolongation --- p.11 / Chapter 2.3 --- The Prolongation Formula --- p.14 / Chapter 3 --- Symmetry Group For the Eikonal Equation --- p.17 / Chapter 4 --- An Optimal System For the Eikonal Equation --- p.25 / Chapter 5 --- Group Invariant Solutions --- p.33 / Chapter 5.1 --- Straight Lines --- p.33 / Chapter 5.2 --- Stationary Solutions --- p.33 / Chapter 5.3 --- Traveling Waves --- p.34 / Chapter 5.4 --- Circles --- p.37 / Chapter 5.5 --- Spirals --- p.38 / Chapter 6 --- Appendix --- p.50 / A Group Analysis for some Geometric Evolution Equations --- p.4 / Bibliography

by Lee Tin Wah. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1998. / Includes bibliographical references (leaves 68-69). / Abstract also in Chinese. / Chapter 1 --- Introduction --- p.5 / Chapter 2 --- Preliminaries --- p.8 / Chapter 2.1 --- Box dimension and Hausdorff dimension --- p.8 / Chapter 2.2 --- Basic properties of dimensions --- p.9 / Chapter 2.3 --- Calculating dimensions --- p.11 / Chapter 3 --- Dimension of graph of the Weierstrass function --- p.14 / Chapter 3.1 --- Calculating dimensions of a graph --- p.14 / Chapter 3.2 --- Weierstrass function --- p.16 / Chapter 3.3 --- An almost everywhere argument --- p.23 / Chapter 3.4 --- Tagaki function --- p.26 / Chapter 4 --- Self-affine mappings --- p.30 / Chapter 4.1 --- Box dimension of self-affine curves --- p.30 / Chapter 4.2 --- Differentability of self-affine curves --- p.35 / Chapter 4.3 --- Tagaki function --- p.42 / Chapter 4.4 --- Hausdorff dimension of self-affine sets --- p.43 / Chapter 5 --- Recurrent set and Weierstrass-like functions --- p.56 / Chapter 5.1 --- Recurrent curves --- p.56 / Chapter 5.2 --- Recurrent sets --- p.62 / Chapter 5.3 --- Weierstrass-like functions from recurrent sets --- p.64 / Bibliography

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In this thesis we unconditionally show that certain K3 surfaces satisfy the Hasse principle. Our method involves the 2-Selmer groups of simultaneous quadratic twists of two elliptic curves, only with places of good or additive reduction. More generally we prove that, given finitely many such elliptic curves defined over a number field (with rational 2-torsion and satisfying some mild conditions) there exists an explicit quadratic extension such that the quadratic twist of each elliptic curve has essential 2-Selmer rank one. Furthermore, given a 2-covering in each of the 2-Selmer groups, the quadratic extension above can be chosen so that the 2-Selmer group of the quadratic twist of each elliptic curve is generated by the given 2-covering and the image of the 2-torsion. Our approach to the Hasse Principle is outlined below and was introduced by Skorobogatov and Swinnerton-Dyer. We also generalize the result proved in their paper. If each elliptic curve has a distinct multiplicative place of bad reduction, then we find a quadratic extension such that the quadratic twist of each elliptic curve has essential 2-Selmer rank one. Furthermore, given a 2-covering in each of the 2-Selmer groups, the quadratic extension above can be chosen so that the 2-Selmer group of the quadratic twist of each elliptic curve is generated by the given 2-covering and the image of the 2-torsion. If we further assume the finiteness of the Shafarevich-Tate groups (of the twisted elliptic curves) then each elliptic curve has Mordell-Weil rank one. If K = Q, then under the above assumptions the analytic rank of each elliptic curves is one. Furthermore, with the assumption on the Shafarevich-Tate group (and K = Q), we describe a single quadratic twist such that each elliptic curve has analytic rank zero and Mordell-Weil rank zero, again under some mild assumptions.

This thesis consists of two unrelated parts. In the first part of this thesis, we give explicit expressions for the Fourier coefficients of Eisenstein series E∗(z, s, χ) twisted by modular symbols ⟨γ, f⟩ in the case where the level of f is prime and equal to the conductor of the Dirichlet character χ. We obtain these expressions by computing the spectral decomposition of an automorphic function closely related to E∗(z, s, χ). We then give applications of these expressions. In particular, we evaluate sums such as Σχ(γ)⟨γ, f⟩, where the sum is over γ ∈ Γ∞\Γ0(N) with c^2 + d^2 < X, with c and d being the lower-left and lower-right entries of γ respectively. This parallels past work of Goldfeld, Petridis, and Risager, and we observe that these sums exhibit different amounts of cancellation than what one might expect. In the second part of this thesis, given an elliptic curve E and a point P in E(R), we investigate the distribution of the points nP as n varies over the integers, giving bounds on the x and y coordinates of nP and determining the natural density of integers n for which nP lies in an arbitrary open subset of {R}^2. Our proofs rely on a connection to classical topics in the theory of Diophantine approximation.

Consider the problem of designing the path of a camera in 3D. As we may identify each camera position with a member of the Euclidean motions, SE(3), the problem may be recast mathematically as constructing interpolating curves on the (non-Euclidean) space SE(3). There exist many ways to formulate this problem, and indeed many solutions. In this thesis we shall examine solutions based on simple geometric constructions, with the goal of discovering well behaved and computable solutions. In affine spaces there exist elegant solutions to the problem of curve design, which are collectively known as the techniques of Computer Aided Geometric Design (CAGD). The approach of this thesis will be the generalization of these methods and an examination of computation on matrix Lie groups. In particular, the Lie groups SO(3) and SE(3) will be examined in some detail.