by Wu Man Ming. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2002. / Includes bibliographical references (leaves 60-62). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.1 / Chapter 2 --- Harmonic functions with linear growth --- p.3 / Chapter 2.1 --- A sharp estimate for dim H1 (M) --- p.3 / Chapter 2.2 --- Linear growth harmonic functions on Kahler manifolds --- p.8 / Chapter 3 --- Harmonic functions of polynomial growth --- p.21 / Chapter 3.1 --- Harmonic sections of polynomial growth --- p.21 / Chapter 3.2 --- Harmonic functions on manifolds with Sobolev in- equality --- p.34 / Chapter 4 --- Harmonic functions on manifolds with nonnegat --- p.ive / sectional curvature --- p.43 / Bibliography --- p.60

Cheung Wing Yi = 介電材料界面附近的局域場分佈 / 張詠怡. / Thesis submitted in: October 2004. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2005. / Includes bibliographical references (leaves 81-85). / Text in English; abstracts in English and Chinese. / Cheung Wing Yi = Jie dian cai liao jie mian fu jin de ju yu chang fen bu / Zhang Yongyi. / Chapter 1 --- Introduction --- p.2 / Chapter 2 --- Fundamentals --- p.10 / Chapter 2.1 --- Local Field --- p.10 / Chapter 2.2 --- Clausius-Mossotti Equation --- p.12 / Chapter 3 --- Multi-layer Formulation --- p.14 / Chapter 3.1 --- Developed Lekner Summation Method --- p.15 / Chapter 3.2 --- Formulation --- p.16 / Chapter 3.2.1 --- Interlayers --- p.16 / Chapter 3.2.2 --- Multi-layers --- p.22 / Chapter 3.3 --- Comparision with Ewald-Kornfeld Formulation --- p.26 / Chapter 3.4 --- Contact with macroscopic concepts --- p.30 / Chapter 4 --- Local Field Distribution near Sharp Interfaces --- p.32 / Chapter 4.1 --- Body-centered Tetragonal Lattices --- p.33 / Chapter 4.2 --- Simple Tetragonal and Body-centered Tetragonal Lattices --- p.39 / Chapter 4.3 --- Effects of Geometric Anisotropy --- p.43 / Chapter 5 --- Local Field Distribution for Graded Materials --- p.52 / Chapter 5.1 --- Bare Polarizability Gradient --- p.53 / Chapter 5.2 --- Temperature Gradient --- p.58 / Chapter 6 --- Optical Response for Drude Dielectric Gradation Profile --- p.63 / Chapter 7 --- Summary --- p.72 / Chapter A --- Lekner summation method --- p.74 / Chapter B --- Ewald-Kornfeld Formulation --- p.78 / Chapter C --- Langevin-Debye Equation --- p.81

Yeung, Hau Man. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2011. / Includes bibliographical references (p. 59-63). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.7 / Chapter 2 --- Work Flows and Kernel Methods --- p.9 / Chapter 2.1 --- Work Flows --- p.9 / Chapter 2.2 --- Frequency Vector --- p.11 / Chapter 2.3 --- Motivation for Kernel Based Distance --- p.12 / Chapter 2.3.1 --- Similarity between sequences --- p.13 / Chapter 2.3.2 --- Distance between sequences --- p.14 / Chapter 2.4 --- Kernels for DNA Sequence --- p.15 / Chapter 2.4.1 --- Kernels based on evolution model --- p.15 / Chapter 2.4.2 --- Kernels based on empirical data --- p.17 / Chapter 2.5 --- Kernels for Peptide Sequence --- p.18 / Chapter 3 --- Dataset for DNA Sequence and Results --- p.25 / Chapter 3.1 --- Dataset and Goal --- p.25 / Chapter 3.1.1 --- Mitochondrial DNA dataset --- p.26 / Chapter 3.1.2 --- 18S ribosomal RNA --- p.28 / Chapter 3.2 --- Results --- p.28 / Chapter 4 --- Dataset for Peptide Sequence and Results --- p.35 / Chapter 4.1 --- Dataset and Goal --- p.36 / Chapter 4.2 --- Classification and Evaluation Methods --- p.39 / Chapter 4.2.1 --- Partition of training and testing datasets --- p.39 / Chapter 4.2.2 --- Classification methods --- p.40 / Chapter 4.3 --- Results --- p.45 / Chapter 4.3.1 --- KNN performs better than the FDSM --- p.45 / Chapter 4.3.2 --- BLOSUM62 performs best and window length not important --- p.46 / Chapter 4.3.3 --- Distance formula (2.4) performs better --- p.49 / Chapter 5 --- Discussion --- p.51 / Chapter 5.1 --- Sequence Length and Window Length --- p.51 / Chapter 5.2 --- Possible Kernels --- p.52 / Chapter 5.3 --- Distance Formulae --- p.53 / Chapter 5.4 --- Protein Structural Problem --- p.54 / Chapter 6 --- Appendix --- p.55 / Chapter 6.1 --- Kernel for Peptide Sequences --- p.55 / Bibliography --- p.59

Wang Jun. / "February 2003." / Thesis (Ph.D.)--Chinese University of Hong Kong, 2003. / Includes bibliographical references (p. 116-124). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Mode of access: World Wide Web. / Abstracts in English and Chinese.

令N = {1, ..., n}. 一組n個隨機變量{Xi : i ∈ N} 的熵函數h是一個2n維的向量，該向量的每個分量h(A) = H(XA);A ⊂ N, 即該組隨機變量的子集的(聯合)熵且空集的熵按傳統看做為0。所有n個隨機變量的熵函數組成的區 域稱為n階熵函數區域，記作Γ* n。熵函數區域Γ* n及其閉包Γ* n的表徵是信息論中著名的開放問題。 / 在本文中，我們研究劃分對稱熵函數。令p = {N₁... ,Nt}為N的 一個t-劃分 。一個熵函數h稱為p-對稱的，若h滿足:對於N的所有子集A，B，對於p的每一 個分塊，只要A和該分塊的交集的基數與B和該分塊交集的基數相等，那麼h(A) = h(B)。所有p-對稱熵函數的集合稱作p-對稱熵函數區域。我們證明p-對稱熵函數區域的 閉包可以由香農型信息不等式完全表徵當且僅當p為1-劃分或者有一個分塊為單元 素集合的2-劃分。 / 劃分對稱熵函數的表徵能應用於那些結構中含有對稱的信息論問題及其相關問題。 / Let N = {1, ..., n}. The entropy function h of a set of n discrete randomvariables {Xi : i ∈ N} is a 2n-dimensional vector whose entries are h(A)H(XA),ACN, the (joint) entropies of the subsets of the set of n randomvariables with H(X) = 0 by convention. The set of all entropy functions for n discrete random variables, denoted by Γ* n , is called the entropy function region for n. Characterization of Γ* n and its closure Γ* n are well-known open problems in information theory. They are important not only because they play key roles in information theory problems but also they are related to other subjects in mathematics and physics. / In this thesis, we consider partition-symmetrical entropy functions. Let p ={N₁... ,Nt} be a t-partition of N. An entropy function h is called p-symmetricalif for all A,B ⊂ N, h(A) = h(B) whenever / The characterization of the partition-symmetrical entropy functions can beuseful for solving some information theory and related problems where symmetryexists in the structure of the problems. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Chen, Qi. / Thesis (Ph.D.) Chinese University of Hong Kong, 2014. / Includes bibliographical references (leaves 70-73). / Abstracts also in Chinese.

Tam Hak Fui = 周期性界面附近的局域场分布 / 谭克奎. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2003. / Includes bibliographical references (leaves 70-72). / Text in English; abstracts in English and Chinese. / Tam Hak Fui = Zhou qi xing jie mian fu jin de ju yu chang fen bu / Tan Kekui. / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Motivation and review on related work --- p.1 / Chapter 1.2 --- Objectives of the thesis --- p.4 / Chapter 2 --- The Green's function formalism (GFF) --- p.6 / Chapter 3 --- Application of GFF to one-dimensional periodic interface --- p.10 / Chapter 3.1 --- The structure Green's function - Greenian --- p.10 / Chapter 3.2 --- Solution by mode expansion --- p.15 / Chapter 3.3 --- Numerical results --- p.16 / Chapter 3.3.1 --- An illustration of the formalism: potential and electric field obtained --- p.16 / Chapter 3.3.2 --- "A numerical integration technique, triangular function and Fourier series as mode function" --- p.21 / Chapter 3.3.3 --- More results for different u --- p.28 / Chapter 3.3.4 --- Geometric resonance and analysis of M --- p.30 / Chapter 3.3.5 --- Computation requirement --- p.33 / Chapter 4 --- Application of GFF to two-dimensional periodic interface --- p.37 / Chapter 4.1 --- The formalism --- p.37 / Chapter 4.2 --- Solution by mode expansion --- p.43 / Chapter 4.3 --- Technical details in summing the series - points to be noticed --- p.44 / Chapter 4.4 --- Numerical results --- p.49 / Chapter 4.4.1 --- Step function as mode function --- p.49 / Chapter 4.4.2 --- The two-dimensional formulation reproduces previous results --- p.51 / Chapter 4.4.3 --- The interface with ripples added --- p.55 / Chapter 4.4.4 --- A truly two-dimensional periodic interface --- p.59 / Chapter 4.4.5 --- Computational limitation --- p.61 / Chapter 4.4.6 --- Geometric resonance --- p.65 / Chapter 5 --- Conclusion --- p.68 / Bibliography --- p.70 / A Proof of two identities --- p.73

Yu, Hok Pun. / Thesis submitted in: October 2008. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2009. / Includes bibliographical references (leaves 55-56). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.6 / Chapter 2 --- Complex Tori --- p.8 / Chapter 2.1 --- Homomorphisms of complex tori --- p.9 / Chapter 2.2 --- Cohomology of Complex Tori --- p.10 / Chapter 3 --- Line bundles on complex tori --- p.11 / Chapter 3.1 --- First Chern classes --- p.11 / Chapter 3.2 --- Semicharacters on line bundles --- p.12 / Chapter 3.3 --- Theorem of the Square --- p.14 / Chapter 4 --- Principally polarized abelian varieties --- p.16 / Chapter 4.1 --- Riemann Relations --- p.17 / Chapter 4.2 --- Characteristics of line bundles --- p.20 / Chapter 4.3 --- Theta Functions --- p.21 / Chapter 4.4 --- The Ox(l) bundle --- p.22 / Chapter 4.5 --- Metric on Ox(l) --- p.23 / Chapter 4.6 --- Abelian Varieties and Elliptic Curves --- p.24 / Chapter 5 --- Isogeny of Abelian Varieties --- p.26 / Chapter 5.1 --- Symmetric Line Bundles --- p.27 / Chapter 5.2 --- Theta Relations --- p.28 / Chapter 5.3 --- Theta Divisors --- p.30 / Chapter 6 --- Jacobians --- p.32 / Chapter 6.1 --- Jacobian as an abelian variety --- p.33 / Chapter 6.2 --- Abel-Jacobi Theorem --- p.36 / Chapter 6.3 --- Torelli´ةs theorem --- p.42 / Chapter 7 --- The Heisenberg Group --- p.43 / Chapter 8 --- Balanced Embedding into the Projective Space --- p.50

The well known binomial coefficient is the building block of Pascal’s triangle. We explore the relationship between functions of the binomial coefficient and Pascal’s triangle, providing proofs of connections between Catalan numbers, determinants, non-intersecting paths, and Baxter permutations.

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