We consider on the interval [0; a], rstly fourth-order di erential operators with eigenvalue parameter dependent boundary conditions and secondly a sixth-order di erential operator with eigenvalue parameter dependent boundary conditions. We associate to each of these problems a quadratic operator pencil with self-adjoint operators. We investigate the spectral proprieties of these problems, the location of the eigenvalues and we explicitly derive the rst four terms of the eigenvalue asymptotics.

by Lau Shui-kong, Francis. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1999. / Includes bibliographical references (leaves 95-98). / Abstracts in English and Chinese. / Chapter 1 --- General Theory of Wavelets --- p.8 / Chapter 1.1 --- Introduction --- p.8 / Chapter 1.2 --- Multiresolution Analysis and Wavelets --- p.9 / Chapter 1.3 --- Orthonormal Bases of Compactly Supported Wavelets --- p.12 / Chapter 1.3.1 --- Example : The Daubechies Wavelets --- p.15 / Chapter 1.4 --- Wavelets in Higher Dimensions --- p.20 / Chapter 1.4.1 --- Tensor product method --- p.20 / Chapter 1.4.2 --- Multiresolution Analysis in Rd --- p.21 / Chapter 1.5 --- Generalization to frames --- p.25 / Chapter 2 --- Wavelet Bases Numerical Algorithm --- p.27 / Chapter 2.1 --- The Algorithm in Wavelet Bases --- p.27 / Chapter 2.1.1 --- Definitions and Notations --- p.28 / Chapter 2.1.2 --- Fast Wavelet Transform --- p.31 / Chapter 2.2 --- Wavelet-Based Quadratures --- p.33 / Chapter 2.3 --- "The Integral Operator, Standard and Non-standard Form" --- p.39 / Chapter 2.3.1 --- The Standard Form --- p.40 / Chapter 2.3.2 --- The Non-standard Form --- p.41 / Chapter 2.4 --- The Calderon-Zygmund Operator and Numerical Cal- culation --- p.45 / Chapter 2.4.1 --- Numerical Algorithm to Construct the Non- standard Form --- p.45 / Chapter 2.4.2 --- Numerical Calculation and Compression of Op- erators --- p.45 / Chapter 2.5 --- Differential Operators in Wavelet Bases --- p.48 / Chapter 3 --- T(l)-Theorem of David and Journe --- p.55 / Chapter 3.1 --- Definitions and Notations --- p.55 / Chapter 3.1.1 --- T(l) Operator --- p.56 / Chapter 3.2 --- The Wavelet Proof of the T(l)-Theorem --- p.59 / Chapter 3.3 --- Proof of the T(l)-Theorem (Continue) --- p.64 / Chapter 3.4 --- Some recent results on the T(l)-Theorem --- p.70 / Chapter 4 --- Singular Values of Compact Pseudodifferential Op- erators --- p.72 / Chapter 4.1 --- Background --- p.73 / Chapter 4.1.1 --- Singular Values --- p.73 / Chapter 4.1.2 --- Schatten Class Ip --- p.73 / Chapter 4.1.3 --- The Ambiguity Function and the Wigner Dis- tribution --- p.74 / Chapter 4.1.4 --- Weyl Correspondence --- p.76 / Chapter 4.1.5 --- Gabor Frames --- p.78 / Chapter 4.2 --- Singular Values of Lσ --- p.82 / Chapter 4.3 --- The Calderon-Vaillancourt Theorem --- p.87 / Chapter 4.3.1 --- Holder-Zygmund Spaces --- p.87 / Chapter 4.3.2 --- Smooth Dyadic Resolution of Unity --- p.88 / Chapter 4.3.3 --- The proof of the Calderon-Vaillancourt The- orem --- p.89 / Bibliography

A análise harmônica e o ramo da matemática que estuda a representação de funções ou sinais como a sobreposição de ondas base. Ela investiga e generaliza as noções das séries de Fourier e da transformação de Fourier. Neste trabalho, investigou-se um teorema de restrição da transformada de Fourier devido a Mitsis e Mockenhaupt (uma generalização do teorema de Stein-Tomas). Foram realizados estudos analíticos sobre o método para operadores integrais oscilatórios, baseado na fase estacionária. Os resultados permitem deduzir o teorema de restrição no plano (em seu caso geral) e o teorema de Carleson-Sjölin. / Harmonic analysis is the mathematical branch that studies the function or signals representation as a base wave overlay. It investigates and generalizes the notions of Fourier series and of the Fourier transform. In this work, was investigated a restriction theorem of the Fourier transform due to Mitsis and Mockenhaupt (a generalization of Stein-Tomas theorem) . Were performed analytic studies on the method for oscillating integral operators, based in the stationary phase. The results allow deducing the restriction theorem on the plane (in the general case) and the Carleson-Sjölin theorem.

by Hon-Ming Ho. / "May 2004." / Thesis (Ph.D.)--Chinese University of Hong Kong, 2004. / Includes bibliographical references (p. 69-71). / 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.

Alon 和Krivelevich (SIAM J. Discrete Math. 15(2): 211-227 (2002)) 證明了如果一個圖是ε -非二部圖，那麼階數為Ỡ(1/ε) 的隨機導出于圖以很大概率是非二部圖。我們進一步猜想，這個導出子圖以很大概率是Ω(ε)-非二部圖。Gonen 和Ron (RANDOM 2007) 證明了當圖的最大度不超過O (εn )時猜想成立。我們將對更一般的情形給出證明，對於任意d，所有d 正則(或幾乎d 正則)的圖猜想成立。 / 假設猜想成立的情況下，我們證明二分屬性是可以被單側誤差在O(1/ε^c ) 時間內檢驗的，其中c 是一個嚴格小於2 的常數，而這個結果也改進了Alon 和Krivelevich 的檢驗算法。由於己知對二分屬性的非適應性的檢驗算法需要Ω(1 /ε²) 的複雜性(Bogdanov 和Trevisan ， CCC 2004) ，我們的結果也得出，假設猜想成立，適應性對檢驗二分屬性是有幫助的。 / 另外一個有很多屬性檢驗問題被廣泛研究的領域是布爾函數。對布爾函數單調性的檢驗也是屬性檢驗的經典問題。給定對布爾函數f: {0,1}{U+207F} → {0,1} 的訪問，在[18]中，證明了檢驗算法複雜性的下界是Ω(√n) 。另一方面，在[21]中，作者們構造了一個複雜性為O(n) 的算法。在本文中，我們刻畫一些單調布爾函數的本質，設計算法并分析其對於一些困難例子的效果。最近，在[14] 中， 一個類似的算法被證明是非適應性，單側誤差，複雜性為Ỡ (n⁵/⁶ ε⁻⁵/³) 的算法。 / Alon and Krivelevich (SIAM J. Discrete Math. 15(2): 211-227 (2002)) show that if a graph is ε-far from bipartite, then the subgraph induced by a random subset of Ỡ (1/ε) vertices is not bipartite with high probability. We conjecture that the induced subgraph is Ω(ε)-far from bipartite with high probability. Gonen and Ron (RANDOM 2007) proved this conjecture in the case when the degrees of all vertices are at most O(εn). We give a more general proof that works for any d-regular (or almost d-regular) graph for arbitrary degree d. / Assuming this conjecture, we prove that bipartiteness is testable with one-sided error in time O(1=ε{U+1D9C}), where c is a constant strictly smaller than two, improving upon the tester of Alon and Krivelevich. As it is known that non-adaptive testers for bipartiteness require Ω(1/ε²) queries (Bogdanov and Trevisan, CCC2004), our result shows, assuming the conjecture, that adaptivity helps in testing bipartiteness. / The other area in which various properties have been well studied is boolean function. Testing monotonicity of Boolean functions is a classical question in property testing. Given oracle access to a Boolean function f : {0, 1}{U+207F} →{0, 1}, in [18], it is shown a lower bound of testing is Ω(√n). On the other hand, in [21], the authors introduced an algorithm to test monotonicity using O(n) queries. In this paper, we characterize some nature of monotone functions, design a tester and analyze the performance on some generalizations of the hard case. Recently, in [14], a similar tester is shown to be a non-adaptive, one-sided error tester making Ỡ (n⁵/⁶ ε⁻⁵/³) queries. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Li, Fan. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2013. / Includes bibliographical references (leaves 76-79). / Abstracts also in Chinese. / Abstract --- p.i / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Property Testing --- p.1 / Chapter 1.2 --- Testing Bipartiteness --- p.4 / Chapter 1.3 --- Testing Monotonicity --- p.7 / Chapter 2 --- Testing Bipartiteness --- p.11 / Chapter 2.1 --- Background --- p.11 / Chapter 2.1.1 --- Our result --- p.11 / Chapter 2.1.2 --- The algorithms of Gonen and Ron --- p.13 / Chapter 2.1.3 --- Our proof --- p.16 / Chapter 2.1.4 --- Notation --- p.19 / Chapter 2.2 --- Splitting the vertices by degree --- p.19 / Chapter 2.3 --- The algorithm for high degree vertices --- p.20 / Chapter 2.4 --- Eliminating the high degree vertices --- p.22 / Chapter 2.5 --- From an XOR game to a bipartite graph --- p.33 / Chapter 2.6 --- Proof of the main theorem --- p.35 / Chapter 2.7 --- Proof of the conjecture for regular graphs --- p.37 / Chapter 3 --- Testing Monotonicity --- p.40 / Chapter 3.1 --- Towards an improved tester --- p.40 / Chapter 3.1.1 --- Properties of Distribution D --- p.42 / Chapter 3.1.2 --- An Alternative Representation of D --- p.46 / Chapter 3.1.3 --- Performance of D on Decreasing Functions --- p.51 / Chapter 3.1.4 --- Functions Containing Ω(2{U+207F}) Disjoint Violating Edges --- p.54 / Chapter 3.2 --- A o(n) Monotonicity Tester [14] and Some Improvements --- p.62 / Chapter 3.2.1 --- A o(n) Monotonicity Tester [14] --- p.62 / Chapter 3.2.2 --- An Alternative Proof of Theorem 3.2.2 --- p.65 / Chapter 4 --- Other Related Results --- p.67 / Chapter 4.1 --- Short Odd Cycles in Graphs that are Far From Bipartiteness --- p.67 / Chapter 4.2 --- Fourier Analysis on Boolean Functions --- p.69 / Bibliography --- p.76

Genetic algorithms are an evolutionary technique that use crossover and mutation operators to solve optimization problems using a survival of the fittest idea. They have been used successfully in a variety of different problems, including the traveling salesman problem. In the traveling salesman problem we wish to find a tour of all nodes in a weighted graph so that the total weight is minimized. The traveling salesman problem is NP-hard but has many real world applications so a good solution would be useful. Many different crossover and mutation operators have been devised for the traveling salesman problem and each give different results. We compare these results and find that operators that use heuristic information or a matrix representation of the graph give the best results.

In this work we tackle the challenge of designing quantum unitary operators which represent solutions to optimization problems. We start with a novel method which combines an evolutionary algorithm known as an Evolution Strategy (ES) with a method to randomly generate unitary operators. With this new method, a quantum operator is represented for the first time using real-valued vectors and can be "evolved" or designed to meet certain target criteria. This criteria could be the solution to an optimization problem. With the ability to evolve quantum operators, we attempt to evolve various known single and multi-qubit quantum gates as well as quantum oracles. We evolve quantum operators which solve instance problems of a known NP-Hard problem and even attempt to evolve a generalized solution operator. We evolve multiple operators with varying size and investigate their properties through eigenanalysis methods as well as by synthesizing them into quantum logic gates using the quantum compiler Qubiter. We also present a new quantum logic algebra which offers a new way to represent quantum circuits and demonstrate its immediate uses in quantum computing.

Since F. Riesz showed in 1909 that the dual of C[0, 1] is BV[0, 1] (the functions of bounded variation on [0, 1] with II g IIBV = V(g)) via the Stieltjes integral, obtaining representations for linear operators in various settings has been a problem of interest. This paper shows the historical manner of representations, the road map type theorems and representations obtained via the v-integral. (44 pages)

George Zames. / "Submitted to the Department of Electrical Engineering, M.I.T., August 22, 1960, in partial fulfillment of the requirements for the degree of Doctor of Science." / Bibliography: p.76. / Army Signal Corps Contract DA 36-039-sc-78108. Dept. of the Army Task 3-99-20-001 and Project 3-99-00-000.

No description available.