Ip Tsz Ho. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2002. / Includes bibliographical references (leaves 86-89). / Chapter 0 --- Introduction --- p.5 / Chapter 1 --- Preliminary --- p.11 / Chapter 1.1 --- Background in analysis --- p.11 / Chapter 1.1.1 --- Holder Continuity --- p.11 / Chapter 1.1.2 --- Hausdorff Measure --- p.12 / Chapter 1.1.3 --- Weak Derivatives --- p.13 / Chapter 1.2 --- Basic Facts of Harmonic Functions --- p.14 / Chapter 1.2.1 --- Harmonic Approximation --- p.14 / Chapter 1.2.2 --- Elliptic Regularity --- p.15 / Chapter 1.3 --- Background in geometry --- p.16 / Chapter 1.3.1 --- Notations and Symbols --- p.16 / Chapter 1.3.2 --- Nearest Point Projection --- p.16 / Chapter 2 --- Monotonicity formula and Regularity of Harmonic maps --- p.17 / Chapter 2.1 --- Energy Minimizing Maps --- p.17 / Chapter 2.2 --- Variational Equations --- p.18 / Chapter 2.3 --- Monotonicity Formula --- p.21 / Chapter 2.4 --- A Technical Lemma --- p.22 / Chapter 2.5 --- Luckhau's Lemma --- p.28 / Chapter 2.6 --- Reverse Poincare Inequality --- p.40 / Chapter 2.7 --- ε-Regularity of Energy Minimizing Maps --- p.45 / Chapter 3 --- Monotonicity Formulae and Vanishing Theorems --- p.52 / Chapter 3.1 --- Stress energy tensor and basic formulae for harmonic p´ؤforms --- p.52 / Chapter 3.2 --- Monotonicity formula --- p.59 / Chapter 3.2.1 --- Monotonicity Formula for Harmonic Maps --- p.64 / Chapter 3.2.2 --- Bochner-Weitzenbock Formula --- p.65 / Chapter 3.3 --- Conservation Law and Vanishing Theorem --- p.68 / Chapter 4 --- On conformally compact Einstein Manifolds --- p.71 / Chapter 4.1 --- Energy Decay of Harmonic Maps with Finite Total Energy --- p.73 / Chapter 4.2 --- Vanishing Theorem of Harmonic Maps --- p.81 / Bibliography --- p.86

R.M. Redheffer. / "June 2, 1948." "This report is a copy of a thesis ... submitted ... for the degree of Doctor of Philosophy in Mathematics at the Massachusetts Institute of Technology." / Bibliography: p. 88. / Army Signal Corps Contract No. W-36-039 sc-32037.

This thesis investigates the relevance between the acoustic resonance and power harmonics on a metal halide lamp. First, a sinusoidal current ranging from 20 kHz to 400 kHz is used to drive a 70 W metal halide lamp. Second, a hybrid-current test circuit is designed to generate a current waveform consisting of a low-frequency square-wave and a high-frequency sinusoidal wave. Both of the frequency and the amplitude can be adjusted independently. The test lamp is deliberately driven at its acoustic-resonance eigen-frequencies to observe the effect of the power spectrum on the degree of the acoustic resonance. The experimental results indicate that the occurrence of acoustic resonance is indeed affected by the DC level and related power harmonics. The power harmonic spectrum that elucidates the initiation of acoustic resonance is deduced from the observations. It is found that the power harmonics that excites acoustic resonance can be divided into three categories. The first is independent of the average lamp power; it excites acoustic resonance only if the magnitude of its power exceeds a specific level. The thresholds of power harmonics belong to the second category are proportional to their DC powers. One can also find those remaining power harmonics belong to the third category. The power harmonic spectrum of the acoustic resonance is demonstrated by driving the test lamp with quasi-square-wave and triangle-wave currents. This work helps advance the understanding of the phenomena and mechanism of acoustic resonance in a metal halide lamp.

Sinusoidal oscillator is intensively used in many applications, such as built-in-self-testing and ADC characterization. An innovative medical application for skin cancer detection employed a technology named bio-impedance spectroscopy, which also requires highly linear sinusoidal-wave as the reference clock. Moreover, the generated sinusoidal signals should be tunable within the frequency range from 10kHz to 10MHz, and quadrature outputs are demanded for coherent demodulation within the system. A design methodology of sinusoidal oscillator named digital-harmonic-cancellation (DHC) technique is presented. DHC technique is realized by summing up a set of square-wave signals with different phase shifts and different summing coefficient to cancel unwanted harmonics. With a general survey of literature, some sinusoidal oscillators based on DHC technique are reviewed and categorized. Also, the mathematical algorithm behind the technique is explained, and non-ideality effect is analyzed based on mathematical calculation. The prototype is fabricated in OnSemi 0.5um CMOS technology. The experimental results of this work show that it can achieve HD2 is -59.74dB and HD3 is -60dB at 0.9MHz, and the frequency is tunable over 0.1MHz to 0.9MHz. The chip consumes area of 0.76mm2, and power consumption at 0.9MHz is 2.98mW. Another design in IBM 0.18um technology is still in the phase of design. The preliminary simulation results show that the 0.18um design can realize total harmonic distortion of -72dB at 10MHz with the power consumption of 0.4mW. The new design is very competitive with state-of-art, which will be finished with layout, submitted for fabrication and measured later.

No description available.

The Fourier algebra A(G) can be viewed as a dual object for the group G and, in turn, for the group algebra L1(G). It is a commutative Banach algebra constructed using the representation theory of the group, and from which the group G may be recovered as its spectrum. When G is abelian, A(G) coincides with L1(G^); for non-abelian groups, it is viewed as a generalization of this object. B. Johnson has shown that G is amenable as a group if and only if L1(G) is amenable as a Banach algebra. Hence, it is natural to expect that the cohomology of A(G) will reflect the amenability of G. The initial hypothesis to this effect is that G is amenable if and only if A(G) is amenable as a Banach algebra. Interestingly, it turns out that A(G) is amenable only when G has an abelian group of finite index, leaving a large class of amenable groups with non-amenable Fourier algebras. The dual of A(G) is a von Neumann algebra (denoted VN(G)); as such, A(G) inherits a natural operator space structure. With this operator space structure, A(G) is a completely contractive Banach algebra, which is the natural operator space analogue of a Banach algebra. By taking this additional structure into account, one recovers the intuition behind the first conjecture: Z.-J. Ruan showed that G is amenable if and only if A(G) is operator amenable. This thesis concerns both the non-amenability of the Fourier algebra in the category of Banach spaces and why Ruan's Theorem is actually the proper analogue of Johnson's Theorem for A(G). We will see that the operator space projective tensor product behaves well with respect to the Fourier algebra, while the Banach space projective tensor product generally does not. This is crucial to explaining why operator amenability is the right sort of amenability in this context, and more generally, why A(G) should be viewed as a completely contractive Banach algebra and not merely a Banach algebra.

Surprisingly, Fourier series on certain fractals can have better convergence properties than classical Fourier series. This is a result of the existence of gaps in the spectrum of the Laplacian. In this work we prove a general criterion for the existence of gaps. Most of the known examples on which the Laplacians admit spectral decimation satisfy the criterion. Then we analyze the infinite family of Vicsek sets, finding an explicit formula for the spectral decimation functions in terms of Chebyshev polynomials. The Laplacians on this infinite family of fractals are also shown to satisfy our criterion and thus have gaps in their spectrum.

The task of constructing a transmission has been provided to us by Digital Engineering, which is a company located in Hovmantorp. After meeting with Digital Engineering we started immediately with the brainstorming to find different ideas with pen and paper. After having presented a number of concepts for the company, we had finally chosen a winner. For the gearbox to be able to work in a good way the tension must never exceed the strain limit of the material. A prototype of the gearbox was produced by TKB Modell. Standard components that are available have been purchased from Sverull AB in Växjö.

The Fourier algebra A(G) can be viewed as a dual object for the group G and, in turn, for the group algebra L1(G). It is a commutative Banach algebra constructed using the representation theory of the group, and from which the group G may be recovered as its spectrum. When G is abelian, A(G) coincides with L1(G^); for non-abelian groups, it is viewed as a generalization of this object. B. Johnson has shown that G is amenable as a group if and only if L1(G) is amenable as a Banach algebra. Hence, it is natural to expect that the cohomology of A(G) will reflect the amenability of G. The initial hypothesis to this effect is that G is amenable if and only if A(G) is amenable as a Banach algebra. Interestingly, it turns out that A(G) is amenable only when G has an abelian group of finite index, leaving a large class of amenable groups with non-amenable Fourier algebras. The dual of A(G) is a von Neumann algebra (denoted VN(G)); as such, A(G) inherits a natural operator space structure. With this operator space structure, A(G) is a completely contractive Banach algebra, which is the natural operator space analogue of a Banach algebra. By taking this additional structure into account, one recovers the intuition behind the first conjecture: Z.-J. Ruan showed that G is amenable if and only if A(G) is operator amenable. This thesis concerns both the non-amenability of the Fourier algebra in the category of Banach spaces and why Ruan's Theorem is actually the proper analogue of Johnson's Theorem for A(G). We will see that the operator space projective tensor product behaves well with respect to the Fourier algebra, while the Banach space projective tensor product generally does not. This is crucial to explaining why operator amenability is the right sort of amenability in this context, and more generally, why A(G) should be viewed as a completely contractive Banach algebra and not merely a Banach algebra.

Surprisingly, Fourier series on certain fractals can have better convergence properties than classical Fourier series. This is a result of the existence of gaps in the spectrum of the Laplacian. In this work we prove a general criterion for the existence of gaps. Most of the known examples on which the Laplacians admit spectral decimation satisfy the criterion. Then we analyze the infinite family of Vicsek sets, finding an explicit formula for the spectral decimation functions in terms of Chebyshev polynomials. The Laplacians on this infinite family of fractals are also shown to satisfy our criterion and thus have gaps in their spectrum.