Thesis (Ph. D.)--University of Wisconsin--Madison, 1980. / Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references (leaves 20-21).

Thesis (Ph. D.)--Oregon State University, 2001. / Typescript (photocopy). Includes bibliographical references (leaves 30). Also available on the World Wide Web.

Thesis--University of Helsinki, 2009. / Includes bibliographical references (p. 109-111).

Thesis (Ph. D.)--University of Wisconsin--Madison, 1980. / Typescript. Vita. Description based on print version record. Includes bibliographical references (leaf 28).

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

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.

This work is devoted to the equation <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Cint_%7BS%7D%0A%5Cfrac%7Bu(y)%20%5C,%20dS(y)%7D%7B%7Cx-y%7C%5E%7BN-1%7D%7D%20=%20f(x)%20%5Ctext%7B,%7D%20%5Cqquad%20%5Cqquad%20x%20%5Cin%20S%20%5Ctext%7B,%7D%0A%5Cqquad%20%5Cqquad%20%5Cqquad%20%5Cqquad%20%5Cqquad%20%5Cqquad%20%5Cqquad%20(1)%0A" /> where S is the graph of a Lipschitz function φ on RN with small Lipschitz constant, and dS is the Euclidian surface measure. The integral in the left-hand side is referred to as a simple layer potential and f is a given function. The main objective is to find a solution u to this equation along with estimates for solutions near points on S. Our analysis is carried out in local Lp-spaces and local Sobolev spaces, and the estimates are given in terms of seminorms. In Paper 1, we consider the case when S is a hyperplane. This gives rise to the classical Riesz potential operator of order one, and we prove uniqueness of solutions in the largest class of functions for which the potential in (1) is defined as an absolutely convergent integral. We also prove an existence result and derive an asymptotic formula for solutions near a point on the surface. Our analysis allows us to obtain optimal results concerning the class of right-hand sides for which a solution to (1) exists. We also apply our results to weighted Lp- and Sobolev spaces, showing that for certain weights, the operator in question is an isomorphism between these spaces. In Paper 2, we present a fixed point theorem for a locally convex space <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Cmathscr%7BX%7D" />, where the topology is given by a family <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5C%7Bp(%20%5C,%20%5Ccdot%20%5C,%20;%20%5Calpha%20)%5C%7D_%7B%5Calpha%20%5Cin%20%5COmega%7D" /> of seminorms. We study the existence and uniqueness of fixed points for a mapping<img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Cmathscr%7BK%7D%20%5C,%20:%20%5C;%20%5Cmathscr%7BD_K%7D%20%5Crightarrow%20%5Cmathscr%7BD_K%7D" /> defined on a set <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Cmathscr%7BD_K%7D%20%5Csubset%20%5Cmathscr%7BX%7D" />. It is assumed that there exists a linear and positive operator K, acting on functions defined on the index set Ω, such that for every <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?u,v%20%5Cin%20%5Cmathscr%7BD_K%7D" />,   <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?p(%5Cmathscr%7BK%7D(u)%20-%20%5Cmathscr%7BK%7D(v)%20%5C,%20;%20%5C,%20%5Calpha%20)%20%0A%5Cleq%20K(p(u-v%20%5C,%20;%20%5C,%20%5Ccdot%20%5C,%20))%20(%5Calpha)%20%5Ctext%7B,%7D%20%5Cqquad%20%5Cqquad%20%5Calpha%20%5Cin%20%5COmega%0A%5Ctext%7B.%7D%0A" /> Under some additional assumptions, one of which is the existence of a fixed point for the operator K + p(<img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Cmathscr%7BK%7D(0)" /> ; · ), we prove that there exists a fixed point of <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Cmathscr%7BK%7D" />. For a class of elements satisfying Kn (p(u ; · ))(α) → 0 as n → ∞, we show that fixed points are unique. This class includes, in particular, the solution we construct in the paper. We give several applications, proving existence and uniqueness of solutions for two types of first and second order nonlinear differential equations in Banach spaces. We also consider pseudodifferential equations with nonlinear terms. In Paper 3, we treat equation (1) in the case when S is a general Lipschitz surface and 1 &lt; p &lt; ∞. Our results are presented in terms of Λ(r), which is the Lipschitz constant of φ on the ball centered at the origin with radius 2r. Estimates of solutions to (1) are provided, which can be used to obtain knowledge about behaviour near a point on S in terms of seminorms. We also show that solutions to (1) are unique if they are subject to certain growth conditions. Examples are given when specific assumptions are placed on Λ. The main tool used for both existence and uniqueness is the fixed point theorem from Paper 2. In Paper 4, we collect some properties and estimates of Riesz potential operators, and also for the operator that was used in Paper 1 and Paper 3 to invert the Riesz potential of order one on RN, for the case when the density function is either radial or has mean value zero on spheres. It turns out that these properties define invariant subspaces of the respective domains of the operators in question.

We here developed a third-order accurate numerical method for scattering of 3D electromagnetic waves by doubly periodic structures. The method is an intuitively simple numerical scheme based on a boundary integral formulation. It involves smoothing the singular Green's functions in the integrands and finding correction terms to the resulting smooth integrals. The analytical method is based on the singular integral methods of J. Thomas Beale, while the scattering problem is motivated by the 2D work of Stephanos Venakides, Mansoor Haider, and Stephen Shipman. The 3D problem was done with boundary element methods by Andrew Barnes. We present a method that is both more straightforward and more accurate. In solving these problems, we have used the M\"{u}ller integral equation formulation of Maxwell's equations, since it is a Fredholm integral equation of the second kind and is well-posed. M\"{u}ller derived his equations for the case of a compact scatterer. We outline the derivation and adapt it to a periodic scatterer. The periodic Green's functions found in the integral equation contain singularities which make it difficult to evaluate them numerically with accuracy. These functions are also very time consuming to evaluate numerically. We use Ewald splitting to represent these functions in a way that can be computed rapidly.We present a method of smoothing the singularity of the Green's function while maintaining its periodicity. We do local analysis of the singularity in order to identify and eliminate the largest sources of error introduced by this smoothing. We prove that with our derived correction terms, we can replace the singular integrals with smooth integrals and only introduce a error that is third order in the grid spacing size. The derivation of the correction terms involves transforming to principal directions using concepts from differential geometry. The correction terms are necessarily invariant under this transformation and depend on geometric properties of the scatterer such as the mean curvature and the differential of the Gauss map. Able to evaluate the integrals to a higher order, we implement a \mbox{GMRES} algorithm to approximate solutions of the integral equation. From these solutions, M\"{u}ller's equations allow us to compute the scattered fields and transmission coefficients. We have also developed acceleration techniques that allow for more efficient computation.We provide results for various scatterers, including a test case for which exact solutions are known. The implemented method does indeed converge with third order accuracy. We present results for which the method successfully resolves Wood's anomaly resonances in transmission. / Dissertation

Thesis (Ph. D.)--University of Chicago, Department of Mathematics, June 2000. / Includes bibliographical references. Also available on the Internet.

Thesis (Ph. D.)--University of Chicago, Dept. of Mathematics, August 2000. / Includes bibliographical references. Also available on the Internet.