We study boundary-contact problems for elliptic equations (and systems) with interfaces that have edge singularities. Such problems represent continuous operators between weighted edge spaces and subspaces with asymptotics. Ellipticity is formulated in terms of a principal symbolic hierarchy, containing interior, transmission, and edge symbols. We construct parametrices, show regularity with asymptotics of solutions in weighted edge spaces and illustrate the results by boundary-contact problems for the Laplacian with jumping coefficients.

In Chapter One the theoretical basis for infinitesimal transformations is presented with particular emphasis on the central theme of this thesis which is the invariance of ordinary differential equations, and their first integrals, under infinitesimal transformations. The differential operators associated with these infinitesimal transformations constitute an algebra under the operation of taking the Lie Bracket. Some of the major results of Lie's work are recalled. The way to use the generators of symmetries to reduce the order of a differential equation and/or to find its first integrals is explained. The chapter concludes with a summary of the state of the art in the mid-seventies just before the work described here was initiated. Chapter Two describes the growing awareness of the algebraic properties of the paradigms of differential equations. This essentially ad hoc period demonstrated that there was value in studying the Lie method of extended groups for finding first integrals and so solutions of equations and systems of equations. This value was emphasised by the application of the method to a class of nonautonomous anharmonic equations which did not belong to the then pantheon of paradigms. The generalised Emden-Fowler equation provided a route to major development in the area of the theory of the conditions for the linearisation of second order equations. This was in addition to its own interest. The stage was now set to establish broad theoretical results and retreat from the particularism of the seventies. Chapters Three and Four deal with the linearisation theorems for second order equations and the classification of intrinsically nonlinear equations according to their algebras. The rather meagre results for systems of second order equations are recorded. In the fifth chapter the investigation is extended to higher order equations for which there are some major departures away from the pattern established at the second order level and reinforced by the central role played by these equations in a world still dominated by Newton. The classification of third order equations by their algebras is presented, but it must be admitted that the story of higher order equations is still very much incomplete. In the sixth chapter the relationships between first integrals and their algebras is explored for both first order integrals and those of higher orders. Again the peculiar position of second order equations is revealed. In the seventh chapter the generalised Emden-Fowler equation is given a more modern and complete treatment. The final chapter looks at one of the fundamental algebras associated with ordinary differential equations, the three element 8£(2, R), which is found in all higher order equations of maximal symmetry, is a fundamental feature of the Pinney equation which has played so prominent a role in the study of nonautonomous Hamiltonian systems in Physics and is the signature of Ermakov systems and their generalisations. / Thesis (Ph.D.)-University of Natal, 1995.

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In this thesis, we consider real matrix functions that depend on two parameters and study the problem of how to detect and approximate parameters' values where the singular values coalesce. We prove several results connecting the existence of coalescing points to the periodic structure of the smooth singular values decomposition computed around the boundary of a domain enclosing the points. This is further used to develop algorithms for the detection and approximation of coalescing points in planar regions. Finally, we present techniques for continuing curves of coalescing singular values of matrices depending on three parameters, and illustrate how these techniques can be used to locate coalescing singular values of complex-valued matrices depending on three parameters.

Inverse problems are problems where causes for desired or an observed effect are to be determined. They lie at the heart of scientific inquiry and technological development, including radar/sonar, medical imaging, geophysical exploration, invisibility cloaking and remote sensing, to name just a few. In this thesis, we focus on the theoretical study and applications of some intriguing inverse problems. Precisely speaking, we are concerned with two typical kinds of problems in the field of wave scattering and nonlocal inverse problem, respectively. The first topic is on the geometric structures of eigenfunctions and their applications in wave scattering theory, in which the conductive transmission eigenfunctions and Laplacian eigenfunctions are considered. For the study on the intrinsic geometric structures of the conductive transmission eigenfunctions, we first present the vanishing properties of the eigenfunctions at corners both in R2 and R3, based on microlocal analysis with the help of a particular type of planar complex geometrical optics (CGO) solution. This significantly extends the previous study on the interior transmission eigenfunctions. Then, as a practical application of the obtained geometric results, we establish a unique recovery result for the inverse problem associated with the transverse electromagnetic scattering by a single far-field measurement in simultaneously determining a polygonal conductive obstacle and its surface conductive parameter. For the study on the geometric structures of Laplacian eigenfunctions, we separately discuss the two-dimensional case and the three-dimensional case. In R2, we introduce a new notion of generalized singular lines of Laplacian eigenfunctions, and carefully investigate these singular lines and the nodal lines. The studies reveal that the intersecting angle between two of those lines is closely related to the vanishing order of the eigenfunction at the intersecting point. We provide an accurate and comprehensive quantitative characterization of the relationship. In R3, we study the analytic behaviors of Laplacian eigenfunctions at places where nodal or generalized singular planes intersect, which is much more complicated. These theoretical findings are original and of significant interest in spectral theory. Moreover, they are applied directly to some physical problems of great importance, including the inverse obstacle scattering problem and the inverse diffraction grating problem. It is shown in a certain polygonal (polyhedral) setup that one can recover the support of the unknown scatterer as well as the surface impedance parameter by finitely many far-field patterns. Our second topic is concerning the fractional partial differential operators and some related nonlocal inverse problems. We present some prelimilary knowledge on fractional Sobolev Spaces and fractional partial differential operators first. Then we focus on the simultaneous recovery results of two interesting nonlocal inverse problems. One is simultaneously recovering potentials and the embedded obstacles for anisotropic fractional Schrödinger operators based on the strong uniqueness property and Runge approximation property. The other one is the nonlocal inverse problem associated with a fractional Helmholtz equation that arises from the study of viscoacoustics in geophysics and thermoviscous modelling of lossy media. We establish several general uniqueness results in simultaneously recovering both the medium parameter and the internal source by the corresponding exterior measurements. The main method utilized here is the low-frequency asymptotics combining with the variational argument. In sharp contrast, these unique determination results are unknown in the local case, which would be of significant importance in thermo- and photo-acoustic tomography.

The program MANDO efficiently performs computations involving pairs of operators, a single operator, and operators applied to functions, saving time and cost over pencil-·and-paper methods. A versatile but compact data structure, defined under SPIT-BOL's facility for creation of datatypes, contains the operators (and functions) and provides a means for systematically referencing their relevant parts. On input, functions and operators are written in a restricted but natural string format, for which the program can readily convert them to the internal data structure. Central to the method of operation of the derivative routine is its ability to differentiate a function written as a string. This allows for a certain compactness in the internal form. To counteract the relative slowness of string processing in the derivative routine, the program keeps a table of derivatives repeated during the processing . The table is checked for the presence of the function and its derivative before the derivative sequence is applied. Some simplification is performed. The simplification relies on the ordering sequence, followed by a sequence which cancels or combines terms that are alike, except, in general, for numerical multipliers.

In this thesis, we define differential operators for Hermitian Jacobi forms and Hermitian modular forms over the Gaussian number field Q(i). In particular, we construct Rankin-Cohen brackets for such spaces of Hermitian Jacobi forms and Hermitian modular forms. As an application, we extend Rankin's method to the case of Hermitian Jacobi forms. Finally we compute Fourier series coefficients of Hermitian modular forms, which allow us to give an example of the first Rankin-Cohen bracket of two Hermitian modular forms. In the appendix, we provide tables of Fourier series coefficients of Hermitian modular forms and also the computer source code that we used to compute such Fourier coefficients.

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This thesis consists of two papers, enumerated by Roman numerals. The main focus is on the spectral theory of <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?n" />-Laplacians. Here, an <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?n" />-Laplacian, for integer <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?n" />, refers to a metric graph equipped with a differential operator whose differential expression is the <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?2n" />-th derivative. In Paper I, a classification of all vertex conditions corresponding to self-adjoint <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?n" />-Laplacians is given, and for these operators, a secular equation is derived. Their spectral asymptotics are analysed using the fact that the secular function is close to a trigonometric polynomial, a type of almost periodic function. The notion of the quasispectrum for <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?n" />-Laplacians is introduced, identified with the positive roots of the associated trigonometric polynomial, and is proved to be unique. New results about almost periodic functions are proved, and using these it is shown that the quasispectrum asymptotically approximates the spectrum, counting multiplicities, and results about asymptotic isospectrality are deduced. The results obtained on almost periodic functions have wider applications outside the theory of differential operators. Paper II deals more specifically with bi-Laplacians (<img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?n=2" />), and a notion of standard conditions is introduced. Upper and lower estimates for the spectral gap --- the difference between the two lowest eigenvalues - for these standard conditions are derived. This is achieved by adapting the methods of graph surgery used for quantum graphs to fourth order differential operators. It is observed that these methods offer stronger estimates for certain classes of metric graphs. A geometric version of the Ambartsumian theorem for these operators is proved.

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