In this thesis, we consider the question of describing the eigenvalues of a distinguished family of invariant differential operators associated to a Lie superalgebra g and a g-module W, called the "Capelli basis", via evaluation of certain classes of supersymmetric functions, called the interpolation super Jack polynomials. Finding the eigenvalues of the Capelli basis is referred to the Capelli Eigenvalue Problem. The eigenvalue formula depends on the chosen parametrization of the highest weight vectors in the decomposition of the superpolynomial algebra P(W), and consequently on the choice of a Borel subalgebra. In this thesis, we give a solution for each conjugacy class of Borel subalgebras, which we call a refined solution to the Capelli Eigenvalue Problem. Given the pair (g, W), we investigate the formulae for the eigenvalues of the Capelli operators associated to the completely reducible and multiplicity-free modules for two cases: diagonal and symmetric cases. In the former case, we show that we can express the eigenvalue of the Capelli operator on the irreducible component of the multiplicity-free decomposition of P(W) as a polynomial function of the b-highest weight of the irreducible component for any Borel subalgebra b. In the latter case, we show with a concrete counterexample that we cannot expect the results to be as strong as in the first case for all Borel subalgebras. We then express the eigenvalue of the Capelli operator on the irreducible component of the multiplicity-free decomposition of P(W) as a polynomial function of a piecewise affine map on the span of b-highest weights of the irreducible submodules of P(W), with respect to different decreasing Borel subalgebras b.

This thesis demonstrates an efficient parallel method of solving the generalized eigenvalue problem, KΦ = M ΦΛ, where K is symmetric and M is symmetric positive-definite, by first converting it to a standard eigenvalue problem, solving the standard eigenvalue problem, and back-transforming the results. An abstraction for parallel dense linear algebra is introduced along with a new algorithm for forming A := U⁻ᵀ K U⁻¹ , where U is the Cholesky factor of M , that is up to twice as fast as the ScaLAPACK implementation. Additionally, large improvements over the PBLAS implementations of general matrix-matrix multiplication and triangular solves with many right-hand sides are shown. Significant performance gains are also demonstrated for Cholesky factorizations, and a case is made for using 2D-cyclic distributions with a distribution blocksize of one. / text

A method that can be used to detect damage in structures is developed. This method is based on the convexity of the geometric interpretation of the eigenvalue problem for undamped positive definite systems. The damage detection scheme establishes various damage scenarios which are used as failure sets. These scenarios are then compared to the structure's actual response by measuring the natural frequencies of the structure and using a Euclideian norm. Mathematical models were developed for application of the method on a cantilever beam. Damage occurring at a single location or in multiple locations was estalished and studied. Experimental verification was performed on serval prismatic beams in which the method provided adequate results. The exact location and extent of damage for several cases was predicted. When the method failed the prediction was very close to the actual condition in the structure. This method is easy to use and does not require a rigorous amount of instrumentation for obtaining the experimental data required in the detection scheme. / Ph. D.

The central theme of this thesis deals with problems related to the question, “Can one hear the shape of a drum?” first posed formally by Mark Kac in 1966. More precisely, can one determine the shape of a membrane with fixed boundary from the spectrum of the associated differential operator? For this paper, Kac received both the Lester Ford Award and the Chauvant Prize of the Mathematical Association of America. This problem has received a great deal of attention in the past forty years and has led to similar questions in completely different contexts such as “Can one hear the shape of a graph associated with the Schrödinger operator?”, “Can you hear the shape of your throat?”, “Can you feel the shape of a manifold with Brownian motion?”, “Can one hear the crack in a beam?”, “Can one hear into the sun?”, etc. Each of these topics deals with inverse eigenvalue problems or related inverse problems. For inverse problems in general, the problem may or may not have a solution, the solution may not be unique, and the solution does not necessarily depend continuously on perturbation of the data. For example, in the case of the drum, it has been shown that the answer to Kac’s question in general is “no.” However, if we restrict the class of drums, then the answer can be yes. This is typical of inverse problems when a priori information and restriction of the class of admissible solutions and/or data are used to make the problem well-posed. This thesis provides an analysis of shapes for which the answer to Kac's question is positive and a variety of interesting questions on this problem and its variants, including cases that remain open. This thesis also provides a synopsis and perspectives of other types of “can one hear” problems mentioned above. Another part of this thesis deals with aspects of direct problems related to musical instruments.

O objetivo desta dissertação é estudar o problema de autovalor de Dirichlet para variedades riemannianas completas. Mais precisamente, pretendemos estudar uma cota por baixo para o -ésimo autovalor de um domínio limitado em uma variedade riemanniana completa. Tal cota é obtida fazendo-se uso de uma fórmula de recorrência de Cheng e Yang e um teorema de Nash. Ademais, pretendemos estudar uma desigualdade universal para os autovalores no espaço hiperbólico. / The goal of this dissertation is to study the Dirichlet eigenvalue problem for a complete riemannian manifold. More accurately, we intend to investigate a lower-bound for the -ℎ eigenvalue on a bounded domain in a complete riemannian manifold. Such a bound is obtained by making use of a recursion formula of Cheng and Yang and Nash’s Theorem. Furthermore, we study a universal inequality for eigenvalues of the Dirichlet eigenvalue problem on a bounded domain in a hyperbolic space (−1).

O objetivo desta dissertação é estudar o problema de autovalor de Dirichlet para variedades riemannianas completas. Mais precisamente, pretendemos estudar uma cota por baixo para o -ésimo autovalor de um domínio limitado em uma variedade riemanniana completa. Tal cota é obtida fazendo-se uso de uma fórmula de recorrência de Cheng e Yang e um teorema de Nash. Ademais, pretendemos estudar uma desigualdade universal para os autovalores no espaço hiperbólico. / The goal of this dissertation is to study the Dirichlet eigenvalue problem for a complete riemannian manifold. More accurately, we intend to investigate a lower-bound for the -ℎ eigenvalue on a bounded domain in a complete riemannian manifold. Such a bound is obtained by making use of a recursion formula of Cheng and Yang and Nash’s Theorem. Furthermore, we study a universal inequality for eigenvalues of the Dirichlet eigenvalue problem on a bounded domain in a hyperbolic space (−1).

O objetivo desta dissertação é estudar o problema de autovalor de Dirichlet para variedades riemannianas completas. Mais precisamente, pretendemos estudar uma cota por baixo para o -ésimo autovalor de um domínio limitado em uma variedade riemanniana completa. Tal cota é obtida fazendo-se uso de uma fórmula de recorrência de Cheng e Yang e um teorema de Nash. Ademais, pretendemos estudar uma desigualdade universal para os autovalores no espaço hiperbólico. / The goal of this dissertation is to study the Dirichlet eigenvalue problem for a complete riemannian manifold. More accurately, we intend to investigate a lower-bound for the -ℎ eigenvalue on a bounded domain in a complete riemannian manifold. Such a bound is obtained by making use of a recursion formula of Cheng and Yang and Nash’s Theorem. Furthermore, we study a universal inequality for eigenvalues of the Dirichlet eigenvalue problem on a bounded domain in a hyperbolic space (−1).

This paper derives explicit eigensolutions of the Laplace operator, whose eigenvalue problem is also called the Helmholtz equation. Specifically, the paper showcases all geometries through which the solutions to the Helmholtz equation can be represented in a finite sinusoidal form. These geometries are the rectangle, the square, the isosceles right triangle, the equilateral triangle, and the hemi-equilateral triangle. As a counterexample, the paper also proves that the parallelogram cannot yield a product form of a solution through the method of separation of variables. The solutions for the isosceles triangle and the hemi-equilateral triangle are derived using symmetric properties of the square and the equilateral triangle.  The paper concludes that symmetry is crucial to solving the Laplacian for these geometries and that this symmetry is also reflected in their respective spectra. However, importantly, the spectrum is unique for the examined geometries.

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Symmetric generalized eigenvalue problems arise in many physical applications and frequently only a few of the eigenpairs are of interest. Typically, the problems are large and sparse, and therefore traditional methods such as the QZ algorithm may not be considered. Moreover, it may be impractical to apply shift-and-invert Lanczos, a favored method for problems of this type, due to difficulties in applying the inverse of the shifted matrix. With these difficulties in mind, Golub and Ye developed an inverse free Krylov subspace algorithm for the symmetric generalized eigenvalue problem. This method does not rely on shift-and-invert transformations for convergence acceleration, but rather a preconditioner is used. The algorithm suffers, however, in the presence of multiple or clustered eigenvalues. Also, it is only applicable to the location of extreme eigenvalues. In this work, we extend the method of Golub and Ye by developing a block generalization of their algorithm which enjoys considerably faster convergence than the usual method in the presence of multiplicities and clusters. Preconditioning techniques for the problems are discussed at length, and some insight is given into how these preconditioners accelerate the method. Finally we discuss a transformation which can be applied so that the algorithm extracts interior eigenvalues. A preconditioner based on a QR factorization with respect to the B-1 inner product is developed and applied in locating interior eigenvalues.