Synthesis of structures is a very difficult task even with only a small number of components that form a system; yet it is the catalyst of innovation. Molecular structures and nanostructures typically have a large number of similar components but different connections, which manifests a more challenging task for their synthesis. <p> This thesis presents a novel method and its related algorithms and computer programs for the synthesis of structures. This novel method is based on several concepts: (1) the structure is represented by a graph and further by the adjacency matrix; and (2) instead of only exploiting the eigenvalue of the adjacency matrix, both the eigenvalue and the eigenvector are exploited; specifically the components of the eigenvector have been found very useful in algorithm development. This novel method is called the Eigensystem method.<p> The complexity of the Eigensystem method is equal to that of the famous program called Nauty in the combinatorial world. However, the Eigensystem method can work for the weighted and both directed and undirected graph, while the Nauty program can only work for the non-weighted and both directed and undirected graph. The cause for this is the different philosophies underlying these two methods. The Nauty program is based on the recursive component decomposition strategy, which could involve some unmanageable complexities when dealing with the weighted graph, albeit no such an attempt has been reported in the literature. It is noted that in practical applications of structure synthesis, weighted graphs are more useful than non-weighted graphs for representing physical systems. <p> Pivoted at the Eigensystem method, this thesis presents the algorithms and computer programs for the three fundamental problems in structure synthesis, namely the isomorphism/automorphism, the unique labeling, and the enumeration of the structures or graphs.

Synthesis of structures is a very difficult task even with only a small number of components that form a system; yet it is the catalyst of innovation. Molecular structures and nanostructures typically have a large number of similar components but different connections, which manifests a more challenging task for their synthesis. <p> This thesis presents a novel method and its related algorithms and computer programs for the synthesis of structures. This novel method is based on several concepts: (1) the structure is represented by a graph and further by the adjacency matrix; and (2) instead of only exploiting the eigenvalue of the adjacency matrix, both the eigenvalue and the eigenvector are exploited; specifically the components of the eigenvector have been found very useful in algorithm development. This novel method is called the Eigensystem method.<p> The complexity of the Eigensystem method is equal to that of the famous program called Nauty in the combinatorial world. However, the Eigensystem method can work for the weighted and both directed and undirected graph, while the Nauty program can only work for the non-weighted and both directed and undirected graph. The cause for this is the different philosophies underlying these two methods. The Nauty program is based on the recursive component decomposition strategy, which could involve some unmanageable complexities when dealing with the weighted graph, albeit no such an attempt has been reported in the literature. It is noted that in practical applications of structure synthesis, weighted graphs are more useful than non-weighted graphs for representing physical systems. <p> Pivoted at the Eigensystem method, this thesis presents the algorithms and computer programs for the three fundamental problems in structure synthesis, namely the isomorphism/automorphism, the unique labeling, and the enumeration of the structures or graphs.

Let X, Y be locally compact Hausdorff spaces. A linear operator T from C0(X,E) to C0(Y,F) is called disjointness preserving if coz(Tf)¡äcoz(Tg) = whenever coz(f)¡äcoz(g) = ∅. We discuss some cases on these compact disjointness preserving operators T and prove that if £f0 is a nonzero point of £m(T), then £f0 is an eigenvalue of T and we find a projection ∏: C0(X,E) ¡÷C0(X,E), such that for Y1 = ∏C0(X;E) and Y2 = (1-∏)C0(X;E), the operator T|Y1 -£f0 is a nilpotent and £f0-T|Y2 is invertible.

The eigenvalue gaps and eigenvalue ratios of the Sturm-Liouville systems have been studied in many papers. Recently, Lavine proved an optimal lower estimate of first eigenvalue gaps for Schrodinger operators with convex potentials. His method uses a variational approach with detailed analysis on different integrals. In 1999, (M.J.) Huang adopted his method to study eigenvalue ratios of vibrating strings. He proved an optimal lower estimate of first eigenvalue ratios with nonnegative densities. In this thesis, we want to generalize the above optimal estimate. The work of Ashbaugh and Benguria helps in attaining our objective. They introduced an approach involving a modified Prufer substitution and a comparison theorem to study the upper bounds of Dirichlet eigenvalue ratios for Schrodinger operators with nonnegative potentials. It is interesting to see that the counterpart of their result is also valid. By Liouville substitution and an approximation theorem, the vibrating strings with concave and positive densities can be transformed to a Schrodinger operator with nonpositive potentials. Thus we have the generalization of Huang's result.

The problem of determining the eigenvalues of a vibrational system having multiple lumped attachments has been investigated extensively. However, most of the research conducted in this field focuses on determining the natural frequencies of the combined system assuming that the characteristics of the combined vibrational system are known (forward problem). A problem of great interest from the point of view of engineering design is the ability to impose certain frequencies on the vibrational system or to avoid certain frequencies by modifying the characteristics of the vibrational system (inverse problem). In this thesis, the effects of adding lumped masses to an Euler-Bernoulli beam on its frequencies and their corresponding mode shapes are investigated for simply-supported as well as fixed-free boundary conditions. This investigation paves the way for proposing a method to impose two frequencies on a system consisting of a beam and a lumped mass by determining the magnitude of the mass as well as its position along the beam.

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 study investigates the vibration of a rotating piezoelectric device that consists of a proof mass that is supported by elastic structures with piezoelectric layers. Vibration of the proof mass causes deformation in the piezoelectric structures and voltages to power electrical loads. The coupled electromechanical equations of motion are derived using Newtonian mechanics and Krichhoff's circuit laws. The free vibration behavior is investigated for devices with identical (tuned) and nonidentical (mistuned) piezoelectric support structures and electrical loads. These devices complex-valued have speed-dependent eigenvalues and eigenvectors as a result of their constant rotation. The imaginary parts of the eigenvalues physically represent the oscillation frequencies of the device. The real parts represent the decay or growth rates of the oscillations, depending on their sign. The complex-valued components of the eigenvectors physically represent the amplitudes and phases of the vibration. The eigenvalue behaviors differ for tuned and mistuned devices. Due to gyroscopic effects, the proof mass in the tuned device only vibrates in either forward or backward decaying circular orbits in single-mode free response. This is proven analytically for all tuned devices. For mistuned devices, the proof mass has decaying elliptical forward and backward orbits. The eigenvalues are shown to be sensitive to changes in the electric load resistances. Closed-form solutions for the eigenvalues are derived for open and close circuits. At high rotation speeds these devices experience critical speeds and instability. Closed-form solutions for the critical speeds are derived. Tuned devices have one degenerate critical speed that separates stable speeds from unstable speeds, where flutter instability occurs. Mistuned devices have two critical speeds. The first critical speed separates stable speeds from speeds where divergence instability occurs. Divergence instability continues for small speeds above the second critical speed. At higher supercritical speeds flutter instability occurs. Transitions between stable and unstable eigenvalues are investigated using root locus diagram. This device has atypical eigenvalue behavior near the critical speeds and stability transitions compared to conventional gyroscopic systems.

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).

El problema de valores propios (tambien llamado de autovalores, o eigenvalues) esta presente en diversas tareas cienficas a traves de la resolucion de ecuaciones diferenciales, analisis de modelos y calculos de funciones matriciales, entre otras muchas aplicaciones. Si los problemas son de dimension moderada (menor a 106), pueden ser abordados mediante los llamados metodos directos, como el algoritmo iterativo QR o el metodo de divide y vencerlas. Sin embargo, si el problema es de gran dimension y solo se requieren unas pocas soluciones (comparado con el tama~no del problema) y con un cierto grado de aproximacion, los metodos iterativos pueden resultar mas eficientes. Ademas los metodos iterativos pueden ofrecer mejores prestaciones en arquitecturas de altas prestaciones, como las de memoria distribuida, en las que existen un cierto numero de nodos computacionales con espacio de memoria propios y solo pueden compartir informacion y sincronizarse mediante el paso de mensajes. Esta tesis aborda la implementacion de metodos de tipo Davidson, destacando Generalized Davidson y Jacobi-Davidson, una clase de metodos iterativos que puede ser competitiva en casos especialmente dificiles como calcular valores propios en el interior del espectro o cuando la factorizacion de matrices es prohibitiva o ineficiente, y solo es posible una factorizacion aproximada. La implementacion se desarrolla en SLEPc (Scalable Library for Eigenvalue Problem Computations), libreria libre destacada en la resolucion de problemas de gran tama~no de valores propios, problemas cuadraticos de valores propios y problemas de valores singulares, entre otros. A su vez, SLEPc se desarrolla bajo el marco de PETSc (Portable, Extensible Toolkit for Scientic Computation), que ofrece implementaciones eficientes de operaciones basicas del algebra lineal, como operaciones con matrices y vectores, resolucion aproximada de sistemas lineales, factorizaciones exactas y aproximadas de matrices, etc. / Romero Alcalde, E. (2012). Parallel implementation of Davidson-type methods for large-scale eigenvalue problems [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/15188 / Palancia

The problem of determining the eigenvalues of a vibrational system having multiple lumped attachments has been investigated extensively. However, most of the research conducted in this field focuses on determining the natural frequencies of the combined system assuming that the characteristics of the combined vibrational system are known (forward problem). A problem of great interest from the point of view of engineering design is the ability to impose certain frequencies on the vibrational system or to avoid certain frequencies by modifying the characteristics of the vibrational system (inverse problem). In this thesis, the effects of adding lumped masses to an Euler-Bernoulli beam on its frequencies and their corresponding mode shapes are investigated for simply-supported as well as fixed-free boundary conditions. This investigation paves the way for proposing a method to impose two frequencies on a system consisting of a beam and a lumped mass by determining the magnitude of the mass as well as its position along the beam.