Ai Jun. / Thesis (Ph.D.)--Chinese University of Hong Kong, 1999. / Includes bibliographical references (p. 100-103). / 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.

In this thesis, we study the inverse nodal problem and inverse spectral problem for various Sturm-Liouville operators, in particular, Hill's operators. We first show that the space of Schr"odinger operators under separated boundary conditions characterized by ${H=(q,al, e)in L^{1}(0,1) imes [0,pi)^{2} : int_{0}^{1}q=0}$ is homeomorphic to the partition set of the space of all admissible sequences $X={X_{k}^{(n)}}$ which form sequences that converge to $q, al$ and $ e$ individually. The definition of $Gamma$, the space of quasinodal sequences, relies on the $L^{1}$ convergence of the reconstruction formula for $q$ by the exactly nodal sequence. Then we study the inverse nodal problem for Hill's equation, and solve the uniqueness, reconstruction and stability problem. We do this by making a translation of Hill's equation and turning it into a Dirichlet Schr"odinger problem. Then the estimates of corresponding nodal length and eigenvalues can be deduced. Furthermore, the reconstruction formula of the potential function and the uniqueness can be shown. We also show the quotient space $Lambda/sim$ is homeomorphic to the space $Omega={qin L^{1}(0,1) : int_{0}^{1}q = 0, q(x)=q(x+1) mbox{on} mathbb{R}}$. Here the space $Lambda$ is a collection of all admissible sequences $X={X_{k}^{(n)}}$ which form sequences that converge to $q$. Finally we show that if the periodic potential function $q$ of Hill's equation is single-well on $[0,1]$, then $q$ is constant if and only if the first instability interval is absent. The same is also valid for convex potentials. Then we show that similar statements are valid for single-barrier and concave density functions for periodic string equation. Our result extends that of M. J. Huang and supplements the works of Borg and Hochstadt.

This research was done to develop tools to predict parametric roll motion of ships in regular and irregular sea and provide guidelines to avoid parametric roll during initial design stage. A post Panamax hull form (modified C11 Hull form, Courtesy of MARIN) was used to study parametric roll in ships. The approach of the study has been to simplify the roll equation of motion to a single degree of freedom equation so as to utilize the tools available to analyze the system retaining the non-linear character of the system. The Hill’ equation is used to develop highly accurate stability boundaries in the Ince-Strutt Diagram. The effect of non-linear damping has also been incorporated into the chart for the first time providing a simple method to predict the bounded roll motion amplitude. Floquet theory is also extended to predict parametric roll motion amplitude. Forward speed of the vessel has been treated as a bifurcation parameter and its effects studied both in head and following sea condition. In the second half of the research, parametric roll of the vessel in irregular sea is investigated using the Volterra Quadratic model. GM variation in irregular sea was obtained using transfer functions of the Volterra model. Heave and pitch coupling to roll motion was also studied using this approach. Sensitivity studies of spectral peak period and significant wave height on roll motion amplitude were also carried out. Forward speed effects were also evaluated using the Volterra approach. Based on the study, the Hill’s equation approach was found to give more accurate prediction of parametric roll in regular sea. The boundaries in the stability chart were more accurately defined by the Hill’s equation. The inclusion of non-linear damping in the stability chart gave reasonably accurate bounded motion amplitude prediction. The Volterra approach was found to be a good analytical prediction tool for parametric roll motion in irregular sea. Using the Volterra model, it was found that there is a high probability of parametric roll when the spectral modal period is close to twice the natural period of roll.

The Hill's equation is the Schrodinger equation $$-y'+qy=la y$$ with a periodic one-dimensional potential function $q$ and coupled with periodic boundary conditions $y(0)=y(1)$, $y'(0)=y'(1)$ or anti-periodic boundary conditions $y(0)=-y(1)$, $y'(0)=-y'(1)$. We study the inverse nodal problem for Hill's equation, in particular the reconstruction problem. Namely, we want to reconstruct the potential function using only nodal data ( zeros of eigenfunctions ). In this thesis, we give a reconstruction formula for $q$ using the periodic nodal data or using anti-periodic nodal data We show that the convergence is pointwise for all $x in (0,1)$ where $q$ is continuous; and pointwise for $a.e.$ $x in (0,1)$ as well as $L^1$ convergence when $qin L^1(0,1)$. We do this by making a translation so that the problem becomes a Dirichlet problem. The idea comes from the work of Coskun and Harris.

SANTOS, Fabiana Alves dos. Espectro de variedades completas e não-compactas. 2017. 39 f. Tese (Doutorado em Matemática) – Centro de Ciências, Universidade Federal do Ceará, Fortaleza, 2017. / Submitted by Andrea Dantas (pgmat@mat.ufc.br) on 2017-09-13T14:03:43Z No. of bitstreams: 1 2017_tese_fasantos.pdf: 609112 bytes, checksum: 0bbcd05e8e335e0ecb00510e212c4e79 (MD5) / Rejected by Rocilda Sales (rocilda@ufc.br), reason: Boa tarde, Estou devolvendo a Tese de FABIANA ALVES DOS SANTOS para que ela corrija alguns itens do trabalho: 1- FICHA CATALOGRÁFICA (refaça a ficha catalográfica colocando seu nome completo) 2- FOLHA DE APROVAÇÃO (substitua a folha de aprovação por uma cópia que não contenha as assinaturas dos membros da banca examinadora, pois, por questões de segurança, não estamos mais publicando os trabalhos com as assinaturas dos membros da banca) 3- ITEM ALEATÓRIO (na página 5, há uma frase aleatória - EBENEZER! - que não se enquadra em nenhum dos itens opcionais de uma Tese. Caso seja uma EPÍGRAFE deve aparecer entre aspas duplas, após à página dos agradecimentos, e com a citação do autor ou fonte de onde foi retirada) 4- TÍTULO DO CAP. 3 (coloque o título do capítulo 3, que aparece no SUMÁRIO e no TÍTULO DO CAPITULO, em letra MAIÚSCULA, NEGRITO e FONTE n 12) Atenciosamente, on 2017-09-13T16:42:12Z (GMT) / Submitted by Andrea Dantas (pgmat@mat.ufc.br) on 2017-09-18T13:52:41Z No. of bitstreams: 1 2017_tese_fasantos.pdf: 12798451 bytes, checksum: 062ab3efa4756ce3a83ed52d9cebcd13 (MD5) / Approved for entry into archive by Rocilda Sales (rocilda@ufc.br) on 2017-09-18T15:16:06Z (GMT) No. of bitstreams: 1 2017_tese_fasantos.pdf: 12798451 bytes, checksum: 062ab3efa4756ce3a83ed52d9cebcd13 (MD5) / Made available in DSpace on 2017-09-18T15:16:07Z (GMT). No. of bitstreams: 1 2017_tese_fasantos.pdf: 12798451 bytes, checksum: 062ab3efa4756ce3a83ed52d9cebcd13 (MD5) Previous issue date: 2017-01-20 / On this work we study the espectrum of Laplace-Beltrami operator on the warped Riemannian manifold Mn = R_r Sn1, whose warping function is smooth, positive, periodic, with period a and satis_es r0 = min r(t) < p n 1a=_. We show that spectrum there no eingevalue, is formed by a union of closed intervals, and, from the peridicity of r, using the classical Hill's Equations Theory, we conclude the existence of gaps. / Neste trabalho caracterizamos o espectro do operador de Laplace-Beltrami na variedade warped Mn = R_r Sn1 cuja função warping _e suave, positiva, periódica, de período a, e satisfaz r0 = min r(t) < p n 1a=_. Mostramos que tal espectro não possui autovalores, é escrito como a união de intervalos e, da periodicidade de r, utilizamos a clássica teoria a cerca dos operados de Hill, e concluímos e existência de gaps no espectro de M.

[EN] This thesis addresses the treatment of perturbed problems with splitting methods. After motivating these problems in Chapter 1, we give a thorough introduction in Chapter 2, which includes the objectives, several basic techniques and already existing methods. In Chapter 3, we consider the numerical integration of non-autonomous separable parabolic equations using high order splitting methods with complex coefficients (methods with real coefficients of order greater than two necessarily have negative coefficients). We propose to consider a class of methods that allows us to evaluate all time dependent operators at real values of the time, leading to schemes which are stable and simple to implement. If the system can be considered as the perturbation of an exactly solvable problem and the flow of the dominant part is advanced using real coefficients, it is possible to build highly efficient methods for these problems. We show the performance of this class of methods for several numerical examples and present some new improved schemes. In Chapter 4, we propose splitting methods for the computation of the exponential of perturbed matrices which can be written as the sum A = D+epsilon*B of a sparse and efficiently exponentiable matrix D with sparse exponential exp(D) and a dense matrix epsilon*B which is of small norm in comparison with D. The predominant algorithm is based on scaling the large matrix A by a small number 2^(-s) , which is then exponentiated by efficient Padé or Taylor methods and finally squared in order to obtain an approximation for the full exponential. In this setting, the main portion of the computational cost arises from dense-matrix multiplications and we present a modified squaring which takes advantage of the smallness of the perturbation matrix B in order to reduce the number of squarings necessary. Theoretical results on local error and error propagation for splitting methods are complemented with numerical experiments and show a clear improvement over existing methods when medium precision is sought. In Chapter 5, we consider the numerical integration of the perturbed Hill's equation. Parametric resonances can appear and this property is of great interest in many different physical applications. Usually, the Hill's equations originate from a Hamiltonian function and the fundamental matrix solution is a symplectic matrix. This is a very important property to be preserved by the numerical integrators. In this chapter we present new sixth-and eighth-order symplectic exponential integrators that are tailored to the Hill's equation. The methods are based on an efficient symplectic approximation to the exponential of high dimensional coupled autonomous harmonic oscillators and yield accurate results for oscillatory problems at a low computational cost. Several numerical examples illustrate the performance of the new methods. Conclusions and pointers to further research are detailed in Chapter 6. / [ES] Esta tesis aborda el tratamiento de problemas perturbados con métodos de escisión (splitting). Tras motivar el origen de este tipo de problemas en el capítulo 1, introducimos los objetivos, varias técnicas básicas y métodos existentes en capítulo 2. En el capítulo 3 consideramos la integración numérica de ecuaciones no autónomas separables y parabólicas usando métodos de splitting de orden mayor que dos usando coeficientes complejos (métodos con coeficientes reales de orden mayor de dos necesariamente tienen coeficientes negativos). Proponemos una clase de métodos que permite evaluar todos los operadores con dependencia temporal en valores reales del tiempo lo cual genera esquemas estables y fáciles de implementar. Si el sistema se puede considerar como una perturbación de un problema resoluble de forma exacta y si el flujo de la parte dominante se avanza usando coeficientes reales, es posible construir métodos altamente eficientes para este tipo de problemas. Demostramos la eficiencia de estos métodos en varios ejemplos numéricos. En el capítulo 4 proponemos métodos de splitting para el cálculo de la exponencial de matrices perturbadas que se pueden escribir como suma A = D + epsilon*B de una matriz dispersa y eficientemente exponenciable con exponencial dispersa exp(D) y una matriz densa epsilon*B de noma pequeña. El algoritmo predominante se basa en escalar la matriz grande con un número pequeño 2^(-s) para poder exponenciar el resultado con métodos eficientes de Padé o Taylor y finalmente obtener la aproximación a la exponencial elevando al cuadrado repetidamente. En este contexto, el coste computacional proviene de las multiplicaciones de matrices densas y presentamos una cuadratura modificada aprovechando la estructura perturbada para reducir el número de productos. Resultados teóricos sobre errores locales y propagación de error para métodos de splitting son complementados con experimentos numéricos y muestran una clara mejora sobre métodos existentes a precisión media. En el capítulo 5, consideramos la integración numérica de la ecuación de Hill perturbada. Resonancias paramétricas pueden aparecer y esta propiedad es de gran interés en muchas aplicaciones físicas. Habitualmente, las ecuaciones de Hill provienen de una función hamiltoniana y la solución fundamental es una matriz simpléctica, una propiedad muy importante que preservar con los integradores numéricos. Presentamos nuevos integradores simplécticos exponenciales de orden seis y ocho tallados a la ecuación de Hills. Estos métodos se basan en una aproximación simpléctica eficiente a la exponencial de osciladores armónicos acoplados de dimensión alta y dan lugar a resultados precisos para problemas oscilatorios a un coste computacional bajo y varios ejemplos numéricos ilustran su rendimiento. Conclusiones e indicadores para futuros estudios se detallan en el capítulo 6. / [CAT] La present tesi està enfocada al tractament de problemes perturbats utilitzant, entre altres, mètodes d'escisió (splitting). Comencem motivant l'oritge d'aquest tipus de problems al capítol 1, i a continuació introduïm el objectius, diferents tècniques bàsiques i alguns mètodes existents al capítol 2. Al capítol 3, consideram la integració numèrica d'equacions no autònomes separables i parabòliques utilitzant mètodes d'splitting d'ordre major que dos utilitzant coeficients complexos (mètodes amb coeficients reials d'ordre major que dos necesariament tenen coeficients negatius). Proposem una clase de mètodes que permeten evaluar tots els operadors amb dependència temporal explícita amb valors reials del temps. Esta forma de procedir genera esquemes estables i fàcils d'implementar. Si el sistema es pot considerar com una perturbació d'un problema exactament resoluble, i la part dominant s'avança utilitzant coeficients reials, es posible construir mètodes altament eficients per aquest tipus de problemes Demostrem la eficiència d'estos mètodes per a diferents exemples numèrics. Al capítol 4, proposem mètodes d'splitting per al càcul de la exponencial de matrius pertorbades que es poden escriure com suma A = D + epsilon*B (una matriu que es pot exponenciar fàcilment i eficientemente, com es el cas d'algunes matrius disperses exp(D), i una matriu densa epsilon*B de norma menuda). L'algorisme predominant es basa en escalar la matriu gran amb un nombre menut 2^(-s) per a poder exponenciar el resultat amb mètodes eficients de Padé o Taylor i finalment obtindre la aproximació a la exponencial elevant al quadrat repetidament. En este context, el cost computacional prové de les multiplicacions de matrius denses i presentem una quadratura modificada aprofitant la estructura de matriu pertorbada per reduir el nombre de productes. Resultats teòrics sobre errors locals i propagació d'error per a mètodes d'splitting son analitzats i corroborats amb experiments numèrics, mostrant una clara millora respecte a mètodes existens quan es busca una precisió moderada. Al capítol 5, considerem la integració numèrica de l'ecuació de Hill pertorbada. En este tipus d'equacions poden apareixer resonàncies paramètriques i esta propietat es de gran interés en moltes aplicacions físiques. Habitualment, les equacions de Hill provenen d'una función hamiltoniana i la solució fonamental es una matriu simplèctica, siguent esta una propietat molt important a preservar pels integradors numèrics. Presentams nous integradors simplèctics exponencials d'orden sis i huit construits especialmente per resoldre l'ecuació de Hill. Estos mètodes es basen en una aproxmiació simplèctica eficient a la exponencial d'osciladors harmònics acoplats de dimensió alta i donen lloc a resultats precisos per a problemas oscilatoris a un cost computacional baix. La eficiencia dels mètodes s'il.lustra en diferents exemples numèrics. Conclusions i indicadors per a futurs estudis es detallen al capítol 6. / Seydaoglu, M. (2016). Splitting methods for autonomous and non-autonomous perturbed equations [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/71358 / TESIS

This thesis presents the results of studies on the problem of subharmonic oscillations in nonlinear Paul trap mass spectrometry. The objective of this thesis is to determine whether the subharmonic oscillations of ions in the trap could in any way affect the quality of mass spectrum in resonance ejection experiments. This is accomplished by studying the existence and stability criteria of these oscillations. This study is done for two casesone in which the auxiliary excitation frequency is close to thrice the ion axial secular frequency and other in which it is close to twice the ion axial secular frequency. Initially, the equations describing the ion motion in the presence of auxiliary excitation are derived. The equations describing the ion motion are then brought into a form easily amenable to analysis by techniques of perturbation theory. The necessary background and definitions to understand the basis for the thesis, along with a survey of results obtained in relevant areas in mass spectrometry and nonlinear dynamics is then developed. The first problem is the study of subharmonic oscillations when the auxiliary excitation frequency is in the vicinity of thrice the ion axial secular frequencies taken up. The application of the multiple time scales technique to the equations describing ion motion gives the slow flow equations, that describe the evolution of amplitudes of axial and radial oscillations. The expression for the steady state amplitudes of these oscillations are then derived. From these expressions the conditions for the existence of the oscillations are obtained in terms of the auxiliary amplitude and the frequency detuning. This is then followed by a detailed stability analysis for the subharmonic oscillation with a given amplitude and phase. The study ends with the discussion of the results obtained, the pertinent numerical studies and the relevance of this study to mass spectrometry. The second study is regarding the problem of subharmonic oscillations when the auxiliary excitation frequency is close to twice the ion axial secular frequency is analyzed. When PoincareLindstedt’s method is applied to the ion motion equations, the amplitude frequency relationship that describes the relation between the steady state subharmonic oscillation amplitude and the frequency detuning is obtained. The variation of the oscillation amplitude with the frequency detuning is then studied. Then follows the analysis of stability. The stability of subharmonic oscillation is analyzed using the results from the standard analysis of Hill’s equation of fourth order. This study ends with the discussion of the results obtained in the context of mass spectrometry. Finally, a summary of the results obtained is discussed.

Les lignes de téléportés monocâbles (télésièges et télécabines) sont très utilisées dans le Monde entier pour transporter des personnes dans les stations de sport d'hiver et les sites touristiques. Leur potentiel de développement est très élevé, en particulier dans des contextes urbains et péri-urbains en tant que moyens de transport en commun. Avec l'augmentation des performances au cours des dernières décennies, notamment en termes de vitesse, capacité de transport et longueur de portées plus élevées, des effets dynamiques inattendus sont apparus. En particulier, des instabilités sous forme d'oscillations verticales de certaines portées, portant le nom de « pompage » dans la profession, se sont manifestées de manière imprévisible sur quelques lignes, ainsi que des balancements importants de véhicules au franchissement de certains pylônes. Les accélérations engendrées par ces deux phénomènes sont potentiellement sources d'inconfort dynamique pour les passagers. Les solutions généralement mises en œuvre pour remédier à cette problématique, basées sur l'empirisme, se sont révélées coûteuses et n'ont fourni que des corrections a posteriori. L'ambition des travaux de thèse présentés dans ce mémoire était d'apporter une dimension prédictive afin d'anticiper les risques d'instabilités dynamiques dès le stade de la conception des lignes. Pour cela, une meilleure compréhension des mécanismes à l'origine du pompage et des forts balancements était nécessaire. Une étude combinant théorie, calculs numériques et essais a été menée tant à l'échelle d'une portée que d'une ligne de téléporté monocâble complète. Le pompage serait finalement dû à une mise en résonance paramétrique de certaines portées par une excitation provenant du franchissement des pylônes par les véhicules répartis de manière équidistante sur l'ensemble de la ligne, et propagée par le câble. Les balancements importants des véhicules viendraient d'une amplification de leur mouvement pendulaire lors du franchissement de deux pylônes successifs. Les paramètres d'influence ont été identifiés et des outils de conception ont été développés pour limiter les risques d'apparition de ces phénomènes dynamiques par des critères qualitatifs. Des modèles de simulation ont également été réalisés avec pour objectif de prédire quantitativement les accélérations ressenties par les passagers et ainsi aboutir à un dimensionnement des lignes vis-à-vis de critères de confort objectifs. Une confrontation des résultats des calculs avec les données expérimentales récoltées au cours de plusieurs campagnes d'essais dynamiques a été faite. Enfin, des idées de dispositifs techniques basés sur des principes de contrôle passif, semi-actif voire actif ont été émises dans le but de réduire l'amplitude des oscillations subies par les véhicules. Les résultats de cette thèse seront exploités par l'entreprise POMA, constructeur français parmi les leaders mondiaux du transport par câble. / Monocable aerial ropeways (chairlifts and gondola lifts) are much used nowadays in the entire world to transport people in ski resorts and touristic areas. Their potential of development is very high, particularly in urban and peri-urban environments as means of public transportation. With the increase of performances during the last decades, such as higher speed, transport capacity and length of spans, unexpected dynamic effects have appeared. In particular, instabilities taking the shape of vertical oscillations on some spans, named \sag-oscillations" in the profession, have occurred unpredictably on a few lines, but also important swings of vehicles when crossing some towers. The accelerations generated by those phenomena are potential sources of dynamic discomfort to passengers. The solutions generally employed to address that issue, based on empiricism, have revealed to be costly and have only brought a posteriori corrections. The ambition of the PhD research works presented in this thesis was to bring a predictive dimension in order to anticipate the risks of dynamic instabilities from the early stage of lines design. For this matter, a better understanding of the mechanisms giving rise to the sag-oscillations and the strong swings was necessary. A study combining theory, numerical calculations and experiments has been conducted both on a span and full monocable aerial ropeway scale. The sag-oscillation instability would finally be due to a parametric resonance of some spans, induced by an excitation coming from the crossing of towers by the equally-spaced vehicles on the whole line, and propagated by the cable. The important swings of the vehicles would come from the amplification of their swaying movement when crossing two consecutive towers. The influence parameters have been identified and some design tools have been developed to limit the risks of emergence of those dynamic phenomena by qualitative criteria. Some simulation models have also been made with the aim to predict quantitatively the accelerations experienced by the passengers and then lead to the dimensioning of lines with respect to objective comfort criteria. A confrontation of the calculation results with the experimental data collected during several dynamic test campaigns was performed. Finally, some ideas of technical devices based on the principles of passive, semi-active or even active control have been expressed in order to reduce the amplitude of the oscillations undergone by the vehicles. The results of these PhD research works will be used by the company POMA, a French manufacturer among the world leaders in cable transportation.