In this thesis, we analyze an adaptive discontinuous finite element method for symmetric second order linear elliptic operators. Moreover, we obtain a fully computable convergence analysis on the broken energy seminorm in first order symmetric interior penalty discontin- uous Galerkin finite element approximations of this problem. The method is formulated on nonconforming meshes made of triangular elements with first order polynomial in two di- mension. We use an estimator which is completely free of unknown constants and provide a guaranteed numerical bound on the broken energy norm of the error. This estimator is also shown to provide a lower bound for the broken energy seminorm of the error up to a constant and higher order data oscillation terms. Consequently, the estimator yields fully reliable, quantitative error control along with efficiency. As a second problem, explicit expression for constants of the inverse inequality are given in 1D, 2D and 3D. Increasing mathematical analysis of finite element methods is motivating the inclusion of mesh dependent terms in new classes of methods for a variety of applications. Several inequalities of functional analysis are often employed in convergence proofs. Inverse estimates have been used extensively in the analysis of finite element methods. It is char- acterized as tools for the error analysis and practical design of finite element methods with terms that depend on the mesh parameter. Sharp estimates of the constants of this inequality is provided in this thesis.

Συνδέουμε τα Συνεχή Κλάσματα με τα Ορθογώνια Πολυώνυμα. Ξεκινώντας από τον Stieltjes και το ομώνυμο "Πρόβλημα Ροπών", φτάνουμε μέχρι τις μέρες μας μελετώντας αυτή τη σχέση με μεθόδους Συναρτησιακής Ανάλυσης. / We study the connection between Continued Fractions and Orthogonal Polynomials. We start from Stieltjes and his "Moment Problem". Then we present Chain sequences, methods of Functional Analysis and the Birth-Death processes.

In this thesis, the main perturbation problems encountered in quantum mechanics have been studied.Since the special functions and orthogonal polynomials appear very extensively in such problems, we emphasize on those topics as well. In this context, the classical quantum mechanical anharmonic oscillators described mathematically by the one-dimensional Schr&uml / odinger equation have been treated perturbatively in both finite and infinite intervals, corresponding to confined and non-confined systems, respectively.

The special functions and, particularly, the classical orthogonal polynomials encountered in many branches of applied mathematics and mathematical physics satisfy a second order differential equation, which is known as the equation of the hypergeometric type. The variable coefficients in this equation of the hypergeometric type are of special structures. Depending on the coefficients the classical orthogonal polynomials associated with the names Jacobi, Laguerre and Hermite can be derived as solutions of this equation. In this thesis, these well known classical polynomials as well as another class of polynomials, which receive less attention in the literature called Bessel polynomials have been studied.

The objective of this research work is to develop an efficient methodology for chip-package cosimulation. In the traditional design flow, the integrated circuit (IC) is first designed followed by the package design. The disadvantage of the conventional sequential design flow is that if there are problems with signal and power integrity after the integration of the IC and the package, it is expensive and time consuming to go back and change the IC layout for a different input/output (IO) pad assignment. To overcome this limitation, a concurrent design flow, where both the IC and the package are designed together, has been recommended by researchers to obtain a fast design closure. The techniques from this research work will enable multiscale cosimulation of the chip and the package making the concurrent design flow paradigm possible. Traditional time-domain techniques, such as the finite-difference time-domain method, are limited by the Courant condition and are not suitable for chip-package cosimulation. The Courant condition gives an upper bound on the time step that can be used to obtain stable simulation results. The smaller the mesh dimension the smaller is the Courant time step. In the case of chip-package cosimulation the on-chip structures require a fine mesh, which can make the time step prohibitively small. An unconditionally stable scheme using Laguerre polynomials has been recommended for chip-package cosimulation. Prior limitations in this method have been overcome in this research work. The enhanced transient simulation scheme using Laguerre polynomials has been named SLeEC, which stands for simulation using Laguerre equivalent circuit. A full-wave EM simulator has been developed using the SLeEC methodology. A scheme for efficient use of full-wave solver for chip-package cosimulation has been proposed. Simulation of the entire chip-package structure using a full-wave solver could be a memory and time-intensive operation. A more efficient way is to separate the chip-package structure into the chip, the package signal-delivery network, and the package power-delivery network; use a full-wave solver to simulate each of these smaller subblocks and integrate them together in the following step, before a final simulation is done on the integrated network. Examples have been presented that illustrate the technique.

Made available in DSpace on 2014-06-11T19:26:56Z (GMT). No. of bitstreams: 0 Previous issue date: 2005-07-21Bitstream added on 2014-06-13T19:55:24Z : No. of bitstreams: 1 milani_ff_me_sjrp.pdf: 539043 bytes, checksum: b4613024414cd9fa758d64376a046176 (MD5) / Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) / O objetivo deste trabalho é estudar os polinômios de Szegõ, que são ortogonais no círculo unitário, e suas relações com certas frações contínuas de Perron-Carathéodory e quadratura no círculo unitário, afim de resolver o problema de momento trigonométrico. Além disso, estudar a utilização dos polinômios de Szegõ na determinação das freqüências de um sinal trigonométrico em tempo discreto xN(m). Para isso, investigamos os polinômios de Szegõ gerados por uma medida N definida através do sinal trigonométrico xN(m), para m = 0, 1, 2,...N -1, e o comportamento dos zeros desses polinômios quando N_8. / The purpose here is to study the orthogonal polynomials on the unit circle, known as Szegõ polynomials, and the relations to Perron- Carathéodory continued fractions, and quadratures on the unit circle in order to solve the trigonometric moment problem. Another purpose is to study how the Szegõ polynomials can be used to determine the frequencies from a discrete time trigonometric signal xN(m). We investigate the Szegõ polynomials associated with a measure N defined by the trigonometric sinal xN(m), m = 0, 1, 2, ...N -1. We study the behaviour of zeros of these polynomials when N 8.

Orientador: Eliana Xavier Linhares de Andrade / Banca: Vanessa Avansini Botta Pirani / Banca: Cleonice Fátima Bracciali / Resumo: O principal objetivo deste trabalho é estudar métodos para construir os numeradores e denominadores parciais da fração contínua que corresponde a duas expansões em série de potências de uma função analítica f(z); em z =0 e em z = 00. / Abstract: The main purpose of this work is to two series expansions of an analytic function f(z); in z =0 and z =00 simultaneously. Furthermore we considered the case when there are zero coefficients in the series and also whwn there is symmetry in the coefficients of the two series. Some examples are given. / Mestre

An isospectral algorithm is one which manipulates a matrix without changing its spectrum. In this thesis we study three interrelated examples of isospectral algorithms, all pertaining to Toeplitz matrices in some fashion, and one directly involving orthogonal polynomials. The first set of algorithms we study come from discretising a continuous isospectral flow designed to converge to a symmetric Toeplitz matrix with prescribed eigenvalues. We analyse constrained, isospectral gradient flow approaches and an isospectral flow studied by Chu in 1993. The second set of algorithms compute the spectral measure of a Jacobi operator, which is the weight function for the associated orthogonal polynomials and can include a singular part. The connection coefficients matrix, which converts between different bases of orthogonal polynomials, is shown to be a useful new tool in the spectral theory of Jacobi operators. When the Jacobi operator is a finite rank perturbation of Toeplitz, here called pert-Toeplitz, the connection coefficients matrix produces an explicit, computable formula for the spectral measure. Generalisation to trace class perturbations is also considered. The third algorithm is the infinite dimensional QL algorithm. In contrast to the finite dimensional case in which the QL and QR algorithms are equivalent, we find that the QL factorisations do not always exist, but that it is possible, at least in the case of pert-Toeplitz Jacobi operators, to implement shifts to generate rapid convergence of the top left entry to an eigenvalue. A fascinating novelty here is that the infinite dimensional matrices are computed in their entirety and stored in tailor made data structures. Lastly, the connection coefficients matrix and the orthogonal transformations computed in the QL iterations can be combined to transform these pert-Toeplitz Jacobi operators isospectrally to a canonical form. This allows us to implement a functional calculus for pert-Toeplitz Jacobi operators.

<div> <br></div><div> We consider the behavior of zeros of random polynomials of the from</div><div> \begin{equation*}</div><div> P_{n,m}(z) := \eta_0\varphi_m^{(m)}(z) + \eta_1 \varphi_{m+1}^{(m)}(z) + \cdots + \eta_n \varphi_{n+m}^{(m)}(z)</div><div> \end{equation*}</div><div> as \( n\to\infty \), where \( m \) is a non-negative integer (most of the work deal with the case \( m =0 \) ), \( \{\eta_n\}_{n=0}^\infty \) is a sequence of i.i.d. Gaussian random variables, and \( \{\varphi_n(z)\}_{n=0}^\infty \) is a sequence of orthonormal polynomials on the unit circle \( \mathbb T \) for some Borel measure \( \mu \) on \( \mathbb T \) with infinitely many points in its support. Most of the work is done by manipulating the density function for the expected number of zeros of a random polynomial, which we call the intensity function.</div>

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