Tese de mestrado integrado. Engenharia Mecânica. Faculdade de Engenharia. Universidade do Porto. 2008

Dissertação de Mestrado, na especialidade de Optimização e Controle, na Faculdade de Ciências da Universidade do Porto

Dissertação apresentada para obtenção do grau de Mestre em Estruturas de Engenharia Civil, na Faculdade de Engenharia da Universidade do Porto, sob a orientação do Prof. Doutor A. Silva Cardoso

Dissertação apresentada para obtenção do grau de Doutor em Ciências, na especialidade de Investigação Operacional, na Universidade de Coimbra

This thesis expands on the notion of linear complexity for a graph as defined by Michael Orrison and David Neel in their paper "The Linear Complexity of a Graph." It considers additional classes of graphs and provides upper bounds for additional types of graphs and graph operations.

The main purpose of this thesis is to obtain surprising identities by counting the representations of quivers over finite fields. A classical result states that the dimension vectors of the absolutely indecomposable representations of a quiver ?? are in one-to-one correspondence with the positive roots of a root system ??, which is infinite in general. For a given dimension vector ?? ??? ??+, the number A??(??, q), which counts the isomorphism classes of the absolutely indecomposable representations of ?? of dimension ?? over the finite field Fq, turns out to be a polynomial in q with integer coefficients, which have been conjectured to be nonnegative by Kac. The main result of this thesis is a multi-variable formal identity which expresses an infinite series as a formal product indexed by ??+ which has the coefficients of various polynomials A??(??, q) as exponents. This identity turns out to be a qanalogue of the remarkable Weyl-Macdonald-Kac denominator identity modulus a conjecture of Kac, which asserts that the multiplicity of ?? is equal to the constant term of A??(??, q). An equivalent form of this conjecture is established and a partial solution is obtained. A new proof of the integrality of A??(??, q) is given. Three Maple programs have been included which enable one to calculate the polynomials A??(??, q) for quivers with at most three nodes. All sample out-prints are consistence with Kac???s conjectures. Another result of this thesis is as follows. Let A be a finite dimensional algebra over a perfect field K, M be a finitely generated indecomposable module over A ???K ??K. Then there exists a unique indecomposable module M??? over A such that M is a direct summand of M??? ???K ??K, and there exists a positive integer s such that Ms = M ??? ?? ?? ?? ??? M (s copies) has a unique minimal field of definition which is isomorphic to the centre of End ??(M???) rad (End ??(M???)). If K is a finite field, then s can be taken to be 1.

Speech coding is a very important area of research in digital signal processing. It is a fundamental element of digital communications and has progressed at a fast pace in parallel to the increase of demands in telecommunication services and capabilities. Most of the speech coders reported in the literature are based on linear prediction (LP) analysis. Code Excited Linear Predictive (CELP) coder is a typical and popular example of this class of coders. This coder performs LP analysis of speech for extracting LP coefficients and employs an analysis-by-synthesis procedure to search a stochastic codebook to compute the excitation signal. The method used for performing LP analysis plays an important role in the design of a CELP coder. The autocorrelation method is conventionally used for LP analysis. Though this works reasonably well for noise-free (clean) speech, its performance goes down when signal is corrupted by noise. Spectral analysis of speech signals in noisy environments is an aspect of speech coding that deserves more attention. This dissertation studies the application of recently proposed robust LP analysis methods for estimating the power spectrum envelope of speech signals. These methods are the moving average, moving maximum and average threshold methods. The proposed methods will be compared to the more commonly used methods of LP analysis, such as the conventional autocorrelation method and the Spectral Envelope Estimation Vocoder (SEEVOC) method. The Linear Predictive Coding (LPC) spectrum calculated from these proposed methods are shown to be more robust. These methods work as well as the conventional methods when the speech signal is clean or has high signal-to-noise ratio. Also, these robust methods give less quantisation distortion than the conventional methods. The application of these robust methods for speech compression using the CELP coder provides better speech quality when compared to the conventional LP analysis methods.

Writers on the business cycle often emphasize that non-linear models are needed to account for certain of its features. Thus it is often said that either the asymmetry of the duration of business cycle expansions and contractions or the variability of these quantities demand a non-linear model. Such comments are rarely made precise however and mostly consist of references to such assertions from the past. Thus the asymmetry in the cycle is mostly accompanied by references to Keynes (1936) and Burns and Mitchell (1946). But these authors were looking at what we call today the classical cycle i.e. movements in the level of GDP, and so the fact that there are long expansions and short contractions can arise simply due to the presence of long-run growth in the economy, and it is not obvious that it has much to do with non-linearity. This thesis aims to introduce various statistics that can be used to characterise the specific shape of the non-linearity observed in macroeconomic time series. Chapter 2 introduces a range of statistics and presents the dating algorithm used in this thesis, which is based on the BBQ algorithm of Harding and Pagan (2002). Chapter 3 tests the adequacy of linear models versus the SETAR model of van Dijk and Franses(2003) and the bounceback model of Kim, Morley and Piger (2005) in capturing observed non-linear features of the data. Chapter 4 extends this work by examining the three state Markov model of Hamilton (1989), again using the ??bounce-back?? model of Kim C., Morley, J. and J. Piger, (2005), and the more complicated ??tension?? model of DeJong, D., Dharmarajan, H., Liesenfeld, R. and Richard, J., (2005). Chapter 4 also extends Chapter 3 by estimating the above mentioned models on US GDP, Australian non-farm GDP, US investment and Australian dwellings investment. They are then simulated in order to gauge the cycle properties. Chapter 5 analyses the business cycle implications of two related multivariate dynamic factor models presented in papers by Kim and Piger (2001, 2002). Finally Chapter 6 concludes.

This thesis proposes an approximate maximum likelihood estimator and likelihood ratio test for parameters in a generalized linear model when two or more random effects are present. Substantial progress in parameter estimation for such models has been made with methods involving generalized least squares based on the approximate marginal mean and covariance matrix. However, tests and confidence intervals based on this approach have been limited to what is provided through asymptotic normality of estimates. The proposed solution is based on maximizing a Laplace approximation to the log-likelihood function. This approximation is remarkably accurate and has previously been demonstrated to work well for obtaining likelihood based estimates and inferences in generalized linear models with a single random effect. This thesis concentrates on extensions to the case of several random effects and the comparison of the likelihood ratio inference from this approximate likelihood analysis to the Wald-like inferences for existing estimators. The shapes of the Laplace approximate and true log-likelihood functions are practically identical, implying that maximum likelihood estimates and likelihood ratio inferences are obtained from the Laplace approximation to the log-likelihood. Use of the Laplace approximation circumvents the need for numerical integration, which can be practically impossible to compute when there are two random effects. However, both the Laplace and exact (via numerical integration) methods require numerical optimization, a sometimes slow process, for obtaining estimates and inferences. The proposed Laplace method for estimation and inference is demonstrated for three real (and some simulated) data sets, along with results from alternative methods which involve use of marginal means and covariances. The Laplace approximate method and another denoted as Restricted Maximum Likelihood (REML) performed rather similarly for estimation and hypothesis testing. The REML approach produced faster analyses and was much easier to implement while the Laplace implementation provided likelihood ratio based inferences rather than those relying on asymptotic normality. / Graduation date: 1995

Graduation date: 1994