This thesis investigates the statistical properties of preliminary test estimators of linear models with normally distributed errors. Specifically, we derive exact expressions for the mean, variance and quadratic risk (i.e. the Mean Square Error) of estimators whose form are determined by the outcome of a statistical test. In the process, some new results on the moments of truncated linear or quadratic forms in normal vectors are established. In the first paper (Paper I), we consider the estimation of the vector of regression coefficients under a model selection procedure where it is assumed that the analyst chooses between two nested linear models by some of the standard model selection criteria. This is shown to be equivalent to estimation under a preliminary test of some linear restrictions on the vector of regression coefficients. The main contribution of Paper I compared to earlier research is the generality of the form of the test statistic; we only assume it to be a quadratic form in the (translated) observation vector. Paper II paper deals with the estimation of the regression coefficients under a preliminary test for homoscedasticity of the error variances. In Paper III, we investigate the statistical properties of estimators, truncated at zero, of variance components in linear models with random effects. Paper IV establishes some new results on the moments of truncated linear and/or quadratic forms in normally distributed vectors. These results are used in Papers I-III. In Paper V we study some algebraic properties of matrices that occur in the comparison of two nested models. Specifically we derive an expression for the inertia (the number of positive, negative and zero eigenvalues) of this type of matrices.

This thesis consists of two papers devoted to the asymptotics of random matrix ensembles and measure valued stochastic processes which can be considered as generalizations of the Gaussian unitary ensemble (GUE) of Hermitian matrices H=A+A†, where the entries of A are independent identically distributed (iid) centered complex Gaussian random variables. In the first paper, a system of interacting diffusing particles on the real line is studied; special cases include the eigenvalue dynamics of matrix-valued Ornstein-Uhlenbeck processes (Dyson's Brownian motion). It is known that the empirical measure process converges weakly to a deterministic measure-valued function and that the appropriately rescaled fluctuations around this limit converge weakly to a Gaussian distribution-valued process. For a large class of analytic test functions, explicit formulae are derived for the mean and covariance functionals of this fluctuation process. The second paper concerns a family of random matrix ensembles interpolating between the GUE and the Ginibre ensemble of n x n matrices with iid centered complex Gaussian entries. The asymptotic spectral distribution in these models is uniform in an ellipse in the complex plane, which collapses to an interval of the real line as the degree of non-Hermiticity diminishes. Scaling limit theorems are proven for the eigenvalue point process at the rightmost edge of the spectrum, and it is shown that a non-trivial transition occurs between Poisson and Airy point process statistics when the ratio of the axes of the supporting ellipse is of order n -1/3. / Denna avhandling består av två vetenskapliga artiklar som handlar om gränsvärdessatser för slumpmatriser och måttvärda stokastiska processer. De modeller som studeras kan betraktas som generaliseringar av den gaussiska unitära ensembeln (GUE) av hermiteska n x n-matriser H=A+A†, där A är en matris vars element är oberoende, likafördelade, centrerade, komplexa normalfördelade stokastiska variabler. I artikel I betraktas ett system av växelverkande diffunderande partiklar på reella linjen, vissa specialfall av denna modell kan tolkas som egenvärdesdynamiken för matrisvärda Ornstein-Uhlenbeck-processer (Dysons brownska rörelse). Sedan tidigare är det känt att den empiriska måttprocessen konvergerar svagt mot en deterministisk måttvärd funktion och att fluktuationerna runt denna gräns, i lämplig skalning, konvergerer svagt mot en distributionsvärd gaussisk process. För en stor klass av analytiska testfunktioner härleds explicita formler för medelvärdes- och kovariansfunktionalerna för denna fluktuationsprocess. Artikel II behandlar en familj av slumpmatrisensembler som interpolerar mellan GUE och Ginibre-ensembeln, bestående av matriser A som ovan. För denna modell är egenvärdena komplexa och asymptotiskt likformigt fördelade i en ellips i komplexa planet. Skalningsgränsvärdessatser för egenvärdet med maximal realdel och för egenvärdespunktprocessen kring detta visas för ett allmänt val av interpolationsparametern i modellen. Då förhållandet mellan axlarna i den asymptotiska ellipsen är av storleksordning n-1/3 uppträder en övergångsfas mellan Airypunktprocess- och Poissonprocessbeteendena, typiska för GUE respektive Ginibre-ensembeln. / QC 20100705

Acetylacetone (AcAc) exists as a mixture of enol and keto tautomers. Besides providing a good example for the study of tautomerization, it is a model system for investigating intramolecular hydrogen bonds in its enol form. Trapping AcAc in the soft para-Hydrogen (pH2 ) environment brings out new opportunities to investigate its properties. Infrared spectra of the samples give a good characterization of the two stable enol and keto tautomers. The keto/enol ratio in solid pH2 is found to be higher than in other matrices. While vibrational bands of keto are narrow, those of enol are broad, reflecting the intrinsic properties of the enol which exhibits three entangled large amplitude motions (two methyl torsions and the intramolecular hydrogen transfer). Surprisingly, narrowing of some of these bands is observed in a slow time evolution. This effect is interpreted as a consequence of nuclear spin conversion in the hydrogen atoms of the methyl groups, giving access to AcAc species differing by their nuclear spin symmetry. This offers new pertinent investigations on the large amplitude motions, especially on the intramolecular hydrogen transfer. AcAc/pH2 samples have been irradiated by UV laser beams. Irradiation at 266 nm induces isomerization from the stable chelated enol form to non chelated conformers, similarly to the case of other matrices. A clear IR signature of the conformers is obtained thanks to the pH2 host. Irradiation at 248 nm induces the enol/keto tautomerization. The kinetics of this interconversion highlights a non-direct process. Fragmentation is clearly observed under irradiation at 193 nm, followed by chemical reaction with the hydrogen host.

University of Macau / Faculty of Science and Technology / Department of Mathematics

University of Macau / Faculty of Science and Technology / Department of Mathematics

In recent centuries the Swedish Tiled Stove has varied in its design with an aesthetic evolutionary history that has not only followed the architectural and artistic ideals but also the taste in stove design. The heat source has varied depending on the regional traditions and where it was located within the country, but the inside of the construction was of a similar five channel system. The purpose of this paper is to study and compare the design of tile matrices made by Emil Petterson with other Swedish stoves.  The following questions will be addressed: What characterized the production of a small scale tile stove workroom during the years 1900-50? Which main styles can be described as Emil Pettersson’s ideal while producing his matrices? And finally, is it possible to study the tile stove matrices as objects of historical and cultural significance, and what difficult methodological issues might occur in the doing of it?  To answer these questions I will present the results of the documentation of Pettersson’s tile matrices with detailed historical categorizations and schedules. The conclusion shows that Pettersson as a stove builder and -producer was characterized by eclectic ideals and a design shape based on more liberal interpretations of contemporary trends. The studying of tile stove matrices showed that, despite its difficulties regarding the inverted shapes, it is a barer of great historical and cultural value and that it plays a key part in the historical context of tile stove ovens.

Nous avons mis en place un dispositif permettant l'acquisition d'échos de photons stimulés infrarouges à l'échelle femtoseconde. Le but est d'examiner la dynamique vibrationnelle aux temps courts de métaux carbonyles (W(CO)₆ and Fe(CO)₅) piégés en matrice cryogénique (4-50 K). Cet environnement solide, issu de la condensation d'un mélange gazeux contenant une impureté et un gaz inerte (N₂, CH₄, Ar, etc.), est propice à l'étude de systèmes dans leur état fondamental. L'excitation d'une vibration moléculaire s'atténue toujours temporellement, ce qui correspond dans le domaine spectral à un élargissement des raies d'absorption. L'étude de la dynamique vibrationnelle vise à examiner les causes physiques à l'origine de cet élargissement spectral. Typiquement, elles sont de trois sortes : phénomènes intramoléculaires, interactions entre molécules piégées et interactions entre la molécule piégée et l'environnement. Les échos de photons permettent de distinguer les contributions homogènes et inhomogènes de l'élargissement spectral et de caractériser les processus de déphasage, de relaxation des populations et de diffusion spectrale. Parmi les résultats obtenus, nous avons mis en évidence l'influence des phonons spécifiques aux matrices moléculaires (ex : libration de N₂ et rotation de CH₄ ) sur le déphasage vibrationnel ainsi que l'influence de la transition de phase du méthane solide à 20 K sur la dynamique vibrationnelle. Nous avons également montré que la dynamique vibrationnelle était dépendante du site cristallographique dans lequel est piégée la molécule. Enfin, en excitant plusieurs modes de vibration simultanément, nous avons pu examiner les couplages intramoléculaires.

The main result of this thesis is the development of a theory of semidefinite facial reduction for the Euclidean distance matrix completion problem. Our key result shows a close connection between cliques in the graph of the partial Euclidean distance matrix and faces of the semidefinite cone containing the feasible set of the semidefinite relaxation. We show how using semidefinite facial reduction allows us to dramatically reduce the number of variables and constraints required to represent the semidefinite feasible set. We have used this theory to develop a highly efficient algorithm capable of solving many very large Euclidean distance matrix completion problems exactly, without the need for a semidefinite optimization solver. For problems with a low level of noise, our SNLSDPclique algorithm outperforms existing algorithms in terms of both CPU time and accuracy. Using only a laptop, problems of size up to 40,000 nodes can be solved in under a minute and problems with 100,000 nodes require only a few minutes to solve.

This thesis develops ways of localizing the Ritz values of non-Hermitian matrices. The restarted Arnoldi method with exact shifts, useful for determining a few desired eigenvalues of a matrix, employs Ritz values to refine eigenvalue estimates. In the Hermitian case, using selected Ritz values produces convergence due to interlacing. No generalization of interlacing exists for non-Hermitian matrices, and as a consequence no satisfactory general convergence theory exists. To study Ritz values, I propose the inverse field of values problem for k Ritz values, which asks if a set of k complex numbers can be Ritz values of a matrix. This problem is always solvable for k = 1 for any complex number in the field of values; I provide an improved algorithm for finding a Ritz vector in this case. I show that majorization can be used to characterize, as well as localize, Ritz values. To illustrate the difficulties of characterizing Ritz values, this work provides a complete analysis of the Ritz values of two 3 × 3 matrices: a Jordan block and a normal matrix. By constructing conditions for localizing the Ritz values of a matrix with one simple, normal, sought-after eigenvalue, this work develops sufficient conditions that guarantee convergence of the restarted Arnoldi method with exact shifts. For general matrices, the conditions provide insight into the subspace dimensions that ensure that shifts do not cluster near the wanted eigenvalue. As Ritz values form the basis for many iterative methods for determining eigenvalues and solving linear systems, an understanding of Ritz value behavior for non-Hermitian matrices has the potential to inform a broad range of analysis.

The main result of this thesis is the development of a theory of semidefinite facial reduction for the Euclidean distance matrix completion problem. Our key result shows a close connection between cliques in the graph of the partial Euclidean distance matrix and faces of the semidefinite cone containing the feasible set of the semidefinite relaxation. We show how using semidefinite facial reduction allows us to dramatically reduce the number of variables and constraints required to represent the semidefinite feasible set. We have used this theory to develop a highly efficient algorithm capable of solving many very large Euclidean distance matrix completion problems exactly, without the need for a semidefinite optimization solver. For problems with a low level of noise, our SNLSDPclique algorithm outperforms existing algorithms in terms of both CPU time and accuracy. Using only a laptop, problems of size up to 40,000 nodes can be solved in under a minute and problems with 100,000 nodes require only a few minutes to solve.