Since direct-sequence code-division multiple-access (DS-CDMA) is an interference-limited random multiple-access scheme, the reduction of co-channel interference with either interference suppression or interference cancellation multiuser receivers and/or power control to prevent detrimental near-far situations is vital for improved performance. Up to date, some contributions investigated randomly-spread asymptotically - large number of users and large bandwidth - large CDMA systems with multiuser receivers and power control via random matrix theoretic and free probability theoretic tools especially over Gaussian and single-path fading channels. As complement within this thesis, we analyze also within the generalized random spreading framework but at finite system sizes and without power control the capacity achievable with linear multichannel multiuser receivers / i.e. RAKE, zero-forcing decorrelator, linear minimum mean-squared error (LMMSE) multiuser receivers, within a single-cell setting over generalized time-varying GWSSUS - Rayleigh/Ricean - fading channels via random matrix theoretic tools. Assuming maximal-ratio combining (MRC) of resolved frequency - multipath - diversity channels due to wideband transmission, the signal-to-interference ratios (SIRs) with multichannel multiuser receivers that set the basis for further derivations are statistically characterized. The information-theoretic ergodic and outage sum-rates spectral efficiencies are then derived and analyzed.

O estudo de matrizes aleatórias na física tradicionalmente ocorre no contexto dos modelos de Wigner e na estatística por modelos de Wishart, que se conectam através do threefold way de Dyson para matrizes aleatórias reais, complexas e de quaternios indexadas respectivamente pelo índice B = 1; 2; 4 de Dyson. Estudos recentes mostraram o caminho para que estes modelos fossem generalizados para valores reais de B, permitindo o estudo de ensembles com índice arbitrário. Neste trabalho, estudamos as propriedades estatísticas destes sistemas e exploramos a física subjacente nos modelos de Wigner e Wishart e investigamos, através de cálculos numéricos, os efeitos de localização nos modelos de geral. Também introduzimos quebras na simetria desta nova forma e estudamos numericamente os resultados da estatística dos sistemas perturbados. / The study of random matrices in physics has traditionally occurred in the context of Wigner models and in statistics by Wishart models, which are connected through Dyson\'s threefold way for real, complex and quaternion random matrices index by the Dyson _ = 1; 2; 4 index, respectively. Recent studies have shown the way by which these models are generalized for real values of _, allowing for the study the ensembles with arbitrary index. In this work, we study the statistical properties of these systems and explore the underlying physics in Wigner\'s and Wishart\'s models through and investigate through numerical calculations the e_ects of localization in general _ models. We also introduce symmetry breaks in this new form and study numerically the results of the statistics of the disturbed systems.

While many university students get introduced to the concept of statistics early in their education, random matrix theory (RMT) usually first arises (if at all) in graduate level classes. This thesis serves as a friendly introduction to RMT, which is the study of matrices with entries following some probability distribution. Fundamental results, such as Gaussian and Wishart ensembles, are introduced and a discussion of how their corresponding eigenvalues are distributed is presented. Two well-studied applications, namely neural networks and PCA, are discussed where we present how RMT can be applied / Medan många stöter på statistik och sannolikhetslära tidigt under sina universitetsstudier så är det sällan slumpmatristeori (RMT) dyker upp förän på forskarnivå. RMT handlar om att studera matriser där elementen följer någon sannolikhetsfördelning och den här uppsatsen presenterar den mest grundläggande teorin för slumpmatriser. Vi introducerar Gaussian ensembles, Wishart ensembles samt fördelningarna för dem tillhörande egenvärdena. Avslutningsvis så introducerar vi hur slumpmatriser kan användas i neruonnät och i PCA.

Le défi du BigData entraîne un besoin pour les algorithmes d'apprentissage automatisé de s'adapter aux données de grande dimension et de devenir plus efficace. Récemment, une nouvelle direction de recherche est apparue qui consiste à analyser les méthodes d’apprentissage dans le régime moderne où le nombre n et la dimension p des données sont grands et du même ordre. Par rapport au régime conventionnel où n>>p, le régime avec n,p sont grands et comparables est particulièrement intéressant, car les performances d’apprentissage dans ce régime restent sensibles à l’ajustement des hyperparamètres, ouvrant ainsi une voie à la compréhension et à l’amélioration des techniques d’apprentissage pour ces données de grande dimension.L'approche technique de cette thèse s'appuie sur des outils avancés de statistiques de grande dimension, nous permettant de mener des analyses allant au-delà de l'état de l’art. La première partie de la thèse est consacrée à l'étude de l'apprentissage semi-supervisé sur des grandes données. Motivés par nos résultats théoriques, nous proposons une alternative supérieure à la méthode semi-supervisée de régularisation laplacienne. Les méthodes avec solutions implicites, comme les SVMs et la régression logistique, sont ensuite étudiées sous des modèles de mélanges réalistes, fournissant des détails exhaustifs sur le mécanisme d'apprentissage. Plusieurs conséquences importantes sont ainsi révélées, dont certaines sont même en contradiction avec la croyance commune. / The BigData challenge induces a need for machine learning algorithms to evolve towards large dimensional and more efficient learning engines. Recently, a new direction of research has emerged that consists in analyzing learning methods in the modern regime where the number n and the dimension p of data samples are commensurately large. Compared to the conventional regime where n>>p, the regime with large and comparable n,p is particularly interesting as the learning performance in this regime remains sensitive to the tuning of hyperparameters, thus opening a path into the understanding and improvement of learning techniques for large dimensional datasets.The technical approach employed in this thesis draws on several advanced tools of high dimensional statistics, allowing us to conduct more elaborate analyses beyond the state of the art. The first part of this dissertation is devoted to the study of semi-supervised learning on high dimensional data. Motivated by our theoretical findings, we propose a superior alternative to the standard semi-supervised method of Laplacian regularization. The methods involving implicit optimizations, such as SVMs and logistic regression, are next investigated under realistic mixture models, providing exhaustive details on the learning mechanism. Several important consequences are thus revealed, some of which are even in contradiction with common belief.

Motivée par les récentes avancées dans l'analyse théorique des performances des algorithmes d'apprentissage automatisé, cette thèse s'intéresse à l'analyse de performances et à l'amélioration de la classification nonsupervisée de données et graphes en grande dimension. Spécifiquement, dans la première grande partie de cette thèse, en s'appuyant sur des outils avancés de la théorie des grandes matrices aléatoires, nous analysons les performances de méthodes spectrales sur des modèles de graphes réalistes et denses ainsi que sur des données en grandes dimensions en étudiant notamment les valeurs propres et vecteurs propres des matrices d'affinités de ces données. De nouvelles méthodes améliorées sont proposées sur la base de cette analyse théorique et démontrent à travers de nombreuses simulations que leurs performances sont meilleures comparées aux méthodes de l'état de l'art. Dans la seconde partie de la thèse, nous proposons un nouvel algorithme pour la détection de communautés hétérogènes entre plusieurs couches d'un graphe à plusieurs types d'interaction. Une approche bayésienne variationnelle est utilisée pour approximer la distribution apostériori des variables latentes du modèle. Toutes les méthodes proposées dans cette thèse sont utilisées sur des bases de données synthétiques et sur des données réelles et présentent de meilleures performances en comparaison aux approches standard de classification dans les contextes susmentionnés. / Spurred by recent advances on the theoretical analysis of the performances of the data-driven machine learning algorithms, this thesis tackles the performance analysis and improvement of high dimensional data and graph clustering. Specifically, in the first bigger part of the thesis, using advanced tools from random matrix theory, the performance analysis of spectral methods on dense realistic graph models and on high dimensional kernel random matrices is performed through the study of the eigenvalues and eigenvectors of the similarity matrices characterizing those data. New improved methods are proposed and are shown to outperform state-of-the-art approaches. In a second part, a new algorithm is proposed for the detection of heterogeneous communities from multi-layer graphs using variational Bayes approaches to approximate the posterior distribution of the sought variables. The proposed methods are successfully applied to synthetic benchmarks as well as real-world datasets and are shown to outperform standard approaches to clustering in those specific contexts.

Ever since its introduction in 1952, the Mean-Variance (MV) portfolio selection theory has remained a centerpiece within the realm of e_cient asset allocation. However, in scienti_c circles, the theory has stirred controversy. A strand of criticism has emerged that points to the phenomenon that Mean-Variance Optimization su_ers from the severe drawback of estimation errors contained in the expected return vector and the covariance matrix, resulting in portfolios that may signi_cantly deviate from the true optimal portfolio. While a substantial amount of e_ort has been devoted to estimating the expected return vector in this context, much less is written about the covariance matrix input. In recent times, however, research that points to the importance of the covariance matrix in MV optimization has emerged. As a result, there has been a growing interest whether MV optimization can be enhanced by improving the estimate of the covariance matrix. Hence, this thesis was set forth by the purpose to investigate whether nancial practitioners and institutions can allocate portfolios consisting of assets in a more e_cient manner by changing the covariance matrix input in mean-variance optimization. In the quest of chieving this purpose, an out-of-sample analysis of MV optimized portfolios was performed, where the performance of ve prominent covariance matrix estimators were compared, holding all other things equal in the MV optimization. The optimization was performed under realistic investment constraints, taking incurred transaction costs into account, and for an investment asset universe ranging from equity to bonds. The empirical _ndings in this study suggest one dominant estimator: the covariance matrix estimator implied by the Gerber Statistic (GS). Speci_cally, by using this covariance matrix estimator in lieu of the traditional sample covariance matrix, the MV optimization rendered more e_cient portfolios in terms of higher Sharpe ratios, higher risk-adjusted returns and lower maximum drawdowns. The outperformance was protruding during recessionary times. This suggests that an investor that employs traditional MVO in quantitative asset allocation can improve their asset picking abilities by changing to the, in theory, more robust GS  ovariance matrix estimator in times of volatile nancial markets.

[ES] El Análisis Matricial proporciona herramientas muy útiles en la Matemática Aplicada. La teoría de matrices inversas generalizadas constituye una de estas herramientas. Su aplicación a otras áreas de las matemáticas y a otras disciplinas es importante. En esta tesis doctoral se definen e investigan nuevas inversas generalizadas, y se encuentran y caracterizan nuevos órdenes parciales definidos a partir de algunas de ellas. Por lo tanto, esta tesis doctoral se enmarca en dos importantes áreas: el Análisis Matricial y la Teoría de Matrices, y el Algebra de la Lógica (Estructuras Algebraicas Ordenadas). En la primera parte de esta tesis se define e investiga una nueva clase de inversas generalizadas híbridas, las inversas GDMP (y dualmente, las MPGD inversas) en el conjunto de matrices cuadradas de índice arbitrario, como una extensión de las inversas DMP a una clase más general. En esta tesis se presentan las nuevas inversas generalizadas GDMP como cierto producto de matrices que involucra las inversas G-Drazin y la inversa de Moore- Penrose. Se investigan sus propiedades mediante diferentes enfoques y se las caracteriza desde diferentes puntos de vista. Como complemento, se proporciona un algoritmo para hallarlas, que además permite encontrar una inversa G-Drazin. El estudio de proyectores es un área importante en diferentes ramas de las Matemáticas y en el Análisis Matricial en particular. La teoría de inversas generalizadas se utiliza como herramienta para analizarlos y operar con ellos. En la segunda parte de esta tesis se estudia el comportamiento de ciertos proyectores oblicuos definidos mediante inversas generalizadas. A partir de la definición de una adecuada relación de equivalencia en conjuntos particulares de matrices complejas, se introduce una nueva clase de matrices inversas generalizadas como el representante "más simple" de cada clase de equivalencia. Además, se representan como combinación de una inversa interior y la inversa de Moore-Penrose. Esta es la razón por la que se las ha denominado inversas 1MP y MP1. De manera similar se introducen las inversas 2MP y sus duales, las MP2. M. Mehdipour y A. Salemi definieron en [53j la inversa CMP de una matriz cuadrada A poniendo el énfasis en la parte core de la propia matriz A. En esta tesis doctoral se realiza un análisis similar, centrando el enfoque en las inversas 2MP. Surgen de esta manera las inversas generalizadas C2MP. La teoría de inversas generalizadas se relaciona estrechamente con la de órdenes parciales. En esta tesis se retoma el estudio, comenzado en [45], de las propiedades del orden diamante en conjuntos de matrices rectangulares. Como una aplicación de las inversas generalizadas 1MP y MP1, se definen dos nuevas relaciones de orden en conjuntos de matrices rectangulares. Esta tesis está organizadas en cuatro capítulos. En el Capítulo 1 se desarrollan algunos antecedentes del tema de la tesis y se presentan los resultados preliminares necesarios para el desarrollo del resto de los capítulos. En el Capítulo 2 se presentan las clases de matrices GDMP y MPGD, se demuestran propiedades de estas inversas y se describe un algoritmo para hallarlas. El Capítulo 3 se aboca al estudio de ciertos proyectores que permiten definir las clases de inversas generalizadas 1MP, MP1, 2MP y MP2. Al tomar un caso particular de inversa exterior, se definen las inversas C2MP. Además, se presentan las inversas definidas en esta tesis como inversas con espacio rango y espacio nulo prescrito. Finalmente, en el Capítulo 4, con la intención de estudiar una aplicación de la teoría de inversas generalizadas, se profundiza el estudio de órdenes parciales, proporcionando nuevas propiedades del orden diamante. También, se presentan e investigan dos nuevas relaciones de orden en el conjunto de matrices rectangulares y se analizan sus propiedades. Algunos de los resultados obtenidos en esta tesis pueden encontrarse en [37, 38, 39, 40, 41j. / [CA] L'Analisi Matricial proporciona eines molt útils en la Matematica Aplicada. La teoria de matrius inverses generalitzades constitueix una d'aquestes eines. La seua aplicació a altres arees de les matematiques i a altres disciplines és important. En aquesta tesi doctoral es defineixen i investiguen noves inverses generalitzades, i es troben i caracteritzen nous ordres parcials definits a partir d'algunes d'elles. Per tant, aquesta tesi doctoral s'emmarca en dues importants arees: l'Analisi Matricial i la Teoria de Matrius, i l' Álgebra de la Lógica (Estructures Algebraiques Ordenades). En la primera part d'aquesta tesi es defineix i investiga una nova classe d'inverses generalitzades híbrides, les invernes GDMP (i dualment, les MPGD invernes) en el conjunt de matrius quadrades d'índex arbitrari, com una extensió de les invernes DMP a una classe més general. En aquesta tesi es presenten les noves invernes generalitzades GDMP com a cert producte de matrius que involucra les invernes G-Drazin i la inversa de Moore-Penrose. S'investiguen les seues propietats mitjanc;ant diferents enfocaments i es caracteritzen des de diferents punts de vista. Com a complement, es proporciona un algorisme per a trabar-les, que a més permet trabar una inversa G-Drazin. L'estudi de projectors és una area important en diferents branques de les Matemati­ ques i en l' Analisi Matricial en particular. La teoría d'inverses generalitzades s'utilitza com a eina per a analitzar-los i operar amb ells. En la segona part d'aquesta tesi s'estudia el comportament d'uns certs projectors oblics definits mitjanc;ant invernes generalitzades. A partir de la definició d'una adequada relació d'equivalencia en conjunts particulars de matrius complexes, s'introdueix una nova classe de matrius invernes generalitzades com el representant "més simple" de cada classe d'equivalencia. A més, es representen com a combinació d'una inversa interior i la inversa de Moore­ Penrose. Aquesta és la raó per la qual se les ha denominades invernes lMP i MPl. De manera similar, es defineixen les inverses 2MP i els seus duals, les MP2. M. Mehdipour i A. Salemi van definir en [53] la inversa CMP d'una matriu quadrada A posant l'emfasi en la part core de la propia matriu A. En aquesta tesi doctoral es realitza una analisi similar, centrant l'enfocament en les inverses 2MP. Sorgeixen d'aquesta manera les inverses generalitzades C2MP. En aquesta tesi es reprén l'estudi, començat a [45], de les propietats de l'ordre diamant en conjunts de matrius rectangulars. Comuna aplicació de les inverses generalitzades lMP i MPl, es defineixen dues noves relacions d'ordre en conjunts de matrius rectangulars. Finalment, es troba una altra caracterització de l'ordre diamant. Aquesta tesis esta organitzada en quatre capítols. En el Capítol 1 es desenvolupen alguns antecedents del tema de la tesi i es presenten els resultats preliminars necessaris per al desenvolupament de la resta dels capítols. En el Capítol 2 es presenten les classes de matrius GDMP i MPGD, es demostren propietats d'aquestes inverses i es descriu un algorisme per a trobar-les. El Capítol 3 es dedica a l'estudi d'uns certs projectors que permeten definir les classes d'inverses generalitzades lMP, MPl, 2MP i MP2. Particularitzant la inversa exterior considerada, es defineixen les inverses C2MP. A més, es presenten les inverses definides en aquesta tesi com a inverses amb espai rang i espai nul prescrit. Finalment, en el Capítol 4, amb la intenció d'estudiar una aplicació de la teoría d'inverses generalitzades, s'aprofundeix en l'estudi d'ordres parcials, proporcionant noves propietats de l'ordre diamant. També, es presenten i investiguen dues noves relacions d'ordre en el conjunt de matrius rectangulars i s'analitzen les seues propietats. Alguns dels resultats obtinguts en aquesta tesi poden trobar-se en [37, 38, 39, 40, 41]. / [EN] The Matrix Analysis provides with very useful tools for the Applied Mathematics. The theory of Generalized Inverse Matrices constitutes one of these tools. Its application is important for other areas of mathematics and other disciplines. In this PhD. thesis, new generalized inverses are defined and investigated, and new partial orders defined by sorne of them are found and characterized. Therefore, this PhD. thesis is based on two important areas: the Matrix Analysis and the Theory of Matrices, and the Algebra of Logic (Ordered Algebraic Structures). In the first part this PhD. thesis, a new kind of hybrid generalized inverse is defined and investigated, the GDMP-inverses (and their duals, the MPGD-inverses), in the setting of square matrices of an arbitrary index, as an extension of the DMP inverses to a more general class. In this PhD. thesis, generalized GDMP-inverses are introduced as a certain product of matrices that involve the G-Drazin inverse and the Moore-Penrose inverse. The pro­ perties are investigated by different methods and characterized from different points of view. As a complement, it is provided an algorithm to compute them, which also allows to find a G-Drazin inverse. The study of projectors is an important area in different branches of Mathematics and particularly in the Matrix Analysis. The theory of generalized inverses is used as a tool to analyze them and operate with them. In the second part of this PhD. thesis, the behaviour of certain oblique projectors defined by generalized inverses is studied. From the definition of an adequate equivalence relation in particular sets of complex matrices, a new class of generalized inverse matrices is introduced as the "simplest" representant of each class of equivalence. Besides, they are represented as a product of an inner inverse and the Moore-Penrose inverse. This is the reason why they have been named lMP and MPl inverses. Both the core inverse and the DMP inverse are expressed as an adequate product involving a specific outer inverse and the Moore-Penrose inverse. Similarly, the 2MP inverses and their duals, the MP2 inverses, are defined. M. Mehdipour and A. Salemi defined in [53] the CMP inverse of a square matrix A, emphasizing the care part of the A matrix itself. In this PhD. thesis, a similar analysis is done, focusing on the care part of 2MP inverses. In this way, the generalized C2MP inverses are investigated. The study of the diamond order properties in sets of rectangular matrices is inves­ tigated in this PhD. thesis. Two new order relations in sets of rectangular matrices are defined as an application of the generalized lMP and MPl inverses. Finally, another characterization of the diamond order is investigated in this PhD. thesis. This PhD. thesis is organized into four chapters. In Chapter 1, sorne introduction of the PhD. thesis topic are developed and the preliminary results needed for the development of the rest of the chapters are presented. In Chapter 2, the classes of GDMP and MPGD matrices are presented, properties of these inverses are proved and an algorithm to find them is described. Chapter 3 is focused on the study of certain projectors that allow to define the classes of generalized lMP, MPl, 2MP and MP2 inverses. When taking a particular case of outer inverse, the C2MP inverses are defined. Moreover, the inverses defined in this PhD. thesis are presented as inverses with prescribed range and null space. Finally, in Chapter 4, the partial orders are studied in more detail, providing new properties of the diamond order, with the purpose of studying an application of the theory of generalized inverses. Finally, two new order relations are presented and investigated in the set of rectangular matrices and their properties are analyzed. Sorne of the results obtained in this PhD. thesis can be found in [37, 38, 39, 40, 41]. / Hernández, MV. (2022). Matrices inversas generalizadas definidas mediante proyectores y su aplicación a órdenes parciales matriciales [Tesis doctoral]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/186007

Le sujet principal de cette thèse est la physique des verres de spin. Les verres de spin ont été introduits au début des années 70 pour décrire alliages magnétiques diluées. Ils ont désormais été considerés pour comprendre le comportement de liquides sousrefroidis. Parmis les systèmes qui peuvent être décrits par le langage des systèmes desordonnés, on trouve les problèmes d’optimisation combinatoire. Dans la première partie de cette thèse, nous considérons les modèles de verre de spin avec intéraction de Kac pour investiguer la phase de basse température des liquides sous-refroidis. Dans les chapitres qui suivent, nous montrons comment certaines caractéristiques des modèles de verre de spin peuvent être obtenues à partir de résultats de la théorie des matrices aléatoires en connection avec la statistique des valeurs extrêmes. Dans la dernière partie de la thèse, nous considérons la connexion entre la théorie desverres de spin et la science computationnelle, et présentons un nouvel algorithme qui peut être appliqué à certains problèmes dans le domaine des finances. / The main subject of this thesis is the physics of spin glasses. After their introduction in the 70s in order to describe dilute magnetic alloys, spin-glass models have been considered prototype models to understand the behavior of supercooled liquids. Among the systems that can be described and analyzed using the language of disordered systems, there are problems of combinatorial optimization. In the first part of the thesis, we consider spin-glass models with Kac interactions in order to investigate the supercooled phase of glass-forming liquids. Afterwards, we show how some features of spin-glass models can be described by ubiquitous results of Random Matrix Theory in connection with Extreme Value Statistics. Finally, from the interaction of spin-glass theory and computer science, we put forward a new algorithm of immediate application in Financial problems.

In der vorliegenden Promotionsarbeit werden optische Eigenschaften von Halbleiterquantengräben untersucht, die mit der Ausbildung von Speckle-Mustern in der exzitonischen Emission zusammenhängen. Die in nichtspekulärer Richtung nach resonanter Anregung von Exzitonen ausgestrahlte Emission enthält Informationen über Unordnung und Streuprozesse in der Probe. Im Kapitel "Spektrale Speckle-Analyse" wird gezeigt, dass Speckles zur Bestimmung des koährenten Anteils verwendet werden können. Außerdem kann die innerhalb des inhomogen verbreiterten Ensembles frequenzaufgelöste Lebensdauer der Exzitonen bestimmt werden. Eine mikroskopische Dichtematrixtheorie wird entwickelt und numerisch gelöst. Es wird eine gute Übereinstimmung mit den gemessenen Daten für unterschiedliche Quantengraben-Dicken und Temperaturen gefunden. Im Kapitel "Schrägliegende Speckles" werden Quantengräben mit mechanischer Verzerrung betrachtet. Die Verzerrung führt zu einer ort-abhängigen Änderung der Emissionsenergie in der Ebene des Quantengrabens und das richtungs- und zeitaufgelöste Specklemuster erfährt eine Drehung. Die theoretische Beschreibung des Rayleigh-Spektrums erlaubt es, diese Drehung mit dem lokalen Wert des Gradienten der Exzitonenergie in Beziehung zu setzen. Numerische Simulationen zeigen allerdings, dass dieser Effekt nicht durch eine Bewegung der Exzitonen entlang des Verzerrungs-Gradienten verursacht wird. Im Kapitel "Nicht-Markovsche Exziton-Phonon Dynamik" die Dichtematrixtheorie, wird jenseits der Markovschen Näherung für die Streuung von Exzitonen an akustischen Phononen numerisch gelöst. Das Absorptionsspektrum besteht aus Lorentz-formige Peaks und breiteren Seitenbändern, die aus der nicht-Markovschen Kopplung stammen. Diese Eigenschaften sind vor allem für die stark lokalisierten Zustände auf der Niederenergie-Seite des Spektrums wichtig, und erlauben eine bessere Deutung von Nahfeld-Experimenten. / In this work, optical properties of semiconductor quantum wells (QW) are investigated, which are relevant for the irregular light pattern (speckle pattern) emitted in nonspecular directions by QW after resonant excitation of the exciton states. This emission contains information on disorder and scattering processes in the sample. In Chapter "Spectral Speckle Analysis", it is shown that Speckles can be used for extraction of the coherent part of the emission, the Resonant Rayleigh Scattering. Furthermore, the frequency resolved lifetime of excitons within an inhomogeneously broadened ensemble can be established. A microscopic density matrix theory for excitons interacting with acoustic phonons is developed and numerically solved. Good agreement with the experimental results for different QW sizes and temperatures is found. In Chapter "Sloped Speckles" QW with mechanical strain are considered. The strain leads to a spatially dependent modification of the emission energy and to a tilting of the direction- and time-resolved speckle pattern. The theoretical description of the RRS relates this tilting to the local value of the spatial gradient of the exciton energy. However, numerical simulations make clear that this effect is not due to exciton motion along the strain gradient. In Chapter "Non-Markovian exciton-phonon dynamics" the density matrix theory is numerically solved beyond the Markov approximation for the interaction between excitons and acoustical phonons. The resulting absorption spectrum consists of Lorentzian peaks on top of broader sidebands originating from the non-Markovian coupling. These features are mostly important for the strongly localized states in the low energy side of the spectrum, suggesting a better interpretation of near-field experiments.

Zusammenfassung in PostScript In dieser Arbeit wird die photoinduzierte Dynamik offener Molekularsysteme unter dem Einfluß intensiver und ultrakurzer Laserpulse untersucht. Die Anregung eines Moleküls durch einen optischen ultrakurzen Laserpuls führt zu Übergängen zwischen verschiedenen elektronischen Zuständen. Dieser Anregungsprozeß wird begleitet von dissipativen Vorgängen wie Energie-- und Phasenrelaxation. Die Beschreibung dieser photoinduzierten Dynamik erfolgt mit Hilfe der Methode der Dichtematrixtheorie. Dabei zeigt die Ableitung der Quanten--Master--Gleichung im Rahmen des Projektionsoperator--Formalismus, daß die wirkenden äußeren Felder einmal direkt im reversiblen Anteil der Bewegungsgleichung auftreten, aber auch einen indirekten Einfluß über den die Dissipation beschreibenden Dissipations--Superoperator ausüben. In dieser Arbeit wird zum ersten Mal die durch ultrakurze Laserpulse induzierte Feldabhängigkeit des Dissipations--Superoperators berücksichtigt. Im Rahmen der Darstellung der Quanten--Master--Gleichung im Floquetbild kann eine anschauliche Deutung dieses feldabhängigen Effektes gegeben werden: die die Dissipation beschreibende frequenzabhängige Spektraldichte der Umgebungsmoden wird feldabhängig bei verschiedenen Frequenzen abgefragt. Analytische Untersuchungen zum Zwei--Niveau--System zeigen, daß die Feldabhängigkeit dann relevant wird, wenn die Pulslänge vergleichbar ist mit der Zeitskala, auf der die Autokorrelationsfunktion der Umgebungsfreiheitsgrade abklingt. Um den Einfluß auf experimentelle Größen zu untersuchen, wird ein zweifarbiges Pump--Test--Experiment zum Laserfarbstoffmolekül IR 125 betrachtet, bei welchem die spektral und zeitlich aufgelöste Transmission auf einer Femtosekunden-- und Pikosekunden--Zeitskala gemessen wurde. Im Rahmen des Modells einer effektiven Schwingungsmode wird eine Anpassungsrechnung an das Experiment vorgenommen. Dabei wird zunächst die Standard-Redfield-Theorie verwendet, um ein Referenzmodell zu gewinnen. Es gelingt, eine gute Übereinstimmung mit dem Experiment zu erreichen. Die exakte Berücksichtigung des Einflusses der internen Konversion zwischen den angeregten elektronischen Zuständen führt zu einem Anstieg der Transmission innnerhalb einer Pikosekunde. Es ist notwendig, die Dichtematrixgleichungen exakt zu lösen, da eine vergleichende Untersuchung mit Hilfe der nichtlinearen Suszeptibilität dritter Ordnung eine deutliche Abweichung zum exakten Resultat zeigt. Ausgehend vom Referenzfall feldunabhängiger Dissipation wird dann die Feldabhängigkeit der Relaxationsraten bestimmt sowie der Einfluß auf Observablen wie der relativen Transmission untersucht. In Übereinstimmung mit den analytischen Ergebnissen zeigt sich, daß der feldabhängige Effekt am größen ausgeprägt ist, wenn die Pulslänge kleiner als die Korrelationszeit der Umgebungsfreiheitsgrade wird und die wirkenden Felder hinreichend intensiv sind.Damit wird eine Kontrolle von Dissipation möglich. Ein Einfluß des feldabhängigen Effektes auf experimentelle Observablen wird vorhergesagt. / abstract in PostScript This thesis investigates the influence of intense and ultrashort laser pulses on the photoinduced dynamics of open molecular systems. The excitation of a molecule by an optical ultrashort laser pulse induces transitions between different electronic states. This excitation process is accompanied by the dissipative processes of energy and vibrational relaxation. This excitation process is described within the method of the density matrix theory. Thereby, the derivation of the quantum master equation in the framework of the projection operator formalism demonstrates that the external fields are present in the reversible part of the equation of motion and also exert an indirect influence by acting on the dissipation superoperator which accounts for dissipation. In this thesis the field--dependency of the dissipation superoperator which is induced by the external fields is considered for the first time. By a representation of the quantum master equation in the Floquet picture, an interpretation of this field--dependent effect can be given: the frequency--dependent spectral density of the environmental modes which describe dissipation is determined at different field--dependent frequencies. Analytical investigations for the two level system demonstrate that the field dependence becomes relevant if the pulse length is comparable with the time scale on which the autocorrelation function of the environmental degrees of freedom decays.To investigate the influence on experimental quantities, a two--color pump--probe experiment for the laser dye molecule IR 125 is considered for which the spectrally and temporally resolved transmission on a femtosecond and picosecond time scale has been measured. Within the model of one effective vibrational mode the experimental data is fitted. The standard Redfield theory is used to provide a reference model. A high degree of concurrence between the theory and the results of the experiment is achieved. The exact treatment of internal conversion between the excited electronic states leads to a rise in transmission within one picosecond. It is necessary to solve the density matrix equations exactly because a comparative investigation with the nonlinear susceptibility of third order leads to a clear viation from the exact result. Starting from the reference case of field--independent dissipation, the field--dependency of the relaxation rates is determined and the influence on observables for example the relative transmission is investigated. The analytical results show that the field--dependent effect is strongest if the pulse length becomes smaller than the correlation time of the environmental modes and if the acting fields are sufficiently strong. Thereby, a control of dissipation becomes possible. An influence of the field--dependent effect on experimental observables is predicted.