Gegenstand dieser Arbeit ist die Untersuchung von statistischen Eigenschaften der ungeordneten Metallen im Rahmen des Anderson-Modells der Lokalisierung. Betrachtet wird ein Elektron auf einem Gitter mit "Nächste-Nachbarn-Hüpfen" und zufälligen potentiellen Gitterplatzenergien. Wegen der Zufälligkeit zeigen die Elektroneigenschaften, zum Beispiel die Eigenenergien und -zustände, irreguläre Fluktuationen, deren Statistik von der Amplitude der Potentialenergie abhängt. Mit steigender Amplitude wird das Elektron immer mehr lokalisiert, was schliesslich zum Metall-Isolator-Übergang führt. In dieser Arbeit wird die Statistik insbesondere im metallischen Bereich untersucht, und dadurch der Einfluss der Lokalisierung an den Eigenschaften des Systems betrachtet. Zuerst wird die Statistik der Matrixelemente des Dipoloperators untersucht. Die numerischen Ergebnisse für das Anderson-Modell werden mit Vorhersagen der semiklassischen Näherung verglichen. Dann wird der spektrale Strukturfaktor betrachtet, der als Fourier-Transformation der zwei-Punkt Zustandsdichtekorrelationsfunktion definiert wird. Dabei werden besonders die nichtuniversellen Abweichungen von den Vorhersagen der Zufallsmatrixtheorie untersucht. Die Abweichungen werden numerisch ermittelt, und danach mit den analytischen Vorhersagen verglichen. Die Statistik der Wellenfunktionen zeigt ebenfalls Abweichungen von der Zufallsmatrixtheorie. Die Abweichungen sind am größten für Statistik der großen Wellenfunktionsamplituden, die sogenannte seltene Ereignisse darstellen. Die analytischen Vorhersagen für diese Statistik sind teilweise widersprüchlich, und deshalb ist es interessant, sie auch numerisch zu untersuchen.

The interest in sustainable investing has increased significantly during recent years. Asset managers and institutional investors are urged to invest more sustainable from their stakeholders, reducing their investment universe. This thesis has found that sustainable investments have a different linear dependence structure compared to the regional markets in Europe and North America, but not in Asia-Pacific. However, the largest drawdowns of an sustainable compliant portfolio has historically been lower compared to the a random market portfolio, especially in Europe and North America. / Intresset för hållbara investeringar har ökat avsevärt de senaste åren. Fondförvaltare och institutionella investerare är, från deras intressenter, manade att investera mer hållbart vilket minskar förvaltarnas investeringsuniversum. Denna uppsats har funnit att hållbara investeringar har en beroendestruktur som är skild från de regionala marknaderna i Europa och Nordamerika, men inte för Asien-Stillahavsregionen. De största värdeminskningarna i en hållbar portfölj har historiskt varit mindre än värdeminskningarna från en slumpmässig marknadsportfölj, framförallt i Europa och Nordamerika.

Random matrix theory (RMT) is a mathematical framework that has found profound applications in physics, particularly in the study of many-body systems. Its success lies in its ability to predict universal statistical properties of complex systems, independent of the specific details. This thesis explores the application of RMT to two classes of many-body systems: quantum and stochastic many-body systems. Within the quantum framework, this work focuses on the Bose-Hubbard system, which is paradigmatic for modeling ultracold atoms in optical traps. According to RMT and the Eigenstate Thermalization Hypothesis (ETH), eigenstate-to-eigenstate fluctuations of expectation values of local observables decay rapidly with the system size in the thermodynamic limit at sufficiently large temperatures. Here, we study these fluctuations in the classical limit of fixed lattice size and increasing boson number. We find that the fluctuations follow the RMT prediction for large system sizes but deviate substantially for small lattices. Partly motivated by these results, the Bose-Hubbard model on three sites is studied in more detail. On few sites, the Bose-Hubbard model is known to be a mixed system, being neither fully chaotic nor integrable. We compare energy-resolved classical and quantum measures of chaos, which show a strong agreement. Deviations from RMT predictions are attributed to the mixed nature of the few-site model. In the context of stochastic systems, generators of Markov processes are studied. The focus is on the spectrum. We present results from two investigations of Markov spectra. First, we investigate the effect of sparsity on the spectrum of random generators. Dense random matrices previously used as a model for generic generators led to very large spectral gaps and therefore to unphysically short relaxation times. In this work, a model of random generators with adjustable sparsity — number of zero matrix elements — is presented, extending the dense framework. It is shown that sparsity leads to longer, more physically realistic relaxation times. Second, the generator spectrum of the Asymmetric Simple Exclusion Process (ASEP), a quintessential model in non-equilibrium statistical mechanics, is analyzed. We investigate the spectral boundary, which is characterized by pronounced spikes. The emergence of these spikes is analyzed from several points of view, including RMT. The results presented in this thesis contribute to the understanding of the applicability of RMT to many-body systems. This thesis highlights successes such as the explanation of “ETH fluctuations” in Bose-Hubbard models, the improvement of random matrix descriptions by introducing sparsity, and the emergence of spikes in the spectral boundary of the ASEP. The latter is a notable case where RMT provides insights even though the ASEP is a Bethe-integrable system. Furthermore, this thesis shows examples of the limits of RMT, exemplified by the results presented for the Bose-Hubbard model with a few sites.

Les futurs systèmes de communication mobile sont caractérisés par un déploiement de plus en plus dense de différents types de points d'accès sans fil. Afin d’atténuer les interférences dans ces systèmes, les techniques aux entrées multiples-sorties multiples (MIMO) ainsi que la coopération entre les émetteurs et/ou les récepteurs sont nécessaires. Les systèmes de communication mobile en deviennent plus complexes, ce qui impose une évolution des outils mathématiques permettant leur analyse. Ceux-ci doivent être capables de prendre en compte les caractéristiques les plus importantes du système, telles que l'affaiblissement de propagation, les interférences et l'information imparfaite d'état du canal. Le but de cette thèse est de développer de tels outils basés sur la théorie des grandes matrices aléatoires et de démontrer leur utilité à l'aide de plusieurs applications pratiques, telles que l'analyse des performances des systèmes « network MIMO » et des systèmes MIMO à grande échelle, la conception de détecteurs de faible complexité à expansion polynomiale, l'étude des techniques de précodage unitaire aléatoire, ainsi que l'analyse de canaux à relais multiples et de canaux à double diffusion. En résumé, les méthodes développées dans ce travail fournissent des approximations déterministes de la performance du système qui deviennent exactes en régime asymptotique avec un nombre illimité d'émetteurs et de récepteurs. Cette approche conduit souvent à des approximations de la performance du système étonnamment simples et précises et permet de tirer d’importantes conclusions sur les paramètres les plus pertinents. / Advanced mobile communication systems are characterized by a dense deployment of different types of wireless access points. Since these systems are primarily limited by interference, multiple-input multiple-output (MIMO) techniques as well as coordinated transmission and detection schemes are necessary to mitigate this limitation. Thus, mobile communication systems become more complex which requires that also the mathematical tools for their theoretical analysis must evolve. These must be able to take the most important system characteristics into account, such as fading, path loss, and interference. The aim of this thesis is to develop such tools based on large random matrix theory and to demonstrate their usefulness with the help of several practical applications, such as the performance analysis of network MIMO and large-scale MIMO systems, the design of low-complexity polynomial expansion detectors, and the study of random beamforming techniques as well as multi-hop relay and double-scattering channels. The methods developed in this work provide deterministic approximations of the system performance which become arbitrarily tight in the large system regime with an unlimited number of transmitting and receiving devices. This leads in many cases to simple and close approximations of the finite-size system performance and allows one to draw relevant conclusions about the most significant parameters. One can think of these methods as a way to provide a deterministic abstraction of the physical layer which substantially reduces the system complexity. Due to this complexity reduction, it is possible to carry out a system optimization which would be otherwise intractable.

Face recognition is a field of biometrics that deals with identification of subjects based on features present in the images of their faces. The factors that make face recognition popular and favorite as compared to other biometric methods are easier operation and ability to identify subjects without their knowledge. With these features, face recognition has become an integral part of the present day security systems, targeting a smart and secure world. There are various factors that de ne the performance of a face recognition system. The most important among them are recognition accuracy of algorithm used and time taken for recognition. Recognition accuracy of the face recognition algorithm gets affected by changes in pose, facial expression and illumination along with occlusions in the images. There have been a number of algorithms proposed to enable recognition under these ambient changes. However, it has been hard to and a single algorithm that can efficiently recognize faces in all the above mentioned conditions. Moreover, achieving real time performance for most of the complex face recognition algorithms on embedded platforms has been a challenge. Real-time performance is highly preferred in critical applications such as identification of crime suspects in public. As available software solutions for FR have significantly large latency in recognizing individuals, they are not suitable for such critical real-time applications. This thesis focuses on real-time aspect of FR, where acceleration of the algorithms is achieved by means of parallel hardware architectures. The major contributions of this work are as follows. We target to design a face recognition system that can identify at most 30 faces in each frame of video at 15 frames per second, which amounts to 450 recognitions per second. In addition, we target to achieve good recognition accuracy along with scalability in terms of database size and input image resolutions. To design a system with these specifications, as a first step, we explore algorithms in literature and come up with a hybrid face recognition algorithm. This hybrid algorithm shows good recognition accuracy on face images with changes in illumination, pose and expressions, and also with occlusions. In addition the computations in the algorithm are modular in nature which are suitable for real-time realizations through parallel processing. The face recognition system consists of a face detection module to detect faces in the input image, which is followed by a face recognition module to identify the detected faces. There are well established algorithms and architectures for face detection in literature which can perform detection at 15 frames per second on video frames. Detected faces of different sizes need to be scaled to the size specified by the face recognition module. To meet the real-time constraints, we propose a hardware architecture for real-time bi-cubic convolution interpolation with dynamic scaling factors. To recognize the resized faces in real-time, a scalable parallel pipelined architecture is designed for the hybrid algorithm which can perform 450 recognitions per second on a database containing grayscale images of at most 450 classes on Virtex 6 FPGA. To provide flexibility and programmability, we extend this design to REDEFINE, a multi-core massively parallel reconfigurable architecture. In this design, we come up with FR specific programmable cores termed Scalable Unit for Region Evaluation (SURE) capable of performing modular computations in the hybrid face recognition algorithm. We replicate SUREs in each tile of REDEFINE to construct a face recognition module termed REDEFINE for Face Recognition using SURE Homogeneous Cores (REFRESH). There is a need to learn new unseen faces on-line in practical face recognition systems. Considering this, for real-time on-line learning of unseen face images, we design tiny processors termed VOP, Processor for Vector Operations. VOPs function as coprocessors to process elements under each tile of REDEFINE to accelerate micro vector operations appearing in the synaptic weight computations. We also explore deep neural networks which operate similar to the processing in human brain and capable of working on very large face databases. We explore the field of Random matrix theory to come up with a solution for synaptic weight initialization in deep neural networks for better classification . In addition, we perform design space exploration of hardware architecture for deep convolution networks and conclude with directions for future work.

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