Non-Markovian dynamics and quantum information probes

This Thesis discusses the phenomenology of the dynamics of open quantum systems marked by non-Markovian memory effects. Non-Markovian open quantum systems are the focal point of a flurry of recent research aiming to answer, e.g., the following questions: What is the characteristic trait of non-Markovian dynamical processes that discriminates it from forgetful Markovian dynamics? What is the microscopic origin of memory in quantum dynamics, and how can it be controlled? Does the existence of memory effects open new avenues and enable accomplishments that cannot be achieved with Markovian processes? These questions are addressed in the publications forming the core of this Thesis with case studies of both prototypical and more exotic models of open quantum systems. In the first part of the Thesis several ways of characterizing and quantifying non-Markovian phenomena are introduced. Their differences are then explored using a driven, dissipative qubit model. The second part of the Thesis focuses on the dynamics of a purely dephasing qubit model, which is used to unveil the origin of non-Markovianity for a wide class of dynamical models. The emergence of memory is shown to be strongly intertwined with the structure of the spectral density function, as further demonstrated in a physical realization of the dephasing model using ultracold quantum gases. Finally, as an application of memory effects, it is shown that non- Markovian dynamical processes facilitate a novel phenomenon of timeinvariant discord, where the total quantum correlations of a system are frozen to their initial value. Non-Markovianity can also be exploited in the detection of phase transitions using quantum information probes, as shown using the physically interesting models of the Ising chain in a transverse field and a Coulomb chain undergoing a structural phase transition.

Adaptive Markov chain Monte Carlo and Bayesian filtering for state space models

This thesis is concerned with the state and parameter estimation in state space models. The estimation of states and parameters is an important task when mathematical modeling is applied to many different application areas such as the global positioning systems, target tracking, navigation, brain imaging, spread of infectious diseases, biological processes, telecommunications, audio signal processing, stochastic optimal control, machine learning, and physical systems. In Bayesian settings, the estimation of states or parameters amounts to computation of the posterior probability density function. Except for a very restricted number of models, it is impossible to compute this density function in a closed form. Hence, we need approximation methods. A state estimation problem involves estimating the states (latent variables) that are not directly observed in the output of the system. In this thesis, we use the Kalman filter, extended Kalman filter, Gauss–Hermite filters, and particle filters to estimate the states based on available measurements. Among these filters, particle filters are numerical methods for approximating the filtering distributions of non-linear non-Gaussian state space models via Monte Carlo. The performance of a particle filter heavily depends on the chosen importance distribution. For instance, inappropriate choice of the importance distribution can lead to the failure of convergence of the particle filter algorithm. In this thesis, we analyze the theoretical Lᵖ particle filter convergence with general importance distributions, where p ≥2 is an integer. A parameter estimation problem is considered with inferring the model parameters from measurements. For high-dimensional complex models, estimation of parameters can be done by Markov chain Monte Carlo (MCMC) methods. In its operation, the MCMC method requires the unnormalized posterior distribution of the parameters and a proposal distribution. In this thesis, we show how the posterior density function of the parameters of a state space model can be computed by filtering based methods, where the states are integrated out. This type of computation is then applied to estimate parameters of stochastic differential equations. Furthermore, we compute the partial derivatives of the log-posterior density function and use the hybrid Monte Carlo and scaled conjugate gradient methods to infer the parameters of stochastic differential equations. The computational efficiency of MCMC methods is highly depend on the chosen proposal distribution. A commonly used proposal distribution is Gaussian. In this kind of proposal, the covariance matrix must be well tuned. To tune it, adaptive MCMC methods can be used. In this thesis, we propose a new way of updating the covariance matrix using the variational Bayesian adaptive Kalman filter algorithm.

Aircraft tracking and classification with vhf passive bistatic radar

Since the times preceding the Second World War the subject of aircraft tracking has been a core interest to both military and non-military aviation. During subsequent years both technology and configuration of the radars allowed the users to deploy it in numerous fields, such as over-the-horizon radar, ballistic missile early warning systems or forward scatter fences. The latter one was arranged in a bistatic configuration. The bistatic radar has continuously re-emerged over the last eighty years for its intriguing capabilities and challenging configuration and formulation. The bistatic radar arrangement is used as the basis of all the analyzes presented in this work. The aircraft tracking method of VHF Doppler-only information, developed in the first part of this study, is solely based on Doppler frequency readings in relation to time instances of their appearance. The corresponding inverse problem is solved by utilising a multistatic radar scenario with two receivers and one transmitter and using their frequency readings as a base for aircraft trajectory estimation. The quality of the resulting trajectory is then compared with ground-truth information based on ADS-B data. The second part of the study deals with the developement of a method for instantaneous Doppler curve extraction from within a VHF time-frequency representation of the transmitted signal, with a three receivers and one transmitter configuration, based on a priori knowledge of the probability density function of the first order derivative of the Doppler shift, and on a system of blocks for identifying, classifying and predicting the Doppler signal. The extraction capabilities of this set-up are tested with a recorded TV signal and simulated synthetic spectrograms. Further analyzes are devoted to more comprehensive testing of the capabilities of the extraction method. Besides testing the method, the classification of aircraft is performed on the extracted Bistatic Radar Cross Section profiles and the correlation between them for different types of aircraft. In order to properly estimate the profiles, the ADS-B aircraft location information is adjusted based on extracted Doppler frequency and then used for Bistatic Radar Cross Section estimation. The classification is based on seven types of aircraft grouped by their size into three classes.

Estimation procedures affect the center of pressure frequency analysis

Even though frequency analysis of body sway is widely applied in clinical studies, the lack of standardized procedures concerning power spectrum estimation may provide unreliable descriptors. Stabilometric tests were applied to 35 subjects (20-51 years, 54-95 kg, 1.6-1.9 m) and the power spectral density function was estimated for the anterior-posterior center of pressure time series. The median frequency was compared between power spectra estimated according to signal partitioning, sampling rate, test duration, and detrending methods. The median frequency reliability for different test durations was assessed using the intraclass correlation coefficient. When increasing number of segments, shortening test duration or applying linear detrending, the median frequency values increased significantly up to 137%. Even the shortest test duration provided reliable estimates as observed with the intraclass coefficient (0.74-0.89 confidence interval for a single 20-s test). Clinical assessment of balance may benefit from a standardized protocol for center of pressure spectral analysis that provides an adequate relationship between resolution and variance. An algorithm to estimate center of pressure power density spectrum is also proposed.

Étude empirique de distributions associées à la Fonction de Pénalité Escomptée

On présente une nouvelle approche de simulation pour la fonction de densité conjointe du surplus avant la ruine et du déficit au moment de la ruine, pour des modèles de risque déterminés par des subordinateurs de Lévy. Cette approche s'inspire de la décomposition "Ladder height" pour la probabilité de ruine dans le Modèle Classique. Ce modèle, déterminé par un processus de Poisson composé, est un cas particulier du modèle plus général déterminé par un subordinateur, pour lequel la décomposition "Ladder height" de la probabilité de ruine s'applique aussi. La Fonction de Pénalité Escomptée, encore appelée Fonction Gerber-Shiu (Fonction GS), a apporté une approche unificatrice dans l'étude des quantités liées à l'événement de la ruine été introduite. La probabilité de ruine et la fonction de densité conjointe du surplus avant la ruine et du déficit au moment de la ruine sont des cas particuliers de la Fonction GS. On retrouve, dans la littérature, des expressions pour exprimer ces deux quantités, mais elles sont difficilement exploitables de par leurs formes de séries infinies de convolutions sans formes analytiques fermées. Cependant, puisqu'elles sont dérivées de la Fonction GS, les expressions pour les deux quantités partagent une certaine ressemblance qui nous permet de nous inspirer de la décomposition "Ladder height" de la probabilité de ruine pour dériver une approche de simulation pour cette fonction de densité conjointe. On présente une introduction détaillée des modèles de risque que nous étudions dans ce mémoire et pour lesquels il est possible de réaliser la simulation. Afin de motiver ce travail, on introduit brièvement le vaste domaine des mesures de risque, afin d'en calculer quelques unes pour ces modèles de risque. Ce travail contribue à une meilleure compréhension du comportement des modèles de risques déterminés par des subordinateurs face à l'éventualité de la ruine, puisqu'il apporte un point de vue numérique absent de la littérature.

Bayesian Inference in Exponential and Pareto populations in the presence of Outliers

This thesis Entitled Bayesian inference in Exponential and pareto populations in the presence of outliers. The main theme of the present thesis is focussed on various estimation problems using the Bayesian appraoch, falling under the general category of accommodation procedures for analysing Pareto data containing outlier. In Chapter II. the problem of estimation of parameters in the classical Pareto distribution specified by the density function. In Chapter IV. we discuss the estimation of (1.19) when the sample contain a known number of outliers under three different data generating mechanisms, viz. the exchangeable model. Chapter V the prediction of a future observation based on a random sample that contains one contaminant. Chapter VI is devoted to the study of estimation problems concerning the exponential parameters under a k-outlier model.

Analytical Studies on Diffusion and lntermittency in Chaotic Maps

The study of simple chaotic maps for non-equilibrium processes in statistical physics has been one of the central themes in the theory of chaotic dynamical systems. Recently, many works have been carried out on deterministic diffusion in spatially extended one-dimensional maps This can be related to real physical systems such as Josephson junctions in the presence of microwave radiation and parametrically driven oscillators. Transport due to chaos is an important problem in Hamiltonian dynamics also. A recent approach is to evaluate the exact diffusion coefficient in terms of the periodic orbits of the system in the form of cycle expansions. But the fact is that the chaotic motion in such spatially extended maps has two complementary aspects- - diffusion and interrnittency. These are related to the time evolution of the probability density function which is approximately Gaussian by central limit theorem. It is noticed that the characteristic function method introduced by Fujisaka and his co-workers is a very powerful tool for analysing both these aspects of chaotic motion. The theory based on characteristic function actually provides a thermodynamic formalism for chaotic systems It can be applied to other types of chaos-induced diffusion also, such as the one arising in statistics of trajectory separation. It was noted that there is a close connection between cycle expansion technique and characteristic function method. It was found that this connection can be exploited to enhance the applicability of the cycle expansion technique. In this way, we found that cycle expansion can be used to analyse the probability density function in chaotic maps. In our research studies we have successfully applied the characteristic function method and cycle expansion technique for analysing some chaotic maps. We introduced in this connection, two classes of chaotic maps with variable shape by generalizing two types of maps well known in literature.

Approximate Approximations and a Boundary Point Method for the Linearized Stokes System

The method of approximate approximations, introduced by Maz'ya [1], can also be used for the numerical solution of boundary integral equations. In this case, the matrix of the resulting algebraic system to compute an approximate source density depends only on the position of a finite number of boundary points and on the direction of the normal vector in these points (Boundary Point Method). We investigate this approach for the Stokes problem in the whole space and for the Stokes boundary value problem in a bounded convex domain G subset R^2, where the second part consists of three steps: In a first step the unknown potential density is replaced by a linear combination of exponentially decreasing basis functions concentrated near the boundary points. In a second step, integration over the boundary partial G is replaced by integration over the tangents at the boundary points such that even analytical expressions for the potential approximations can be obtained. In a third step, finally, the linear algebraic system is solved to determine an approximate density function and the resulting solution of the Stokes boundary value problem. Even not convergent the method leads to an efficient approximation of the form O(h^2) + epsilon, where epsilon can be chosen arbitrarily small.

Zur Konvergenz der Randpunktmethode

Das von Maz'ya eingeführte Approximationsverfahren, die Methode der näherungsweisen Näherungen (Approximate Approximations), kann auch zur numerischen Lösung von Randintegralgleichungen verwendet werden (Randpunktmethode). In diesem Fall hängen die Komponenten der Matrix des resultierenden Gleichungssystems zur Berechnung der Näherung für die Dichte nur von der Position der Randpunkte und der Richtung der äußeren Einheitsnormalen in diesen Punkten ab. Dieses numerisches Verfahren wird am Beispiel des Dirichlet Problems für die Laplace Gleichung und die Stokes Gleichungen in einem beschränkten zweidimensionalem Gebiet untersucht. Die Randpunktmethode umfasst drei Schritte: Im ersten Schritt wird die unbekannte Dichte durch eine Linearkombination von radialen, exponentiell abklingenden Basisfunktionen approximiert. Im zweiten Schritt wird die Integration über den Rand durch die Integration über die Tangenten in Randpunkten ersetzt. Für die auftretende Näherungspotentiale können sogar analytische Ausdrücke gewonnen werden. Im dritten Schritt wird das lineare Gleichungssystem gelöst, und eine Näherung für die unbekannte Dichte und damit auch für die Lösung der Randwertaufgabe konstruiert. Die Konvergenz dieses Verfahrens wird für glatte konvexe Gebiete nachgewiesen.

Estimators and Tests based on Likelihood-Depth with Application to Weibull Distribution, Gaussian and Gumbel Copula

In dieser Arbeit werden mithilfe der Likelihood-Tiefen, eingeführt von Mizera und Müller (2004), (ausreißer-)robuste Schätzfunktionen und Tests für den unbekannten Parameter einer stetigen Dichtefunktion entwickelt. Die entwickelten Verfahren werden dann auf drei verschiedene Verteilungen angewandt. Für eindimensionale Parameter wird die Likelihood-Tiefe eines Parameters im Datensatz als das Minimum aus dem Anteil der Daten, für die die Ableitung der Loglikelihood-Funktion nach dem Parameter nicht negativ ist, und dem Anteil der Daten, für die diese Ableitung nicht positiv ist, berechnet. Damit hat der Parameter die größte Tiefe, für den beide Anzahlen gleich groß sind. Dieser wird zunächst als Schätzer gewählt, da die Likelihood-Tiefe ein Maß dafür sein soll, wie gut ein Parameter zum Datensatz passt. Asymptotisch hat der Parameter die größte Tiefe, für den die Wahrscheinlichkeit, dass für eine Beobachtung die Ableitung der Loglikelihood-Funktion nach dem Parameter nicht negativ ist, gleich einhalb ist. Wenn dies für den zu Grunde liegenden Parameter nicht der Fall ist, ist der Schätzer basierend auf der Likelihood-Tiefe verfälscht. In dieser Arbeit wird gezeigt, wie diese Verfälschung korrigiert werden kann sodass die korrigierten Schätzer konsistente Schätzungen bilden. Zur Entwicklung von Tests für den Parameter, wird die von Müller (2005) entwickelte Simplex Likelihood-Tiefe, die eine U-Statistik ist, benutzt. Es zeigt sich, dass für dieselben Verteilungen, für die die Likelihood-Tiefe verfälschte Schätzer liefert, die Simplex Likelihood-Tiefe eine unverfälschte U-Statistik ist. Damit ist insbesondere die asymptotische Verteilung bekannt und es lassen sich Tests für verschiedene Hypothesen formulieren. Die Verschiebung in der Tiefe führt aber für einige Hypothesen zu einer schlechten Güte des zugehörigen Tests. Es werden daher korrigierte Tests eingeführt und Voraussetzungen angegeben, unter denen diese dann konsistent sind. Die Arbeit besteht aus zwei Teilen. Im ersten Teil der Arbeit wird die allgemeine Theorie über die Schätzfunktionen und Tests dargestellt und zudem deren jeweiligen Konsistenz gezeigt. Im zweiten Teil wird die Theorie auf drei verschiedene Verteilungen angewandt: Die Weibull-Verteilung, die Gauß- und die Gumbel-Copula. Damit wird gezeigt, wie die Verfahren des ersten Teils genutzt werden können, um (robuste) konsistente Schätzfunktionen und Tests für den unbekannten Parameter der Verteilung herzuleiten. Insgesamt zeigt sich, dass für die drei Verteilungen mithilfe der Likelihood-Tiefen robuste Schätzfunktionen und Tests gefunden werden können. In unverfälschten Daten sind vorhandene Standardmethoden zum Teil überlegen, jedoch zeigt sich der Vorteil der neuen Methoden in kontaminierten Daten und Daten mit Ausreißern.

Distributions on the simplex

The simplex, the sample space of compositional data, can be structured as a real Euclidean space. This fact allows to work with the coefficients with respect to an orthonormal basis. Over these coefficients we apply standard real analysis, inparticular, we define two different laws of probability trought the density function and we study their main properties

The Dirichlet distribution with respect to the Aitchison measure on the simplex - a first approach

The algebraic-geometric structure of the simplex, known as Aitchison geometry, is used to look at the Dirichlet family of distributions from a new perspective. A classical Dirichlet density function is expressed with respect to the Lebesgue measure on real space. We propose here to change this measure by the Aitchison measure on the simplex, and study some properties and characteristic measures of the resulting density

Sobre les relacions lineals d'energia lliure: mesures de semblança molecular quàntica sobre funcions de densitat electrònica modificad

En aquest treball es presenta l'ús de funcions de densitat electrònica de forat de Fermi per incrementar el paper que pren una regió molecular concreta, considerada com a responsable de la reactivitat molecular, tot i mantenir la mida de la funció de densitat original. Aquestes densitats s'utilitzen per fer mesures d'autosemblança molecular quàntica i es presenten com una alternativa a l'ús de fragments moleculars aillats en estudis de relació entre estructura i propietat. El treball es complementa amb un exemple pràctic, on es correlaciona l'autosemblanca molecular a partir de densitats modificades amb l'energia d'una reacció isodòsmica

Fuzzy set Theory and Quantum Chemistry

Es discuteixen breument algunes consideracions sobre l'aplicació de la Teoria dels Conjunts difusos a la Química quàntica. Es demostra aqui que molts conceptes químics associats a la teoria són adequats per ésser connectats amb l'estructura dels Conjunts difusos. També s'explica com algunes descripcions teoriques dels observables quàntics es potencien tractant-les amb les eines associades als esmentats Conjunts difusos. La funció densitat es pren com a exemple de l'ús de distribucions de possibilitat al mateix temps que les distribucions de probabilitat quàntiques

Sobre els mapes de semblança quàntica molecular

Partint de les definicions usuals de Mesures de Semblança Quàntica (MSQ), es considera la dependència d'aquestes mesures respecte de la superposició molecular. Pel cas particular en qnè els sistemes comparats siguin una molècula i un Àtom i que les mesures es calculin amb l'aproximació EASA, les MSQ esdevenen funcions de les tres coordenades de l'espai. Mantenint fixa una de les tres coordenades, es pot representar fàcilment la variació del valor de semblança en un pla determinat, i obtenir els anomenats mapes de semblança. En aquest article, es comparen els mapes de semblança obtinguts amb diferents MSQ per a sistemes senzills