Understanding the bearing response of the footings on layered soils has always been a challenge for researchers. Due to the limitations of analytical and empirical solutions it had been difficult to understand the true bearing behavior. Some researchers have tried solving this problem by numerical analysis and have found some success. In this study the numerical analysis approach has been applied using a commercial tool FLAC (Fast Lagrangian Analysis of Continua) to study the bearing response of surface footings on layered clays. First, small deformation analyses were taken up to study the undrained bearing response of strip and circular footings resting on a horizontally layered strong over a soft clay foundation, and then over soft over strong clay foundation. In the end application of large strain mode of FLAC was explored to investigate the large deformation behavior of the strip footing resting on the surface of a strong over soft clay foundation. All models were run by applying velocity loading and a elastic-perfectly plastic Tresca yield criterion has been used. The results are compared with published Finite Element Method (FEM) results, and with analytical, empirical and semi-empirical solutions. It was found that bearing capacity results from the present small-strain FLAC analyses agree well with the FEM results. However, these results in most of the cases tend to differ (as much as 49% for certain layered clay foundations) from those predicted with analytical, empirical and semi-empirical solutions, mainly due to the assumptions made in these solutions. Since no such assumptions are made in the present FLAC analyses, the results and the methodology of this thesis can be applied to predict the bearing capacity of the practical problems. Application of the large-strain mode of FLAC to study the large deformation of shallow foundations has pointed out a limitation of FLAC in completing such analyses. However, it is observed from the early trends of these analyses that whereas the small deformation analysis may under estimate the ultimate bearing capacity for certain cases of layered foundations where the upper clay is moderately stiffer than the lower clay layer, it might also over predict the ultimate bearing capacity for other cases when the upper clay is very stiff in comparison to the lower clay layer.

The purpose of this thesis is to advance the understanding of strong-field QED processes in short laser pulses. The processes of non-linear one-photon and two-photon Compton scattering are studied, that is the scattering of photons in the interaction of relativistic electrons with ultra-short high-intensity laser pulses. These investigations are done in view of the present and next generation of ultra-high intensity optical lasers which are supposed to achieve unprecedented intensities of the order of 10^24 W/cm^2 and beyond, with pulse lengths in the order of some femtoseconds. The ultra-high laser intensity requires a non-perturbative description of the interaction of charged particles with the laser field to allow for multi-photon interactions, which is beyond the usual perturbative expansion of QED organized in powers of the fine structure constant. This is achieved in strong-field QED by employing the Furry picture and non-perturbative solutions of the Dirac equation in the presence of a background laser field as initial and final state wave functions, as well as the laser dressed Dirac-Volkov propagator. The primary objective is a realistic description of scattering processes with regard to the finite laser pulse duration beyond the common approximation of infinite plane waves, which is made necessary by the ultra-short pulse length of modern high-intensity lasers. Non-linear finite size effects are identified, which are a result of the interplay between the ultra-high intensity and the ultra-short pulse length. In particular, the frequency spectra and azimuthal photon emission spectra are studied emphasizing the differences between pulsed and infinite laser fields. The proper description of the finite temporal duration of the laser pulse leads to a regularization of unphysical infinities (due to the infinite plane-wave description) of the laser-dressed Dirac-Volkov propagator and in the second-order strong-field process of two-photon Compton scattering. An enhancement of the two-photon process is found in strong laser pulses as compared to the corresponding weak-field process in perturbative QED.

Understanding the bearing response of the footings on layered soils has always been a challenge for researchers. Due to the limitations of analytical and empirical solutions it had been difficult to understand the true bearing behavior. Some researchers have tried solving this problem by numerical analysis and have found some success. In this study the numerical analysis approach has been applied using a commercial tool FLAC (Fast Lagrangian Analysis of Continua) to study the bearing response of surface footings on layered clays. First, small deformation analyses were taken up to study the undrained bearing response of strip and circular footings resting on a horizontally layered strong over a soft clay foundation, and then over soft over strong clay foundation. In the end application of large strain mode of FLAC was explored to investigate the large deformation behavior of the strip footing resting on the surface of a strong over soft clay foundation. All models were run by applying velocity loading and a elastic-perfectly plastic Tresca yield criterion has been used. The results are compared with published Finite Element Method (FEM) results, and with analytical, empirical and semi-empirical solutions. It was found that bearing capacity results from the present small-strain FLAC analyses agree well with the FEM results. However, these results in most of the cases tend to differ (as much as 49% for certain layered clay foundations) from those predicted with analytical, empirical and semi-empirical solutions, mainly due to the assumptions made in these solutions. Since no such assumptions are made in the present FLAC analyses, the results and the methodology of this thesis can be applied to predict the bearing capacity of the practical problems. Application of the large-strain mode of FLAC to study the large deformation of shallow foundations has pointed out a limitation of FLAC in completing such analyses. However, it is observed from the early trends of these analyses that whereas the small deformation analysis may under estimate the ultimate bearing capacity for certain cases of layered foundations where the upper clay is moderately stiffer than the lower clay layer, it might also over predict the ultimate bearing capacity for other cases when the upper clay is very stiff in comparison to the lower clay layer.

Cold bosonic atoms trapped in optical lattices formed by standing wave interference patterns of multiple laser beams constitute excellent emulators of models of strongly correlated quantum systems of bosons. In this thesis, we develop and deploy strong-coupling expansion (i.e., an expansion in terms of the ratio of the inter-site hopping amplitude of the bosons to the strength of their interactions) techniques for studying the properties of three different instances of such systems. In the first instance, we have used strong coupling expansion techniques to calculate the density pro le for bosonic atoms trapped in an optical lattice with an overall harmonic trap at finite temperatures and large on site interaction in the presence of super fluid regions. Our results match well with quantum Monte Carlo simulations at finite temperature. We present calculations for the entropy per particle as a function of temperature which can be used to calibrate the temperature in experiments. Our calculations for the scaled density in the vacuum-to-super fluid transition agree well with the experimental data for appropriate temperatures. We also discuss issues connected with the demonstration of universal quantum critical scaling in the experiments. Experimental realizations of “atomtronic" Josephson junctions have recently been created in annular traps in relative rotation with respect to potential barriers that generate the weak links. If these devices are additionally subjected to optical lattice potentials, then they can incorporate strong-coupling Mott physics within the design, which can modify the behaviour and can allow for interesting new configurations of system generated barriers and of super fluid ow patterns. we have examined theoretically the behavior of a Bose super fluid in an optical lattice in the presence of an annular trap and a barrier across the annular region which acts as a Josephson junction. As the fluid is rotated relative to the barrier, it generates circulating super-currents until, at larger speeds of rotation, it develops phase slips which are typically accompanied by vortices. We use a finite temperature strong-coupling expansion about the mean- held solution of the Bose Hubbard model to calculate various properties of the device. In addition, we discuss some of the rich behavior that can result when there are Mott regions within the system. Rubidium-Cesium dipolar molecule formation through Feshbach resonance is an area of great current interest, for, the dipolar molecules, once formed, interact via v long range dipolar forces, leading to possibilities of novel phases. Experimentalists currently make such systems mostly using trial and error, and the resulting efficiencies for molecule formation tend to be low. With a goal to assist cold-atom experimentalists to achieve higher e ciencies of molecule formation, we have estimated the trap parameters for Rb and Cs atoms in a 3D optical lattice required to create single occupancy per site Mott phase for both the species in the same regions of the trap. We thus identify the ne tuning of the external magnetic held near Rb-Cs Feshbach resonance required to achieve highest probability for creating single Rb-Cs Feshbach molecules in the system. We have used the Falicov-Kimball model to describe the relevant system and strong-coupling expansions about the mean- held solution to calculate the density pro les for both species and efficiency for molecule formation, determined by overlapping regions of single occupancy for both Rb and Cs, up to second order in the expansion. We also calculate the entropy per particle which serves as an estimation of the temperature in the experimental system

As software and hardware systems grow more complex and we begin to rely more on their correctness and reliability, it becomes exceedingly important to formally verify certain properties of these systems. If done naïvely, verifying a system can easily require exponentially more work than running it, in order to account for all possible executions. However, there are often symmetries or other properties of a system that can be exploited to reduce the amount of necessary work. In this thesis, we present a number of approaches that do this in the context of the CSP model checker FDR. CSP is named for Communicating Sequential Processes, or parallel combinations of state machines with synchronised communications. In the FDR model, the component processes are typically converted to explicit state machines while their parallel combination is evaluated lazily during model checking. Our contributions are motivated by this model but applicable to other models as well. We first address the scalability of the component machines by proposing a lazy compiler for a subset of CSP_M selected to model parameterised state machines. This is a typical case where the state space explosion can make model checking impractical, since the size of the state space is exponential in the number and size of the parameters. A lazy approach to evaluating these systems allows only the reachable subset of the state space to be explored. As an example, in studying security protocols, it is common to model an intruder parameterised by knowledge of each of a list of facts; even a relatively small 100 facts results in an intractable 2¹⁰⁰ states, but the rest of the system can ensure that only a small number of these states are reachable. Next, we address the scalability of the overall combination by presenting novel algorithms for bisimulation reduction with respect to strong bisimulation, divergence- respecting delay bisimulation, and divergence-respecting weak bisimulation. Since a parallel composition is related to the Cartesian product of its components, performing a relatively time-consuming bisimulation reduction on the components can reduce its size significantly; an efficient bisimulation algorithm is therefore very desirable. This thesis is motivated by practical implementations, and we discuss an implementation of each of the proposed algorithms in FDR. We thoroughly evaluate their performance and demonstrate their effectiveness.

Cette thèse s'intéresse à deux types de systèmes de différents degrés de liberté en interaction, et soumis à des fluctuations quantiques.Dans le premier projet abordé dans le manuscrit, on établit un diagramme de phase d'électrons en interactions dans un cristal bidimensionnel à géométrie kagome. Ce diagramme de phase est dressé en fonction de deux paramètres étant les interactions coulombiennes entre électrons sur un même atome pour le premier, et sur des atomes plus proches voisins pour le second. Les énergies caractéristiques de ces deux interactions sont quantifiées par rapport à une énergie de référence étant celle des fluctuations quantiques. On met alors en évidence quatre phases dont deux sont nouvelles, alors que les deux autres font le lien avec la littérature déjà existante, et sont en accord avec cette dernière. Ces deux nouvelles phases émergent lorsque l'énergie de répulsion coulombienne entre électrons sur un même atome domine devant l’énergie caractéristique des fluctuations quantiques. En présence d’une forte répulsion coulombienne entre électrons sur des atomes plus proches voisins, les charges électroniques ne peuvent se délocaliser pour former des ondes de Bloch et sont soumis à ce que l’on appelle une contrainte locale de charge. Apparaissent alors sous la compétition de ces deux interactions coulombiennes, des modes unidimensionnels collectifs le long des chaines d’atomes antiferromagnétiquement ordonnées. Ces modes ont la particularité d’être stabilisés à la fois par les fluctuations des degrés de liberté de spin, et de charge des électrons. La seconde de ces nouvelles phases émerge lorsque la répulsion coulombienne entre électrons sur des atomes voisins devient faible devant les fluctuations quantiques. La contrainte locale est alors relâchée et les électrons forment des ondes de Bloch le long de ce qui s’apparente à des bulles quantiques unidimensionnelles et polarisées en spin. Ces bulles sont alors piégées dans un cristal d’électrons inversement polarisés, avec lesquels elles sont en interaction antiferromagnétique.Le second projet porte sur l’étude d’un aimant moléculaire de Terbium Double-Decker. Cette molécule peut être modélisée par trois degrés de liberté interagissant en cascade les uns avec les autres. Le premier d’entre eux est un degré de liberté de spin nucléaire porté par le noyau de l’ion terbium de la molécule. Ce spin nucléaire est en interaction d’échange avec un degré de liberté de spin électronique porté par les électrons de l’ion terbium. Enfin, en première approximation, ce spin électronique génère un champ dipolaire auquel sont soumis les deux ligands de l’aimant moléculaire. Ces deux ligands sont couplés à deux électrodes de source et de drain, assurant le transport d’électrons uniques à travers ces deniers. Le tout forme donc un transistor à électron unique dans lequel les ligands servent de boîte quantique. Par mesure de magnéto-conductance, il est donc possible par une lecture en cascade, de remonter à l’état du spin électronique et du spin nucléaire. La première étape du projet a donc consisté à établir un modèle décrivant l’aimant moléculaire couplé à ces deux électrodes, afin de prédire les mesures de conductance réalisées au travers du transistor lors des thèses de Stefen Thiele et Clément Godfrin. Les résultats théoriques et expérimentaux obtenus sont en accord quantitatifs.D’autres part, à l’aide de champs électriques radio-fréquences, il est possible de manipuler expérimentalement et de façon cohérente le spin nucléaire. Cette manipulation cohérente du spin nucléaire se fait par l’intermédiaire du nuage électronique de l’ion, et permet ainsi d’être en mesure de réaliser un algorithme quantique sur le spin nucléaire de l’ion terbium. La réalisation d’un programme de simulation a permis de guider la réalisation expérimentale de l’algorithme de Grover, lequel a été implémenté avec succès au cours de la thèse de Clément Godfrin. / This thesis focuses on two different spin and charge systems, interacting under the effect of quantum fluctuations.The first project highlights the phase diagram of interacting electrons on a kagome lattice. This diagram is driven by two Coulomb repulsions. The first is a on site repulsion, and the second a nearest neighbor one. These two repulsions are in competition with quantum fluctuations of electronic charges. Four phases are depicted, two are unknown and the two other are in agreement with the literature. The two new phases are stabilized in the strong on site repulsion regime. When nearest neighbor repulsions are strong enough to induce a charge local constraint, the system enters in a so called Heisenberg-Loop Phase. These loops are antiferromagnetically arranged and can be described by a Heisenberg-like model in which both charge and spin play surprisingly a role in the exchange interaction. The second new phase is stabilized in the regime where nearest neighbor interactions are too weak to maintain the local constraint. Then, half of the electrons are delocalized in unidimensional Bloch states similar to quantum polarized electronic bubbles. These bubbles are trapped in an inversely polarized electronic cristal formed by the other electrons. This peculiar phase is favored by both quantum charge fluctuations in the bubbles, and antiferromagnetic exchanges between their electrons and the cristal ones.The second project deals with a Terbium Double-Decker molecular magnet. This molecule is modeled by three interacting degrees of freedom. The first is a nuclear spin of the Terbium ion, and the second is the electronic spin of this same ion. The two spins interact via a magnetic exchange.In a first approximation, the effect of the electronic spin is to induce a dipolar field. Finally, the last degree of freedom is carried by two ligands under the influence of the dipolar field. The ligands play the role of a read-out quantum dot, and by conductance measurements through this last one, we can probe the electronic spin and then, the nuclear spin. The first step of this project highlights the modeling of the global system. Then numerical computations are depicted and are in a quantitative agreement with the experimental measurements realized during the thesis of Stefan Thiele and Clément Godfrin.On the other hand, by applying electrical Radio Frequency Fields, we can drive quantum fluctuations on the nuclear spin. This quantum manipulation of the spin is realized by the dynamic deformation of the electron cloud under the effect of the Radio Frequency Field. As a result, we are able to implement a Grover Quantum Algorithm on the nuclear field. This thesis focuses on the realization of a simulation program that was a tool used by Clément Godfrin to successfully implement the Grover Algorithm.

Magister Artium - MA / Social networking sites are used on a daily basis, to communicate with friends we have known for quite some time as well as make new friends from all over the globe - a global phenomenon. According to Aparicio (2011) the use of social networking sites have given way to a new “social dynamic” where friendships are formed with individuals from diverse backgrounds and geographical locations. Larsen (2007) continues to state that social network sites make for the creation of new friendships and the maintenance of new relationships. Using social networking sites to develop relationships provides us with new social skills, but through constant use of these sites we lose valuable interpersonal skills learnt through the use of face to face interaction (Aparicio, 2011). This thesis investigated the notion of social networking sites, specifically focusing on interpersonal relationships and language use within the networking context. The social networking site in question is that of Twitter, as majority of existing studies in this area focuses on the more popular Facebook. The main objective was to determine whether social networking sites, specifically Twitter, influence the development and maintenance of interpersonal relationships and language use. Participants included a group of 11 males and 11 females (22 in total), ranging from the ages of 17 to 33 and from different geographical locations (e.g. United Kingdom, South Africa, Tokyo, and so on). These participants frequently keep in contact with each other. Four ways in which Twitter has been adapted to emulate face to face communication have been found, namely: (1) the use of Paralinguistic and Prosodic Features to imitate speech, (2) Ustream, although not prominent within the data collected, is used to make up for the lack of face to face communication.This, however, is one-way; only one user provides a video link while those communicating with him or her (as there can be more than one) would type messages, (3) as expected a variety of shortenings can be found within the data collected. Shortenings imitate speech among the younger generation, and (4) participants make use of an informal register, as the most common type of relationship found on Twitter is that of friendships. Both strong and weak ties exist in the collected data; with weak ties being the majority. It is possible for weak ties to become strong ties. All online relationships start off as weak and gradually, over time, become strong ties. This is done through establishing trust between participants and communicating on a regular basis. Paolillo (1999) found that online relationships manifests as both weak and strong. However, “online ties are not ‘branded’ as weak ties” and these ties differ in quality; “those who have regular contact have strong ties and those with less frequent contact have weaker ties weak”. Social support is evident in the collected data and possible in online, textbased communication. In is manifested in four types of support, namely: instrumental, emotional, informational, and appraisal. The most common type of support found in the collected data is that of informational support. With regards to support activation strategies, most tweet fall under the ‘ask’ and ‘cry’ types of strategies. Also, considering the amount of emoticons found in the data, little or no emoticons were found in the activation strategies. There are also more indirect support activation strategies as opposed to direct. This could possibly be due to the fact that majority of the ties are ‘in the weak stage’. Textese has not been adapted in anyway; the same elements used by texters and IMers are used by tweeters, such as initialisms, phonetic spellings and contractions. Although present in the Twitter data, elements of textese did not occur as frequently as that found in e.g. Bieswanger’s (2007) and Thurlow’s (2003) studies; however more elements of Twitter language was found. If anything, the characteristics of textese are well-suited for Twitter; as shortened forms of words would make it easier for users to maintain a character count below the imposed limitation and it promotes the idea of writing quick and concise messages instead of filling message space with irrelevant content. With regards to the difference in the way male and female participants use language in terms of the linguistic characteristics of textese and the language unique to Twitter and the use of paralinguistic and prosodic features, it can be said that females tend to use these characteristics more than males do.

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Title: Stochastic Differential Equations with Gaussian Noise Author: Josef Janák Department: Department of Probability and Mathematical Statistics Supervisor: Prof. RNDr. Bohdan Maslowski, DrSc., Department of Probability and Mathematical Statistics Abstract: Stochastic partial differential equations of second order with two un- known parameters are studied. The strongly continuous semigroup (S(t), t ≥ 0) for the hyperbolic system driven by Brownian motion is found as well as the formula for the covariance operator of the invariant measure Q (a,b) ∞ . Based on ergodicity, two suitable families of minimum contrast estimators are introduced and their strong consistency and asymptotic normality are proved. Moreover, another concept of estimation using "observation window" is studied, which leads to more families of strongly consistent estimators. Their properties and special cases are descibed as well as their asymptotic normality. The results are applied to the stochastic wave equation perturbed by Brownian noise and illustrated by several numerical simula- tions. Keywords: Stochastic hyperbolic equation, Ornstein-Uhlenbeck process, invariant measure, paramater estimation, strong consistency, asymptotic normality.

Title: Multivariate stochastic dominance and its application in portfolio optimization Problems Author: Barbora Petrová Department: Department of Probability and Mathematical Statistics Supervisor: doc. RNDr. Ing. Miloš Kopa, Ph.D., Department of Probability and Mathematical Statistics Abstract: This thesis discusses the concept of multivariate stochastic dominance, which serves as a tool for ordering random vectors, and its possible usage in dynamic portfolio optimization problems. We strictly focus on different types of the first-order multivariate stochastic dominance for which we describe their generators in the sense of von Neumann-Morgenstern utility functions. The first one, called strong multivariate stochastic dominance, is generated by all nondecreasing multivariate utility functions. The second one, called weak multivariate stochastic dominance, is defined by relation between survival functions, and the last one, called the first-order linear multivariate stochastic dominance, applies the first-order univariate stochastic dominance notion to linear combinations of marginals. We focus on the main characteristics of these types of stochastic dominance, their relationships as well as their relation to the cumulative and marginal distribution functions of considered random vectors. Formulated...