Cette thèse présente des résultats nouveaux dans deux sujets de la physique statistique du désordre: les modèles aux energies aléatoires logarithmiquement corrélées (logREMs), et la transition de localisation dans les matrices aléatoires à longues portées.Dans la première partie consacrée aux logREMs, nous montrons comment décrire leurs points communs et les données spécifiques aux modèles particuliers. Ensuite nous appliquons la méthode de la brisure de symétrie des répliques pour les étudier en general, et en déduirons la transition vitreuse et le processus des minima, en termes de processus de Poisson décorés. Nous présentons également une série d'application des polynômes de Jack à la prédiction exactes des observables dans le modèle circulaire et ses variants. Finalement, nous décrivons les progrès récents sur la connexion exacte entre les logREMs et la théorie conforme de Liouville.La seconde partie a pour but d'introduire une nouvelle classe de matrices aléatoires à bandes, dite la classe des distributions larges; elle ressemble essentiellement aux matrices creuses. Nous étudions d'abord un modèle particulier de la classe, les matrices aléatoires Bêta, qui sont inspirées par une correspondence exacte à un modèle statistique récemment étudié, celui de la dynamique épidémique. A l'aide des arguments analytiques appuyés sur la correspondence et des simulations numériques, nous montrons l'existence des transitions de localisation avec des valeurs propres critiques dans le régime des paramètres dit d'exponentielle étirée. Ensuite, en utilisant une approche de renormalisation et de diagonalisation par blocs, nous soutenons que les transitions de localisation sont en général présentes dans la class des distributions larges. / This thesis presents original results in two domains of disordered statistical physics: logarithmic correlated Random Energy Models (logREMs), and localization transition in long-range random matrices.In the first part devoted to logREMs, we show how to characterise their common properties and model--specific data. Then we develop their replica symmetry breaking treatment, which leads to the freezing scenario of their free energy distribution and the general description of their minima process, in terms of decorated Poisson point process. We also report a series of new applications of the Jack polynomials in the exact predictions of some observables in the circular model and its variants. Finally, we present the recent progress on the exact connection between logREMs and the Liouville conformal field theory.The goal of the second part is to introduce and study a new class of banded random matrices, the broadly distributed class, which is characterid an effective sparseness. We will first study a specific model of the class, the Beta Banded random matrices, inspired by an exact mapping to a recently studied statistical model of long--range first--passage percolation/epidemics dynamics. Using analytical arguments based on the mapping and numerics, we show the existence of localisation transitions with mobility edges in the ``stretch--exponential'' parameter--regime of the statistical models. Then, using a block--diagonalization renormalization approach, we argue that such localization transitions occur generically in the broadly distributed class.

Le but à long terme de mon travail de recherche est de trouver une alternative viable à la méthode conforme, qui nous permettrait de mieux comprendre la structure géométrique de l'espace des solutions des équations de contrainte. L'avantage du modèle de Maxwell (the drift model) par rapport aux modèles plus classiques est la présence des paramètres supplémentaires. Le prix à payer, par contre, sera que la complexité analytique du système correspondant. Ma thèse a été structuré en deux parties : a. Existence sous la condition de petitesse des données initiales. Nous avons montré que le système de Maxwell est raisonnable dans le sens où nous pouvons le résoudre, malgré sa forte nonliniarité, sous des conditions de petitesse sur ses coefficients, en dimension 3, 4 et 5. Par conséquent, l'ensemble des solutions est non-vide. b. Stabilité Nous montrons la stabilité des solutions du système: ce résultat est obtenu en dimension 3,4 et 5, dans le cas où la métrique est conformément plate, et le drift et petit / The long-term goal of my work is to find a viable alternative to the conformal method, which would allow us to better understand the geometry of the space of solutions of the constraint equations. The advantage of Maxwell's model (the drift model) is the presence of additional parameters. Its downside, however, is that it proves to be much more difficult from an analytic standpoint. My thesis is structued in two parts: a. Existence under suitable smallness conditions. We show that Maxwell's system is sufficiently reasonable: it can be solved even given the presence of focusing non linearities. We prove this under smallness conditions of its coefficients, and in dimensions 3,4 and 5. An immediate consequence is that the set of solutions is non-empty. b. Stability. We verify that the solutions of the system are stable: this result holds in dimensions 3,4 and 5, when the metric is conformally flat and the drift is small

Conformal geometric algebra is defined in the thesis. Representations of geometric objects and possibilities of their geometric transformations are presented. Conformal geometric algebra is applied to the calculation of forward kinematics of a robotic manipulator UR10 from Universal Robots. It is also applied to determine the position of the machine based on the location and rotation of two cameras. Then it is used in an inverse task, where based on records from the two cameras, dimensions of the UR10 manipulator and possibilities of its movement, the mutual position of these cameras is determined. And consequently the possibilities of their location in space. Finally, the derived procedures are implemented in a custom program created in the CluCalc environment, using which a sample example verifying the correctness of these procedures is calculated.

We introduce the formalism of categorical symmetries, and examine how these appear in quantum field theories. We discuss rational conformal field theories and their Verlinde lines, with the critical Ising model as an example. We introduce Wess Zumino Witten models and affine Lie algebras. An algorithm for the fusion rules is presented. We use bosonization to realise two dimensional adjoint SU(N) QCD as a WZW coset model plus a kinetic term for the gauge field. We argue that the infrared theory has degenerate vacua acted upon by a non-negative integer valued matrix representation of a categorical symmetry. We compute generators for these matrices for gauge groups SU(3) and SU(4).

The diploma thesis deals with the analysis of the effects surface texturing on starved contact. Experimental verification was realized on apparatus simulating contact between a steel ball and glass disc using colorimetric interferometry and high-speed camera. The work deals with lubrication regimes, problems with starved contacts and surface texturing which can reduce effects of starvation on non-conformal surface contacts.

<div>As small satellites are increasingly used in the space industry, creative solutions for the use of their limited volume will be required. Conformal tanks are one idea to better make use of this volume. These tanks are non-traditionally shaped and non-axisymmetric. Because slosh can have detrimental effects on a spacecraft, it should be understood. However, slosh in these more complicated geometries has not been thoroughly investigated in the past.</div><div><br></div><div>This research looks at slosh within six geometries, five of which are conformal tanks. These geometries are evaluated in both an experiment and using CFD simulations. It was found that the total slosh motion appears to be the sum of slosh behavior along each dimension. Slosh along a line of symmetry will have center of mass movement that stays along that line. Slosh off the line of symmetry will deviate from that line unless slosh frequency is the same in each direction.</div>

Following Exner et al. (Commun. Math. Phys. 26 (2014), no. 2, 531–541), we prove new Lieb-Thirring inequalities for a general class of self-adjoint, second order differential operators with matrix-valued potentials, acting in one space-dimension. This class contains, but is not restricted to, the magnetic and non-magnetic Schrödinger operators. We consider the three cases of functions defined on all reals, all positive reals, and an interval, respectively, and acquire three different kinds of bounds. We also investigate the spectral properties of a family of operators from conformal field theory, by proving an asymptotic phase-space bound on the eigenvalue counting function and establishing a number of spectral inequalities. These bound the Riesz-means of eigenvalues for these operators, together with each individual eigenvalue, and are applied to a few physically interesting examples. / Vi följer Exner et al. (Commun. Math. Phys. 26 (2014), nr. 2, 531–541) och bevisar nya Lieb-Thirring-olikheter för generella, andra gradens självadjungerade differentialoperatorer med matrisvärda potentialfunktioner, verkandes i en rumsdimension. Dessa innefattar och generaliserar de magnetiska och icke-magnetiska Schrödingeroperatorerna. Vi betraktar tre olika fall, med funktioner definierade på hela reella axeln, på den positiva reella axeln, samt på ett interval. Detta resulterar i tre sorters olikheter.  Vidare undersöker vi spektralegenskaperna för en klass operatorer från konform fältteori, genom att asymptotiskt begränsa antalet egenvärden med ett fasrymdsuttryck, samt genom att bevisa ett antal spektralolikheter. Dessa begränsar Riesz-medelvärdena för operatorerna, samt varje enskilt egenvärde, och tillämpas på ett par fysikaliskt intressanta exempel.

Microscopy imaging based techniques, such as the Cell Painting assay, could be used to generate images that visualize the Mechanism of Action (MoA) of a drug, which could be of great use in drug development. In order to extract information and predict the MoA of a new compound from these images we need powerful image analysis tools. The purpose with this project is to further develop a Deep Learning model to predict the MoA of different drugs from Cell Painting images using Convolutional Neural Networks (CNNs) and Conformal Prediction. The specific task was to compare the accuracy of different CNN architectures and to compare the efficiency of different nonconformity functions.  During the project the CNN architectures ResNet50, ResNet101 and DenseNet121 were compared as well as the nonconformity functions Inverse Probability, Margin and a combination of them both. No significant difference in accuracy between the CNNs and no difference in efficiency between the nonconformity functions was measured. The results showed that the model could predict the MoA of a compound with high accuracy when all compounds were used both in training, validation and test of the model, which validates the implementations. However, it is desirable for the model to be able to predict the MoA of a new compound if the model has been trained on other compounds with the same MoA. This could not be confirmed through this project and the model needs to be further investigated and tested with another dataset in order to be used for that purpose.

No description available.

<p>This dissertation consists of two parts. In the first part, we discuss the boundary and domain wall theories of the generalized quantum double lattice realization of the two-dimensional topological orders based on Hopf algebras. The boundary Hamiltonian and domain wall Hamiltonian are constructed by using Hopf algebra pairings and generalized quantum double. The algebraic data behind the gapped boundary and domain wall are comodule algebras and bicomodule algebras, respectively. The topological excitations in the boundary and domain wall are classified by bimodules over these algebras. Finally, via the Hopf tensor network representation of the quantum many-body states, we solve the ground state of the model in the presence of the boundary and domain wall.</p> <p><br></p> <p>In the second part, we introduce the weak Hopf algebra extension of symmetry, which arises naturally in anyonic quantum systems, and we establish weak Hopf symmetry breaking theory based on the fusion closed set of anyons. We present a thorough investigation of the quantum double model based on weak Hopf algebras, including the topological excitations and ribbon operators, and show that the vacuum sector of the model has weak Hopf symmetry. The gapped boundary and domain wall theories are also established. We show that the gapped boundary is algebraically determined by a comodule algebra, or equivalently, a module algebra, and the gapped domain wall is determined by the bicomodule algebra, or equivalently, a bimodule algebra. We also introduce the weak Hopf tensor network states, by which we solve the weak Hopf quantum double models on closed and open surfaces. Lastly, we discuss the duality of the quantum double phases.</p>