TY - THES
AB - This thesis is concerned with the inference of current population structure based on geo-referenced genetic data. The underlying idea is that population structure affects its spatial genetic structure. Therefore, genotype information can be utilized to estimate important demographic parameters such as migration rates. These indirect estimates of population structure have become very attractive, as genotype data is now widely available. However, there also has been much concern about these approaches. Importantly, genetic structure can be influenced by many complex patterns, which often cannot be disentangled. Moreover, many methods merely fit heuristic patterns of genetic structure, and do not build upon population genetics theory. Here, I describe two novel inference methods that address these shortcomings. In Chapter 2, I introduce an inference scheme based on a new type of signal, identity by descent (IBD) blocks. Recently, it has become feasible to detect such long blocks of genome shared between pairs of samples. These blocks are direct traces of recent coalescence events. As such, they contain ample signal for inferring recent demography. I examine sharing of IBD blocks in two-dimensional populations with local migration. Using a diffusion approximation, I derive formulas for an isolation by distance pattern of long IBD blocks and show that sharing of long IBD blocks approaches rapid exponential decay for growing sample distance. I describe an inference scheme based on these results. It can robustly estimate the dispersal rate and population density, which is demonstrated on simulated data. I also show an application to estimate mean migration and the rate of recent population growth within Eastern Europe. Chapter 3 is about a novel method to estimate barriers to gene flow in a two dimensional population. This inference scheme utilizes geographically localized allele frequency fluctuations - a classical isolation by distance signal. The strength of these local fluctuations increases on average next to a barrier, and there is less correlation across it. I again use a framework of diffusion of ancestral lineages to model this effect, and provide an efficient numerical implementation to fit the results to geo-referenced biallelic SNP data. This inference scheme is able to robustly estimate strong barriers to gene flow, as tests on simulated data confirm.
AU - Ringbauer, Harald
ID - 200
TI - Inferring recent demography from spatial genetic structure
ER -
TY - THES
AB - We describe arrangements of three-dimensional spheres from a geometrical and topological point of view. Real data (fitting this setup) often consist of soft spheres which show certain degree of deformation while strongly packing against each other. In this context, we answer the following questions: If we model a soft packing of spheres by hard spheres that are allowed to overlap, can we measure the volume in the overlapped areas? Can we be more specific about the overlap volume, i.e. quantify how much volume is there covered exactly twice, three times, or k times? What would be a good optimization criteria that rule the arrangement of soft spheres while making a good use of the available space? Fixing a particular criterion, what would be the optimal sphere configuration? The first result of this thesis are short formulas for the computation of volumes covered by at least k of the balls. The formulas exploit information contained in the order-k Voronoi diagrams and its closely related Level-k complex. The used complexes lead to a natural generalization into poset diagrams, a theoretical formalism that contains the order-k and degree-k diagrams as special cases. In parallel, we define different criteria to determine what could be considered an optimal arrangement from a geometrical point of view. Fixing a criterion, we find optimal soft packing configurations in 2D and 3D where the ball centers lie on a lattice. As a last step, we use tools from computational topology on real physical data, to show the potentials of higher-order diagrams in the description of melting crystals. The results of the experiments leaves us with an open window to apply the theories developed in this thesis in real applications.
AU - Iglesias Ham, Mabel
ID - 201
TI - Multiple covers with balls
ER -
TY - THES
AB - Nowadays, quantum computation is receiving more and more attention as an alternative to the classical way of computing. For realizing a quantum computer, different devices are investigated as potential quantum bits. In this thesis, the focus is on Ge hut wires, which turned out to be promising candidates for implementing hole spin quantum bits. The advantages of Ge as a material system are the low hyperfine interaction for holes and the strong spin orbit coupling, as well as the compatibility with the highly developed CMOS processes in industry. In addition, Ge can also be isotopically purified which is expected to boost the spin coherence times. The strong spin orbit interaction for holes in Ge on the one hand enables the full electrical control of the quantum bit and on the other hand should allow short spin manipulation times. Starting with a bare Si wafer, this work covers the entire process reaching from growth over the fabrication and characterization of hut wire devices up to the demonstration of hole spin resonance. From experiments with single quantum dots, a large g-factor anisotropy between the in-plane and the out-of-plane direction was found. A comparison to a theoretical model unveiled the heavy-hole character of the lowest energy states. The second part of the thesis addresses double quantum dot devices, which were realized by adding two gate electrodes to a hut wire. In such devices, Pauli spin blockade was observed, which can serve as a read-out mechanism for spin quantum bits. Applying oscillating electric fields in spin blockade allowed the demonstration of continuous spin rotations and the extraction of a lower bound for the spin dephasing time. Despite the strong spin orbit coupling in Ge, the obtained value for the dephasing time is comparable to what has been recently reported for holes in Si. All in all, the presented results point out the high potential of Ge hut wires as a platform for long-lived, fast and fully electrically tunable hole spin quantum bits.
AU - Watzinger, Hannes
ID - 49
TI - Ge hut wires - from growth to hole spin resonance
ER -
TY - THES
AB - In this thesis we will discuss systems of point interacting fermions, their stability and other spectral properties. Whereas for bosons a point interacting system is always unstable this ques- tion is more subtle for a gas of two species of fermions. In particular the answer depends on the mass ratio between these two species. Most of this work will be focused on the N + M model which consists of two species of fermions with N, M particles respectively which interact via point interactions. We will introduce this model using a formal limit and discuss the N + 1 system in more detail. In particular, we will show that for mass ratios above a critical one, which does not depend on the particle number, the N + 1 system is stable. In the context of this model we will prove rigorous versions of Tan relations which relate various quantities of the point-interacting model. By restricting the N + 1 system to a box we define a finite density model with point in- teractions. In the context of this system we will discuss the energy change when introducing a point-interacting impurity into a system of non-interacting fermions. We will see that this change in energy is bounded independently of the particle number and in particular the bound only depends on the density and the scattering length. As another special case of the N + M model we will show stability of the 2 + 2 model for mass ratios in an interval around one. Further we will investigate a different model of point interactions which was discussed before in the literature and which is, contrary to the N + M model, not given by a limiting procedure but is based on a Dirichlet form. We will show that this system behaves trivially in the thermodynamic limit, i.e. the free energy per particle is the same as the one of the non-interacting system.
AU - Moser, Thomas
ID - 52
TI - Point interactions in systems of fermions
ER -
TY - THES
AB - The most common assumption made in statistical learning theory is the assumption of the independent and identically distributed (i.i.d.) data. While being very convenient mathematically, it is often very clearly violated in practice. This disparity between the machine learning theory and applications underlies a growing demand in the development of algorithms that learn from dependent data and theory that can provide generalization guarantees similar to the independent situations. This thesis is dedicated to two variants of dependencies that can arise in practice. One is a dependence on the level of samples in a single learning task. Another dependency type arises in the multi-task setting when the tasks are dependent on each other even though the data for them can be i.i.d. In both cases we model the data (samples or tasks) as stochastic processes and introduce new algorithms for both settings that take into account and exploit the resulting dependencies. We prove the theoretical guarantees on the performance of the introduced algorithms under different evaluation criteria and, in addition, we compliment the theoretical study by the empirical one, where we evaluate some of the algorithms on two real world datasets to highlight their practical applicability.
AU - Zimin, Alexander
ID - 68
TI - Learning from dependent data
ER -