Partial reconstruction of the trajectories of a discretely observed branching diffusion with immigration and an application to inference

In this treatise we consider finite systems of branching particles where the particles move independently of each other according to d-dimensional diffusions. Particles are killed at a position dependent rate, leaving at their death position a random number of descendants according to a position dependent reproduction law. In addition particles immigrate at constant rate (one immigrant per immigration time). A process with above properties is called a branching diffusion withimmigration (BDI). In the first part we present the model in detail and discuss the properties of the BDI under our basic assumptions. In the second part we consider the problem of reconstruction of the trajectory of a BDI from discrete observations. We observe positions of the particles at discrete times; in particular we assume that we have no information about the pedigree of the particles. A natural question arises if we want to apply statistical procedures on the discrete observations: How can we find couples of particle positions which belong to the same particle? We give an easy to implement 'reconstruction scheme' which allows us to redraw or 'reconstruct' parts of the trajectory of the BDI with high accuracy. Moreover asymptotically the whole path can be reconstructed. Further we present simulations which show that our partial reconstruction rule is tractable in practice. In the third part we study how the partial reconstruction rule fits into statistical applications. As an extensive example we present a nonparametric estimator for the diffusion coefficient of a BDI where the particles move according to one-dimensional diffusions. This estimator is based on the Nadaraya-Watson estimator for the diffusion coefficient of one-dimensional diffusions and it uses the partial reconstruction rule developed in the second part above. We are able to prove a rate of convergence of this estimator and finally we present simulations which show that the estimator works well even if we leave our set of assumptions.

Chiral properties of two-flavour QCD at zero and non-zero temperature

Lattice Quantum Chromodynamics (LQCD) is the preferred tool for obtaining non-perturbative results from QCD in the low-energy regime. It has by nowrnentered the era in which high precision calculations for a number of phenomenologically relevant observables at the physical point, with dynamical quark degrees of freedom and controlled systematics, become feasible. Despite these successes there are still quantities where control of systematic effects is insufficient. The subject of this thesis is the exploration of the potential of todays state-of-the-art simulation algorithms for non-perturbativelyrn$\mathcal{O}(a)$-improved Wilson fermions to produce reliable results in thernchiral regime and at the physical point both for zero and non-zero temperature. Important in this context is the control over the chiral extrapolation. Thisrnthesis is concerned with two particular topics, namely the computation of hadronic form factors at zero temperature, and the properties of the phaserntransition in the chiral limit of two-flavour QCD.rnrnThe electromagnetic iso-vector form factor of the pion provides a platform to study systematic effects and the chiral extrapolation for observables connected to the structure of mesons (and baryons). Mesonic form factors are computationally simpler than their baryonic counterparts but share most of the systematic effects. This thesis contains a comprehensive study of the form factor in the regime of low momentum transfer $q^2$, where the form factor is connected to the charge radius of the pion. A particular emphasis is on the region very close to $q^2=0$ which has not been explored so far, neither in experiment nor in LQCD. The results for the form factor close the gap between the smallest spacelike $q^2$-value available so far and $q^2=0$, and reach an unprecedented accuracy at full control over the main systematic effects. This enables the model-independent extraction of the pion charge radius. The results for the form factor and the charge radius are used to test chiral perturbation theory ($\chi$PT) and are thereby extrapolated to the physical point and the continuum. The final result in units of the hadronic radius $r_0$ is rn$$ \left\langle r_\pi^2 \right\rangle^{\rm phys}/r_0^2 = 1.87 \: \left(^{+12}_{-10}\right)\left(^{+\:4}_{-15}\right) \quad \textnormal{or} \quad \left\langle r_\pi^2 \right\rangle^{\rm phys} = 0.473 \: \left(^{+30}_{-26}\right)\left(^{+10}_{-38}\right)(10) \: \textnormal{fm} \;, $$rn which agrees well with the results from other measurements in LQCD and experiment. Note, that this is the first continuum extrapolated result for the charge radius from LQCD which has been extracted from measurements of the form factor in the region of small $q^2$.rnrnThe order of the phase transition in the chiral limit of two-flavour QCD and the associated transition temperature are the last unkown features of the phase diagram at zero chemical potential. The two possible scenarios are a second order transition in the $O(4)$-universality class or a first order transition. Since direct simulations in the chiral limit are not possible the transition can only be investigated by simulating at non-zero quark mass with a subsequent chiral extrapolation, guided by the universal scaling in the vicinity of the critical point. The thesis presents the setup and first results from a study on this topic. The study provides the ideal platform to test the potential and limits of todays simulation algorithms at finite temperature. The results from a first scan at a constant zero-temperature pion mass of about 290~MeV are promising, and it appears that simulations down to physical quark masses are feasible. Of particular relevance for the order of the chiral transition is the strength of the anomalous breaking of the $U_A(1)$ symmetry at the transition point. It can be studied by looking at the degeneracies of the correlation functions in scalar and pseudoscalar channels. For the temperature scan reported in this thesis the breaking is still pronounced in the transition region and the symmetry becomes effectively restored only above $1.16\:T_C$. The thesis also provides an extensive outline of research perspectives and includes a generalisation of the standard multi-histogram method to explicitly $\beta$-dependent fermion actions.

Molecular genetic investigation of the Neolithic population history in the western Carpathian Basin

Der Fokus dieser Dissertation ist die populationsgenetische Analyse der neolithischen Bevölkerungswechsel in den 6.-5. Jahrtausende vor Christus, die im westlichen Karpatenbecken stattfanden. Die Zielsetzung der Studie war, mittels der Analyse von mitochondrialer und Y-chromosomaler aDNA, den Genpool der sechs neolithischen und kupferzeitlichen Populationen zu untersuchen und die daraus resultierenden Ergebnisse mit anderen prähistorischen und modernen genetischen Daten zu vergleichen.rnInsgesamt wurden 323 Individuen aus 32 ungarischen, kroatischen und slowakischen Fundplätzen beprobt und bearbeitet in den archäogenetischen Laboren der Johannes Gutenberg-Universität in Mainz. Die DNA Ergebnisse wurden mit verschiedenen populationsgenetischen Methoden ausgewertet. Vergleichsdaten von prähistorischen und modernen eurasiatischen Populationen wurden dazu gesammelt.rnDie HVS-I Region der mitochondrialen DNA konnten bei 256 Individuen reproduziert und authentifiziert werden (mit einer Erfolgsrate von 85.9%). Die Typisierung der HVS-II Region war in 80 Fällen erfolgreich. Testend alle gut erhaltene Proben, die Y-chromosomale Haplogruppe konnte in 33 männlichen Individuen typisiert werden.rnDie neolithischen, mitochondrialen Haplogruppen deuten auf eine hohe Variabilität des maternalen Genpools hin. Sowohl die mitochondrialen als auch die Y-chromosomalen Daten lassen Rückschlüsse auf eine nah-östliche bzw. südwestasiatische Herkunft der frühen Bauern zu. Die Starčevo- und linearbandkermaischen-Populationen in westlichem Karpatenbecken (letztere abgekürzt als LBKT) und die linearbandkermaischen-Population in Mitteleuropa (LBK) haben so starke genetische Ähnlichkeit, dass die Verbreitung der LBK nach Mitteleuropa mit vorangegangenen Wanderungsereignissen zu erklären ist. Die Transdanubische aDNA Daten zeigen hohe Affinität zu den publizierten prähistorischen aDNA Datensätzen von Mitteleuropa aus den 6.-4. Jahrtausende vor Chr. Die maternal-genetische Variabilität der Starčevo-Population konnte auch innerhalb der nachfolgenden Populationen Transdanubiens festgestellt werden. Nur kleinere Infiltrationen und Immigrationsereignissen konnten während der Vinča-, LBKT-, Sopot- und Balaton-Lasinja-Kultur in Transdanubien identifiziert werden. Zwischen den transdanubischen Regionen konnten mögliche genetische Unterschiede nur in der LBKT und in der Lengyel-Periode beobachtet werden, als sich die nördlichen Gruppen von den südlichen Populationen trennten. rnDie festgestellte Heterogenität der mtDNA in Zusammenhang mit der Y-chromosomalen Homogenität in den Starčevo- und LBK-Populationen, weisen auf patrilokale Residenzregeln und patrilineare Abstammungsregeln in den ersten Bauergemeinschaften hin. rnObwohl die hier präsentierten Daten einen großen Fortschritt in der Forschung von aDNA und Neolithikum des Karpatenbeckens und Mitteleuropas bedeuten, werfen sie auch mehrere Fragen auf, deren Beantwortung durch zukünftige Genomforschungen erbracht werden könnte.