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- On the geometry, topology and approximation of amoebas (2013)
- We investigate multivariate Laurent polynomials f \in \C[\mathbf{z}^{\pm 1}] = \C[z_1^{\pm 1},\ldots,z_n^{\pm 1}] with varieties \mathcal{V}(f) restricted to the algebraic torus (\C^*)^n = (\C \setminus \{0\})^n. For such Laurent polynomials f one defines the amoeba \mathcal{A}(f) of f as the image of the variety \mathcal{V}(f) under the \Log-map \Log : (\C^*)^n \to \R^n, (z_1,\ldots,z_n) \mapsto (\log|z_1|, \ldots, \log|z_n|). I.e., the amoeba \mathcal{A}(f) is the projection of the variety \mathcal{V}(f) on its (componentwise logarithmized) absolute values. Amoebas were first defined in 1994 by Gelfand, Kapranov and Zelevinksy. Amoeba theory has been strongly developed since the beginning of the new century. It is related to various mathematical subjects, e.g., complex analysis or real algebraic curves. In particular, amoeba theory can be understood as a natural connection between algebraic and tropical geometry. In this thesis we investigate the geometry, topology and methods for the approximation of amoebas. Let \C^A denote the space of all Laurent polynomials with a given, finite support set A \subset \Z^n and coefficients in \C^*. It is well known that, in general, the existence of specific complement components of the amoebas \mathcal{A}(f) for f \in \C^A depends on the choice of coefficients of f. One prominent key problem is to provide bounds on the coefficients in order to guarantee the existence of certain complement components. A second key problem is the question whether the set U_\alpha^A \subseteq \C^A of all polynomials whose amoeba has a complement component of order \alpha \in \conv(A) \cap \Z^n is always connected. We prove such (upper and lower) bounds for multivariate Laurent polynomials supported on a circuit. If the support set A \subset \Z^n satisfies some additional barycentric condition, we can even give an exact description of the particular sets U_\alpha^A and, especially, prove that they are path-connected. For the univariate case of polynomials supported on a circuit, i.e., trinomials f = z^{s+t} + p z^t + q (with p,q \in \C^*), we show that a couple of classical questions from the late 19th / early 20th century regarding the connection between the coefficients and the roots of trinomials can be traced back to questions in amoeba theory. This yields nice geometrical and topological counterparts for classical algebraic results. We show for example that a trinomial has a root of a certain, given modulus if and only if the coefficient p is located on a particular hypotrochoid curve. Furthermore, there exist two roots with the same modulus if and only if the coefficient p is located on a particular 1-fan. This local description of the configuration space \C^A yields in particular that all sets U_\alpha^A for \alpha \in \{0,1,\ldots,s+t\} \setminus \{t\} are connected but not simply connected. We show that for a given lattice polytope P the set of all configuration spaces \C^A of amoebas with \conv(A) = P is a boolean lattice with respect to some order relation \sqsubseteq induced by the set theoretic order relation \subseteq. This boolean lattice turns out to have some nice structural properties and gives in particular an independent motivation for Passare's and Rullgard's conjecture about solidness of amoebas of maximally sparse polynomials. We prove this conjecture for special instances of support sets. A further key problem in the theory of amoebas is the description of their boundaries. Obviously, every boundary point \mathbf{w} \in \partial \mathcal{A}(f) is the image of a critical point under the \Log-map (where \mathcal{V}(f) is supposed to be non-singular here). Mikhalkin showed that this is equivalent to the fact that there exists a point in the intersection of the variety \mathcal{V}(f) and the fiber \F_{\mathbf{w}} of \mathbf{w} (w.r.t. the \Log-map), which has a (projective) real image under the logarithmic Gauss map. We strengthen this result by showing that a point \mathbf{w} may only be contained in the boundary of \mathcal{A}(f), if every point in the intersection of \mathcal{V}(f) and \F_{\mathbf{w}} has a (projective) real image under the logarithmic Gauss map. With respect to the approximation of amoebas one is in particular interested in deciding membership, i.e., whether a given point \mathbf{w} \in \R^n is contained in a given amoeba \mathcal{A}(f). We show that this problem can be traced back to a semidefinite optimization problem (SDP), basically via usage of the Real Nullstellensatz. This SDP can be implemented and solved with standard software (we use SOSTools and SeDuMi here). As main theoretic result we show that, from the complexity point of view, our approach is at least as good as Purbhoo's approximation process (which is state of the art).

- A stochastic model for the joint evaluation of burstiness and regularity in oscillatory spike trains (2013)
- The thesis provides a stochastic model to quantify and classify neuronal firing patterns of oscillatory spike trains. A spike train is a finite sequence of time points at which a neuron has an electric discharge (spike) which is recorded over a finite time interval. In this work, these spike times are analyzed regarding special firing patterns like the presence or absence of oscillatory activity and clusters (so called bursts). These bursts do not have a clear and unique definition in the literature. They are often fired in response to behaviorally relevant stimuli, e.g., an unexpected reward or a novel stimulus, but may also appear spontaneously. Oscillatory activity has been found to be related to complex information processing such as feature binding or figure ground segregation in the visual cortex. Thus, in the context of neurophysiology, it is important to quantify and classify these firing patterns and their change under certain experimental conditions like pharmacological treatment or genetical manipulation. In neuroscientific practice, the classification is often done by visual inspection criteria without giving reproducible results. Furthermore, descriptive methods are used for the quantification of spike trains without relating the extracted measures to properties of the underlying processes. For that reason, a doubly stochastic point process model is proposed and termed 'Gaussian Locking to a free Oscillator' - GLO. The model has been developed on the basis of empirical observations in dopaminergic neurons and in cooperation with neurophysiologists. The GLO model uses as a first stage an unobservable oscillatory background rhythm which is represented by a stationary random walk whose increments are normally distributed. Two different model types are used to describe single spike firing or clusters of spikes. For both model types, the distribution of the random number of spikes per beat has different probability distributions (Bernoulli in the single spike case or Poisson in the cluster case). In the second stage, the random spike times are placed around their birth beat according to a normal distribution. These spike times represent the observed point process which has five easily interpretable parameters to describe the regularity and the burstiness of the firing patterns. It turns out that the point process is stationary, simple and ergodic. It can be characterized as a cluster process and for the bursty firing mode as a Cox process. Furthermore, the distribution of the waiting times between spikes can be derived for some parameter combination. The conditional intensity function of the point process is derived which is also called autocorrelation function (ACF) in the neuroscience literature. This function arises by conditioning on a spike at time zero and measures the intensity of spikes x time units later. The autocorrelation histogram (ACH) is an estimate for the ACF. The parameters of the GLO are estimated by fitting the ACF to the ACH with a nonlinear least squares algorithm. This is a common procedure in neuroscientific practice and has the advantage that the GLO ACF can be computed for all parameter combinations and that its properties are closely related to the burstiness and regularity of the process. The precision of estimation is investigated for different scenarios using Monte-Carlo simulations and bootstrap methods. The GLO provides the neuroscientist with objective and reproducible classification rules for the firing patterns on the basis of the model ACF. These rules are inspired by visual inspection criteria often used in neuroscientific practice and thus support and complement usual analysis of empirical spike trains. When applied to a sample data set, the model is able to detect significant changes in the regularity and burst behavior of the cells and provides confidence intervals for the parameter estimates.

- Algebraische und konforme Quantenfeldtheorie (1999)
- Verbesserte Auflage der im Logos-Verlag, Berlin, 1997 mit der ISBN 3-931216-49-7 erschienenen Diplomarbeit. Version vom 5 August 1999.

- Yule- und Kontaktprozess auf Small World Graphen (2008)
- Mit den Small World Graphen stehen seit Ende der Neunzigerjahre Modelle für soziale und ähnliche Netzwerke, die im Vergleich zu Erdös-Rényi-Graphen stärker Cluster ausbilden, zur Verfügung. Wir betrachten die Konstruktion dieser Graphen und untersuchen zwei der Modelle genauer im Zusammenhang mit stochastischen Prozessen. Das stetige Modell betrachten wir hinsichtlich dem Abstand zweier Knoten. Der interessanteste Aspekt hierbei ist, dass man bei der Konstruktion des Graphen die entfernten Nachbarn mithilfe der Poissonverteilung wählt und in der Folge einen Yule-Prozess auf dem Graphen erhält. Auf der Bollobás-Chung Small World lassen wir den Kontaktprozess ablaufen und untersuchen diesen bezüglich seiner Überlebenswahrscheinlichkeit. Wir sehen, dass er auf diesem Graphen zwei Phasenübergänge aufweist. Oberhalb des ersten überlebt er für immer mit positiver Wahrscheinlichkeit, oberhalb des zweiten ist zudem der Knoten, auf dem der Kontaktprozess gestartet ist, stets mit positiver Wahrscheinlichkeit infiziert. Schließlich betrachten wir die Zeitdauer, die ein leicht modifizierter, superkritischer Kontaktprozess auf der Small World unter bestimmten Voraussetzungen überlebt. Die wesentliche Dynamik, die wir hierbei ausmachen können, ist, dass auf ein Absinken der Infektionen mit hoher Wahrscheinlichkeit wieder eine Verdopplung der Infektionen folgt.

- Detecting rate changes in spike trains (2010)
- Neuronal activity in the brain is often investigated in the presence of stimuli, termed externally driven activity. This stimulus-response-perspective has long been focussed on in order to find out how the nervous system responds to different stimuli. The neuronal response consists of baseline activity, so called spontaneous activity1, and activity which is caused by the stimulus. The baseline activity is often considered as constant over time which allows the identification of the stimulus-evoked part of the neuronal response by averaging over a set of trials. However, during the last years it has been recognized that own dynamics of the nervous system plays an important role in information processing. As a consequence, spontaneous activity is no longer regarded only as background ’noise’ and its role in cortical processing is reconsidered. Therefore, the study of spontaneous firing pattern gains more importance as these patterns may shape neuronal responses to a larger extent as previously thought. For example, recent findings suggest that prestimulus activity can predict a person’s visual perception performance on a single trial basis (Hanslmayr et al., 2007). In this context, Ringach (2009) remarks that one can learn much about even the quiescent state of the brain which “underlies the importance of understanding cortical responses as the fusion of ongoing activity and sensory input”. Taking into account that spontaneous activity reflects anything else but noise, new challenges arise when analysing neuronal data. In this thesis one of these problems related to the analysis of neuronal activity will be adressed, namely the nonstationarity of firing rates. The present work consists of four chapters. First of all the introduction gives neurophysiological background information to get an idea of neuronal information processing. Afterwords the theory of point processes is provided which forms the basis for modeling neuronal spiking data. In the last section of the introduction a statement of the problem is given. Chapter 2 proposes an easily applicable statistical method for the detection of nonstationarity. It is applied to simulations and to real data in order to show its capabilities. Thereafter, four other approaches are presented which provide useful illustrations concerning the nonstationarity of the firing rate but share the problem that one cannot make objective statements on the basis of their results. They were developed in the course of establishing a suitable method. In chapter 4 the results are discussed and suggestions for further study are given.

- Entropy increase in switching systems (2013)
- The relation between the complexity of a time-switched dynamics and the complexity of its control sequence depends critically on the concept of a non-autonomous pullback attractor. For instance, the switched dynamics associated with scalar dissipative affine maps has a pullback attractor consisting of singleton component sets. This entails that the complexity of the control sequence and switched dynamics, as quantified by the topological entropy, coincide. In this paper we extend the previous framework to pullback attractors with nontrivial components sets in order to gain further insights in that relation. This calls, in particular, for distinguishing two distinct contributions to the complexity of the switched dynamics. One proceeds from trajectory segments connecting different component sets of the attractor; the other contribution proceeds from trajectory segments within the component sets. We call them “macroscopic” and “microscopic” complexity, respectively, because only the first one can be measured by our analytical tools. As a result of this picture, we obtain sufficient conditions for a switching system to be more complex than its unswitched subsystems, i.e., a complexity analogue of Parrondo’s paradox.

- Pólya urns via the contraction method (2013)
- In this thesis, the asymptotic behaviour of Pólya urn models is analyzed, using an approach based on the contraction method. For this, a combinatorial discrete time embedding of the evolution of the composition of the urn into random rooted trees is used. The recursive structure of the trees is used to study the asymptotic behavior using ideas from the contraction method. The approach is applied to a couple of concrete Pólya urns that lead to limit laws with normal distributions, with non-normal limit distributions, or with asymptotic periodic distributional behavior. Finally, an approach more in the spirit of earlier applications of the contraction method is discussed for one of the examples. A general transfer theorem of the contraction method is extended to cover this example, leading to conditions on the coefficients of the recursion that are not only weaker but also in general easier to check.

- On the existence and uniqueness of Glosten-Milgrom price processes (2013)
- We study the price-setting problem of market makers under perfect competition in continuous time. Thereby we follow the classic Glosten-Milgrom model that defines bid and ask prices as the expectation of a true value of the asset given the market makers partial information that includes the customers trading decisions. The true value is modeled as a Markov process that can be observed by the customers with some noise at Poisson times. We analyze the price-setting problem by solving a non-standard filtering problem with an endogenous filtration that depends on the bid and ask price process quoted by the market maker. Under some conditions we show existence and uniqueness of the price processes. In a different setting we construct a counterexample to uniqueness. Further, we discuss the behavior of the spread by a convergence result and simulations.

- A Gaussian limit process for optimal FIND algorithms (2014)
- We consider versions of the FIND algorithm where the pivot element used is the median of a subset chosen uniformly at random from the data. For the median selection we assume that subsamples of size asymptotic to c⋅nα are chosen, where 0<α≤12, c>0 and n is the size of the data set to be split. We consider the complexity of FIND as a process in the rank to be selected and measured by the number of key comparisons required. After normalization we show weak convergence of the complexity to a centered Gaussian process as n→∞, which depends on α. The proof relies on a contraction argument for probability distributions on càdlàg functions. We also identify the covariance function

- Hierarchical equilibria of branching populations (2004)
- The objective of this paper is the study of the equilibrium behavior of a population on the hierarchical group ΩN consisting of families of individuals undergoing critical branching random walk and in addition these families also develop according to a critical branching process. Strong transience of the random walk guarantees existence of an equilibrium for this two-level branching system. In the limit N→∞ (called the hierarchical mean field limit), the equilibrium aggregated populations in a nested sequence of balls B(N)ℓ of hierarchical radius ℓ converge to a backward Markov chain on R+. This limiting Markov chain can be explicitly represented in terms of a cascade of subordinators which in turn makes possible a description of the genealogy of the population.