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Thu, 03 Jul 2014 13:02:59 +0100Thu, 03 Jul 2014 13:02:59 +0100Habilitationsordnung der Mathematisch-Naturwissenschaftlichen Fachbereiche der Johann Wolfgang Goethe-Universität Frankfurt am Main vom 4. Februar 1992 : genehmigt durch Beschluss des Präsidiums der Johann Wolfgang Goethe-Universität Frankfurt am Main am 19. November 2013
http://publikationen.stub.uni-frankfurt.de/frontdoor/index/index/docId/27478
otherhttp://publikationen.stub.uni-frankfurt.de/frontdoor/index/index/docId/27478Fri, 07 Mar 2014 13:02:59 +0100Fachspezifischer Anhang zur SPoL (Teil III): Studienfach Mathematik im Studiengang L3 vom 19.2.2013 : genehmigt durch das Präsidium am 01.10.2013
http://publikationen.stub.uni-frankfurt.de/frontdoor/index/index/docId/33124
otherhttp://publikationen.stub.uni-frankfurt.de/frontdoor/index/index/docId/33124Thu, 06 Mar 2014 15:32:46 +0100Coalescent trees and their lengths
http://publikationen.stub.uni-frankfurt.de/frontdoor/index/index/docId/33130
The work presented in this thesis is devoted to two classes of mathematical population genetics models, namely the Kingman-coalescent and the Beta-coalescents. Chapters 2, 3 and 4 of the thesis include results concerned with the first model, whereas Chapter 5 presents contributions to the second class of models.Iulia-Andra Dahmerdoctoralthesishttp://publikationen.stub.uni-frankfurt.de/frontdoor/index/index/docId/33130Tue, 25 Feb 2014 11:06:03 +0100Hierarchical equilibria of branching populations
http://publikationen.stub.uni-frankfurt.de/frontdoor/index/index/docId/32889
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.Donald A. Dawson; Luis G. Gorostiza; Anton Wakolbingerarticlehttp://publikationen.stub.uni-frankfurt.de/frontdoor/index/index/docId/32889Wed, 29 Jan 2014 16:22:58 +0100Alpha-stable branching and beta-coalescents
http://publikationen.stub.uni-frankfurt.de/frontdoor/index/index/docId/32891
We determine that the continuous-state branching processes for which the genealogy, suitably time-changed, can be described by an autonomous Markov process are precisely those arising from $\alpha$-stable branching mechanisms. The random ancestral partition is then a time-changed $\Lambda$-coalescent, where $\Lambda$ is the Beta-distribution with parameters $2-\alpha$ and $\alpha$, and the time change is given by $Z^{1-\alpha}$, where $Z$ is the total population size. For $\alpha = 2$ (Feller's branching diffusion) and $\Lambda = \delta_0$ (Kingman's coalescent), this is in the spirit of (a non-spatial version of) Perkins' Disintegration Theorem. For $\alpha =1$ and $\Lambda$ the uniform distribution on $[0,1]$, this is the duality discovered by Bertoin & Le Gall (2000) between the norming of Neveu's continuous state branching process and the Bolthausen-Sznitman coalescent.
We present two approaches: one, exploiting the `modified lookdown construction', draws heavily on Donnelly & Kurtz (1999); the other is based on direct calculations with generators.Matthias Birkner; Jochen Blath; Marcella Capaldo; Alison M. Etheridge; Martin Möhle; Jason Schweinsberg; Anton Wakolbingerarticlehttp://publikationen.stub.uni-frankfurt.de/frontdoor/index/index/docId/32891Wed, 29 Jan 2014 15:48:01 +0100On the Total External Length of the Kingman Coalescent
http://publikationen.stub.uni-frankfurt.de/frontdoor/index/index/docId/32892
We prove asymptotic normality of the total length of external branches in the Kingman coalescent. The proof uses an embedded Markov chain, which can be described as follows: Take an urn with black balls. Empty it step by step according to the rule: In each step remove a randomly chosen pair of balls and replace it by one red ball. Finally remove the last remaining ball. Then the numbers of red balls form a Markov chain with an unexpected property: It is time-reversible.Svante Janson; Götz Kerstingarticlehttp://publikationen.stub.uni-frankfurt.de/frontdoor/index/index/docId/32892Wed, 29 Jan 2014 15:36:28 +0100On the asymptotic internal path length and the asymptotic Wiener index of random split trees
http://publikationen.stub.uni-frankfurt.de/frontdoor/index/index/docId/32893
The random split tree introduced by Devroye (1999) is considered. We derive a second order expansion for the mean of its internal path length and furthermore obtain a limit law by the contraction method. As an assumption we need the splitter having a Lebesgue density and mass in every neighborhood of 1. We use properly stopped homogeneous Markov chains, for which limit results in total variation distance as well as renewal theory are used. Furthermore, we extend this method to obtain the corresponding results for the Wiener index.Götz Olaf Munsoniusarticlehttp://publikationen.stub.uni-frankfurt.de/frontdoor/index/index/docId/32893Wed, 29 Jan 2014 15:22:38 +0100Upper large deviations for Branching Processes in Random Environment with heavy tails
http://publikationen.stub.uni-frankfurt.de/frontdoor/index/index/docId/32894
ranching Processes in Random Environment (BPREs) $(Z_n:n\geq0)$ are the generalization of Galton-Watson processes where \lq in each generation' the reproduction law is picked randomly in an i.i.d. manner. The associated random walk of the environment has increments distributed like the logarithmic mean of the offspring distributions. This random walk plays a key role in the asymptotic behavior. In this paper, we study the upper large deviations of the BPRE $Z$ when the reproduction law may have heavy tails. More precisely, we obtain an expression for the limit of $-\log \mathbb{P}(Z_n\geq \exp(\theta n))/n$ when $n\rightarrow \infty$. It depends on the rate function of the associated random walk of the environment, the logarithmic cost of survival $\gamma:=-\lim_{n\rightarrow\infty} \log \mathbb{P}(Z_n>0)/n$ and the polynomial rate of decay $\beta$ of the tail distribution of $Z_1$. This rate function can be interpreted as the optimal way to reach a given "large" value. We then compute the rate function when the reproduction law does not have heavy tails. Our results generalize the results of B\"oinghoff $\&$ Kersting (2009) and Bansaye $\&$ Berestycki (2008) for upper large deviations. Finally, we derive the upper large deviations for the Galton-Watson processes with heavy tails.Vincent Bansaye; Christian Böinghoffarticlehttp://publikationen.stub.uni-frankfurt.de/frontdoor/index/index/docId/32894Wed, 29 Jan 2014 14:43:58 +0100Delay equations driven by rough paths
http://publikationen.stub.uni-frankfurt.de/frontdoor/index/index/docId/32895
In this article, we illustrate the flexibility of the algebraic integration formalism introduced in M. Gubinelli, J. Funct. Anal. 216, 86-140, 2004, Math. Review 2005k:60169 http://www.ams.org/mathscinet-getitem?mr=2005k:60169 , by establishing an existence and uniqueness result for delay equations driven by rough paths. We then apply our results to the case where the driving path is a fractional Brownian motion with Hurst parameter $H>1/3$.
Andreas Neuenkirch; Ivan Nourdin; Samy Tindelarticlehttp://publikationen.stub.uni-frankfurt.de/frontdoor/index/index/docId/32895Wed, 29 Jan 2014 14:17:28 +0100The longtime behavior of branching random walk in a catalytic medium
http://publikationen.stub.uni-frankfurt.de/frontdoor/index/index/docId/32890
Consider a countable collection of particles located on a countable group, performing a critical branching random walk where the branching rate of a particle is given by a random medium fluctuating both in space and time. Here we study the case where the time-space random medium (called catalyst) is also a critical branching random walk evolving autonomously while the local branching rate of the reactant process is proportional to the number of catalytic particles present at a site. The catalyst process and the reactant process typically have different underlying motions.Andreas Greven; Achim Klenke; Anton Wakolbingerarticlehttp://publikationen.stub.uni-frankfurt.de/frontdoor/index/index/docId/32890Wed, 29 Jan 2014 14:03:52 +0100Trickle-down processes and their boundaries
http://publikationen.stub.uni-frankfurt.de/frontdoor/index/index/docId/32896
It is possible to represent each of a number of Markov chains as an evolving sequence of connected subsets of a directed acyclic graph that grow in the following way: initially, all vertices of the graph are unoccupied, particles are fed in one-by-one at a distinguished source vertex, successive particles proceed along directed edges according to an appropriate stochastic mechanism, and each particle comes to rest once it encounters an unoccupied vertex. Examples include the binary and digital search tree processes, the random recursive tree process and generalizations of it arising from nested instances of Pitman's two-parameter Chinese restaurant process, tree-growth models associated with Mallows' ϕ model of random permutations and with Schützenberger's non-commutative q-binomial theorem, and a construction due to Luczak and Winkler that grows uniform random binary trees in a Markovian manner. We introduce a framework that encompasses such Markov chains, and we characterize their asymptotic behavior by analyzing in detail their Doob-Martin compactifications, Poisson boundaries and tail σ-fields.Steven Neil Evans; Rudolf Grübel; Anton Wakolbingerarticlehttp://publikationen.stub.uni-frankfurt.de/frontdoor/index/index/docId/32896Mon, 27 Jan 2014 09:57:23 +0100A Gaussian limit process for optimal FIND algorithms
http://publikationen.stub.uni-frankfurt.de/frontdoor/index/index/docId/32751
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 functionHenning Sulzbach; Ralph Neininger; Michael Drmotaarticlehttp://publikationen.stub.uni-frankfurt.de/frontdoor/index/index/docId/32751Mon, 27 Jan 2014 09:23:17 +0100On the existence and uniqueness of Glosten-Milgrom price processes
http://publikationen.stub.uni-frankfurt.de/frontdoor/index/index/docId/32690
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.Matthias Riedeldoctoralthesishttp://publikationen.stub.uni-frankfurt.de/frontdoor/index/index/docId/32690Tue, 21 Jan 2014 10:55:12 +0100Pólya urns via the contraction method
http://publikationen.stub.uni-frankfurt.de/frontdoor/index/index/docId/32284
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.Margarete Knapedoctoralthesishttp://publikationen.stub.uni-frankfurt.de/frontdoor/index/index/docId/32284Tue, 19 Nov 2013 16:18:54 +0100Algorithmische Aspekte des Lokalen Lovász Lemmas
http://publikationen.stub.uni-frankfurt.de/frontdoor/index/index/docId/29896
Im Rahmen dieser Arbeit wird der aktuelle Stand auf dem Gebiet des Lokalen Lovász Lemmas (LLL) beschrieben und ein Überblick über die Arbeiten zu konstruktiven Beweisen und Anwendungen gegeben. Ausgehend von Jószef Becks Arbeit zu einer algorithmischen Herangehensweise, haben sich in den letzten Jahren im Umfeld von Moser und Tardos und ihren Arbeiten zu einem konstruktiven Beweis des LLL eine erneute starke Beschäftigung mit dem Thema und eine Fülle von Verbesserungen entwickelt.
In Kapitel 1 wird als Motivation eine kurze Einführung in die probabilistische Methode gegeben. Mit der First- und Second Moment Method werden zwei einfache Vorgehensweisen vorgestellt, die die Grundidee dieses Beweisprinzips klar werden lassen. Von Paul Erdős eröffnet, beschreibt es Wege, Existenzbeweise in nicht-stochastischen Teilgebieten der Mathematik mithilfe stochastischer Überlegungen zu führen. Das Lokale Lemma als eine solche Überlegung entstammt dieser Idee.
In Kapitel 2 werden verschiedene Formen des LLL vorgestellt und bewiesen, außerdem wird anhand einiger Anwendungsbeispiele die Vorgehensweise bei der Verwendung des LLL veranschaulicht.
In Kapitel 3 werden algorithmische Herangehensweisen beschrieben, die geeignet sind, von der (mithilfe des LLL gezeigten) Existenz gewisser Objekte zur tatsächlichen Konstruktion derselben zu gelangen.
In Kapitel 4 wird anhand von Beispielen aus dem reichen Schatz neuerer Veröffentlichungen gezeigt, welche Bewegung nach der Arbeit von Moser und Tardos entstanden ist. Dabei beleuchtet die Arbeit nicht nur einen anwendungsorientierten Beitrag von Haeupler, Saha und Srinivasan, sondern auch einen Beitrag Terence Taos, der die Beweistechnik Mosers aus einem anderen Blickwinkel beleuchtet.Arne Moritz Harffdiplomthesishttp://publikationen.stub.uni-frankfurt.de/frontdoor/index/index/docId/29896Wed, 06 Nov 2013 09:12:55 +0100Entropy increase in switching systems
http://publikationen.stub.uni-frankfurt.de/frontdoor/index/index/docId/31440
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.José M. Amigó; Peter E. Kloeden; Ángel Giménezarticlehttp://publikationen.stub.uni-frankfurt.de/frontdoor/index/index/docId/31440Mon, 19 Aug 2013 14:52:17 +0200Detecting rate changes in spike trains
http://publikationen.stub.uni-frankfurt.de/frontdoor/index/index/docId/30093
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.Marietta Tillmanndiplomthesishttp://publikationen.stub.uni-frankfurt.de/frontdoor/index/index/docId/30093Fri, 21 Jun 2013 12:59:23 +0200A stochastic model for the joint evaluation of burstiness and regularity in oscillatory spike trains
http://publikationen.stub.uni-frankfurt.de/frontdoor/index/index/docId/29970
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.Markus Bingmerdoctoralthesishttp://publikationen.stub.uni-frankfurt.de/frontdoor/index/index/docId/29970Fri, 31 May 2013 09:53:26 +0200On the geometry, topology and approximation of amoebas
http://publikationen.stub.uni-frankfurt.de/frontdoor/index/index/docId/29898
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).Timo de Wolffdoctoralthesishttp://publikationen.stub.uni-frankfurt.de/frontdoor/index/index/docId/29898Fri, 17 May 2013 12:31:07 +0200Fachspezifischer Anhang zur SPoL (Teil III): Studienfach Mathematik im Studiengang L5 : Stand: August 2009
http://publikationen.stub.uni-frankfurt.de/frontdoor/index/index/docId/28696
otherhttp://publikationen.stub.uni-frankfurt.de/frontdoor/index/index/docId/28696Sat, 02 Feb 2013 17:35:35 +0100Fachspezifischer Anhang zur SPoL (Teil III): Studienfach Mathematik im Studiengang L3 : Stand: August 2009
http://publikationen.stub.uni-frankfurt.de/frontdoor/index/index/docId/28683
otherhttp://publikationen.stub.uni-frankfurt.de/frontdoor/index/index/docId/28683Sat, 02 Feb 2013 17:13:39 +0100Fachspezifischer Anhang zur SPoL (Teil III): Studienfach Mathematik im Studiengang L2 : Stand: August 2009
http://publikationen.stub.uni-frankfurt.de/frontdoor/index/index/docId/28663
otherhttp://publikationen.stub.uni-frankfurt.de/frontdoor/index/index/docId/28663Sat, 02 Feb 2013 16:35:15 +0100Habilitationsordnung der Mathematisch–Naturwissenschaftlichen Fachbereiche der Johann Wolfgang Goethe-Universität Frankfurt am Main vom 04.02.1992 (ABl. 1992, S.816 ff.), zuletzt geändert am 28. April 2002 (StAnz. 41/2003, S. 4024 – 4025) : genehmigt durch Beschluss des Präsidiums der Johann Wolfgang Goethe-Universität Frankfurt am Main am 27. Januar 2009 ; hier: Änderung bzw. Ergänzung
http://publikationen.stub.uni-frankfurt.de/frontdoor/index/index/docId/28265
otherhttp://publikationen.stub.uni-frankfurt.de/frontdoor/index/index/docId/28265Sat, 02 Feb 2013 11:04:51 +0100Promotionsordnung der Mathematisch-Naturwissenschaftlichen Fachbereiche der Johann Wolfgang Goethe-Universität in Frankfurt am Main vom 26. Mai 1993 (ABL.1/94, S. 21) zuletzt geändert am 05.09.2007 (Uni-Report 13.11.2008) : genehmigt durch Beschluss des Präsidiums der Johann Wolfgang Goethe-Universität Frankfurt am Main am 27. Januar 2009 ; hier: Änderung
http://publikationen.stub.uni-frankfurt.de/frontdoor/index/index/docId/28264
otherhttp://publikationen.stub.uni-frankfurt.de/frontdoor/index/index/docId/28264Sat, 02 Feb 2013 10:58:31 +0100Promotionsordnung der Mathematisch-Naturwissenschaftlichen Fachbereiche der Johann Wolfgang Goethe-Universität Frankfurt am Main vom 26. Mai 1993 (ABL.1/94, S. 21) zuletzt geändert am 27. Januar 2009 (Uni-Report 1. April 2009) : genehmigt durch Beschluss des Präsidiums der Johann Wolfgang Goethe-Universität Frankfurt am Main am 16. Juni 2009
http://publikationen.stub.uni-frankfurt.de/frontdoor/index/index/docId/28249
otherhttp://publikationen.stub.uni-frankfurt.de/frontdoor/index/index/docId/28249Fri, 01 Feb 2013 09:03:12 +0100