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- 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

- Coalescent trees and their lengths (2014)
- 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.

- A multiple filter test for the detection of rate changes in renewal processes with varying variance (2014)
- The thesis provides novel procedures in the statistical field of change point detection in time series. Motivated by a variety of neuronal spike train patterns, a broad stochastic point process model is introduced. This model features points in time (change points), where the associated event rate changes. For purposes of change point detection, filtered derivative processes (MOSUM) are studied. Functional limit theorems for the filtered derivative processes are derived. These results are used to support novel procedures for change point detection; in particular, multiple filters (bandwidths) are applied simultaneously in oder to detect change points in different time scales.

- Partial symmetries of solutions to nonlinear elliptic and parabolic problems in bounded radial domains (2014)
- We consider a class of nonautonomous nonlinear competitive parabolic systems on bounded radial domains under Neumann or Dirichlet boundary conditions. We show that, if the initial profiles satisfy a reflection inequality with respect to a hyperplane, then bounded positive solutions are asymptotically (in time) foliated Schwarz symmetric with respect to antipodal points. Additionally, a related result for (positive and sign changing solutions) of scalar equations with Neumann or Dirichlet boundary conditions is given. The asymptotic shape of solutions to cooperative systems is also discussed.

- Nonnegative polynomials and sums of squares : boundary structure, symmetries and sparsity (2014)
- The cones of nonnegative polynomials and sums of squares arise as central objects in convex algebraic geometry and have their origin in the seminal work of Hilbert ([Hil88]). Depending on the number of variables n and the degree d of the polynomials, Hilbert famously characterizes all cases of equality between the cone of nonnegative polynomials and the cone of sums of squares. This equality precisely holds for bivariate forms, quadratic forms and ternary quartics ([Hil88]). Since then, a lot of work has been done in understanding the difference between these two cones, which has major consequences for many practical applications such as for polynomial optimization problems. Roughly speaking, minimizing polynomial functions (constrained as well as unconstrained) can be done efficiently whenever certain nonnegative polynomials can be written as sums of squares (see Section 2.3 for the precise relationship). The underlying reason is the fundamental difference that checking nonnegativity of polynomials is an NP-hard problem whenever the degree is greater or equal than four ([BCSS98]), whereas checking whether a polynomial can be written as a sum of squares is a semidefinite feasibility problem (see Section 2.2). Although the complexity status of the semidefinite feasibility problem is still an open problem, it is polynomial for fixed number of variables. Hence, understanding the difference between nonnegative polynomials and sums of squares is highly desirable both from a theoretical and a practical viewpoint.

- Ein Multiple-Filter-Test zur Detektion von Varianzänderungen in Erneuerungsprozessen (2014)
- In der Arbeit wird ein Testverfahren zum Prüfen der Varianzhomogenität der Lebenszeiten eines Erneuerungsprozesses entwickelt. Das Verfahren basiert auf der "Filtered-Derivative"-Methode. Zur Herleitung des Annahmebereichs werden zunächst Bootstrap-Permutationen genutzt, bevor zu einer asymptotischen Methode übergangen wird. Ein entsprechender funktionaler Grenzwertsatz wird skizziert. Aufbauend auf dem Test wird ein Multiple-Filter-Algorithmus zur genauen Detektion der Varianz-Change-Points besprochen. Schließlich folgt die Inklusion von vorher detektierten Ratenänderungen in das Verfahren. Der Test und der Algorithmus werden in Simulationsstudien evaluiert. Abschließend erfolgt eine Anwendung auf EEG-Daten.

- A new relaxation technique for polynomial optimization and spectrahedral geometry problems (2014)
- This work is concerned with two topics at the intersection of convex algebraic geometry and optimization. We develop a new method for the optimization of polynomials over polytopes. From the point of view of convex algebraic geometry the most common method for the approximation of polynomial optimization problems is to solve semidefinite programming relaxations coming from the application of Positivstellensätze. In optimization, non-linear programming problems are often solved using branch and bound methods. We propose a fused method that uses Positivstellensatz-relaxations as lower bounding methods in a branch and bound scheme. By deriving a new error bound for Handelman's Positivstellensatz, we show convergence of the resulting branch and bound method. Through the application of Positivstellensätze, semidefinite programming has gained importance in polynomial optimization in recent years. While it arises to be a powerful tool, the underlying geometry of the feasibility regions (spectrahedra) is not yet well understood. In this work, we study polyhedral and spectrahedral containment problems, in particular we classify their complexity and introduce sufficient criteria to certify the containment of one spectrahedron in another one.

- 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).