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We compute genus-0 Belyi maps with prescribed monodromy and strictly verify the computed results. Among the computed examples are almost simple primitive groups that satisfy the rational rigidity criterion yielding polynomials with prescribed Galois groups over Q(t). We also give an explicit version of a theorem of Magaard, which lists all sporadic groups occurring as composition factors of monodromy groups of rational functions.
We generalize a theorem by Titchmarsh about the mean value of Hardy’s \(Z\)-function at the Gram points to the Hecke \(L\)-functions, which in turn implies the weak Gram law for them. Instead of proceeding analogously to Titchmarsh with an approximate functional equation we employ a different method using contour integration.
In the generalized Nash equilibrium problem not only the cost function of a player depends on the rival players' decisions, but also his constraints. This thesis presents different iterative methods for the numerical computation of a generalized Nash equilibrium, some of them globally, others locally superlinearly convergent. These methods are based on either reformulations of the generalized Nash equilibrium problem as an optimization problem, or on a fixed point formulation. The key tool for these reformulations is the Nikaido-Isoda function. Numerical results for various problem from the literature are given.
In my Ph.D. thesis "On the geometry and parametrization of almost invariant subspaces and observer theory" I consider the set of almost conditioned invariant subspaces of fixed dimension for a given fixed linear finite-dimensional time-invariant observable control system in state space form. Almost conditioned invariant subspaces were introduced by Willems. They generalize the concept of a conditioned invariant subspace requiring the invariance condition to hold only up to an arbitrarily small deviation in the metric of the state space. One of the goals of the theory of almost conditioned invariant subspaces was to identify the subspaces appearing as limits of sequences of conditioned invariant subspaces. An example due to {\"O}zveren, Verghese and Willsky, however, shows that the set of almost conditioned invariant subspaces is not big enough. I address this question in a joint paper with Helmke and Fuhrmann (Towards a compactification of the set of conditioned invariant subspaces, Systems and Control Letters, 48(2):101-111, 2003). Antoulas derived a description of conditioned invariant subspaces as kernels of permuted and truncated reachability matrices of controllable pairs of the appropriate size. This description was used by Helmke and Fuhrmann to construct a diffeomorphism from the set of similarity classes of certain controllable pairs onto the set of tight conditioned invariant subspaces. In my thesis I generalize this result to almost conditioned invariant subspaces describing them in terms of restricted system equivalence classes of controllable triples. Furthermore, I identify the controllable pairs appearing in the kernel representations of conditioned invariant subspaces as being induced by corestrictions of the original system to the subspace. Conditioned invariant subspaces are known to be closely related to partial observers. In fact, a tracking observer for a linear function of the state of the observed system exists if and only if the kernel of that function is conditioned invariant. In my thesis I show that the system matrices of the observers are in fact the corestrictions of the observed system to the kernels of the observed functions. They in turn are closely related to partial realizations. Exploring this connection further, I prove that the set of tracking observer parameters of fixed size, i.e. tracking observers of fixed order together with the functions they are tracking, is a smooth manifold. Furthermore, I construct a vector bundle structure for the set of conditioned invariant subspaces of fixed dimension together with their friends, i.e. the output injections making the subspaces invariant, over that manifold. Willems and Trentelman generalized the concept of a tracking observer by including derivatives of the output of the observed system in the observer equations (PID-observers). They showed that a PID-observer for a linear function of the state of the observed system exists if and only if the kernel of that function is almost conditioned invariant. In my thesis I replace PID-observers by singular systems, which has the advantage that the system matrices of the observers coincide with the matrices appearing in the kernel representations of the subspaces. In a second approach to the parametrization of conditioned invariant subspaces Hinrichsen, M{\"u}nzner and Pr{\"a}tzel-Wolters, Fuhrmann and Helmke and Ferrer, F. Puerta, X. Puerta and Zaballa derived a description of conditioned invariant subspaces in terms of images of block Toeplitz type matrices. They used this description to construct a stratification of the set of conditioned invariant subspaces of fixed dimension into smooth manifolds. These so called Brunovsky strata consist of all the subspaces with fixed restriction indices. They constructed a cell decomposition of the Brunovsky strata into so called Kronecker cells. In my thesis I show that in the tight case this cell decomposition is induced by a Bruhat decomposition of a generalized flag manifold. I identify the adherence order of the cell decomposition as being induced by the reverse Bruhat order.
Background
It is hypothesized that because of higher mast cell numbers and mediator release, mastocytosis predisposes patients for systemic immediate-type hypersensitivity reactions to certain drugs including non-steroidal anti-inflammatory drugs (NSAID).
Objective
To clarify whether patients with NSAID hypersensitivity show increased basal serum tryptase levels as sign for underlying mast cell disease.
Methods
As part of our allergy work-up, basal serum tryptase levels were determined in all patients with a diagnosis of NSAID hypersensitivity and the severity of the reaction was graded. Patients with confirmed IgE-mediated hymenoptera venom allergy served as a comparison group.
Results
Out of 284 patients with NSAID hypersensitivity, 26 were identified with basal serum tryptase > 10.0 ng/mL (9.2%). In contrast, significantly (P = .004) more hymenoptera venom allergic patients had elevated tryptase > 10.0 ng/mL (83 out of 484; 17.1%). Basal tryptase > 20.0 ng/mL was indicative for severe anaphylaxis only in venom allergic subjects (29 patients; 4x grade 2 and 25x grade 3 anaphylaxis), but not in NSAID hypersensitive patients (6 patients; 4x grade 1, 2x grade 2).
Conclusions
In contrast to hymenoptera venom allergy, NSAID hypersensitivity do not seem to be associated with elevated basal serum tryptase levels and levels > 20 ng/mL were not related to increased severity of the clinical reaction. This suggests that mastocytosis patients may be treated with NSAID without special precautions.
In this paper, we prove an asymptotic formula for the sum of the values of the periodic zeta-function at the nontrivial zeros of the Riemann zeta-function (up to some height) which are symmetrical on the real line and the critical line. This is an extension of the previous results due to Garunkštis, Kalpokas, and, more recently, Sowa. Whereas Sowa's approach was assuming the yet unproved Riemann hypothesis, our result holds unconditionally.
In the following dissertation we consider three preconditioners of algebraic multigrid type, though they are defined for arbitrary prolongation and restriction operators, we consider them in more detail for the aggregation method. The strengthened Cauchy-Schwarz inequality and the resulting angle between the spaces will be our main interests. In this context we will introduce some modifications. For the problem of the one-dimensional convection we obtain perfect theoretical results. Although this is not the case for more complex problems, the numerical results we present will show that the modifications are also useful in these situation. Additionally, we will consider a symmetric problem in the energy norm and present a simple rule for algebraic aggregation.
On the Fragility Index
(2011)
The Fragility Index captures the amount of risk in a stochastic system of arbitrary dimension. Its main mathematical tool is the asymptotic distribution of exceedance counts within the system which can be derived by use of multivariate extreme value theory. Thereby the basic assumption is that data comes from a distribution which lies in the domain of attraction of a multivariate extreme value distribution. The Fragility Index itself and its extension can serve as a quantitative measure for tail dependence in arbitrary dimensions. It is linked to the well known extremal index for stochastic processes as well the extremal coefficient of an extreme value distribution.
Beatty sets (also called Beatty sequences) have appeared as early as 1772 in the astronomical studies of Johann III Bernoulli as a tool for easing manual calculations and - as Elwin Bruno Christoffel pointed out in 1888 - lend themselves to exposing intricate properties of the real irrationals. Since then, numerous researchers have explored a multitude of arithmetic properties of Beatty sets; the interrelation between Beatty sets and modular inversion, as well as Beatty sets and the set of rational primes, being the central topic of this book. The inquiry into the relation to rational primes is complemented by considering a natural generalisation to imaginary quadratic number fields.
In this dissertation, we develop and analyze novel optimizing feedback laws for control-affine systems with real-valued state-dependent output (or objective) functions. Given a control-affine system, our goal is to derive an output-feedback law that asymptotically stabilizes the closed-loop system around states at which the output function attains a minimum value. The control strategy has to be designed in such a way that an implementation only requires real-time measurements of the output value. Additional information, like the current system state or the gradient vector of the output function, is not assumed to be known. A method that meets all these criteria is called an extremum seeking control law. We follow a recently established approach to extremum seeking control, which is based on approximations of Lie brackets. For this purpose, the measured output is modulated by suitable highly oscillatory signals and is then fed back into the system. Averaging techniques for control-affine systems with highly oscillatory inputs reveal that the closed-loop system is driven, at least approximately, into the directions of certain Lie brackets. A suitable design of the control law ensures that these Lie brackets point into descent directions of the output function. Under suitable assumptions, this method leads to the effect that minima of the output function are practically uniformly asymptotically stable for the closed-loop system. The present document extends and improves this approach in various ways.
One of the novelties is a control strategy that does not only lead to practical asymptotic stability, but in fact to asymptotic and even exponential stability. In this context, we focus on the application of distance-based formation control in autonomous multi-agent system in which only distance measurements are available. This means that the target formations as well as the sensed variables are determined by distances. We propose a fully distributed control law, which only involves distance measurements for each individual agent to stabilize a desired formation shape, while a storage of measured data is not required. The approach is applicable to point agents in the Euclidean space of arbitrary (but finite) dimension. Under the assumption of infinitesimal rigidity of the target formations, we show that the proposed control law induces local uniform asymptotic (and even exponential) stability. A similar statement is also derived for nonholonomic unicycle agents with all-to-all communication. We also show how the findings can be used to solve extremum seeking control problems.
Another contribution is an extremum seeking control law with an adaptive dither signal. We present an output-feedback law that steers a fully actuated control-affine system with general drift vector field to a minimum of the output function. A key novelty of the approach is an adaptive choice of the frequency parameter. In this way, the task of determining a sufficiently large frequency parameter becomes obsolete. The adaptive choice of the frequency parameter also prevents finite escape times in the presence of a drift. The proposed control law does not only lead to convergence into a neighborhood of a minimum, but leads to exact convergence. For the case of an output function with a global minimum and no other critical point, we prove global convergence.
Finally, we present an extremum seeking control law for a class of nonholonomic systems. A detailed averaging analysis reveals that the closed-loop system is driven approximately into descent directions of the output function along Lie brackets of the control vector fields. Those descent directions also originate from an approximation of suitably chosen Lie brackets. This requires a two-fold approximation of Lie brackets on different time scales. The proposed method can lead to practical asymptotic stability even if the control vector fields do not span the entire tangent space. It suffices instead that the tangent space is spanned by the elements in the Lie algebra generated by the control vector fields. This novel feature extends extremum seeking by Lie bracket approximations from the class of fully actuated systems to a larger class of nonholonomic systems.