## Mathematisch-Naturwissenschaftliche Fakultät: Suche

Anzeige der Dokumente 1-10 von 25

#### Approximation Algorithms for Traveling Salesman Problems

(2020-04-15)

The traveling salesman problem is the probably most famous problem in combinatorial optimization. Given a graph G and nonnegative edge costs, we want to find a closed walk in G that visits every vertex at least once and ...

#### Sedimentation of particle suspensions in Stokes flows

(2020-05-14)

In this thesis, we consider problems arising from the physical phenomenon of particle sedimentation. We focus on non-Brownian particles in fluids at zero Reynolds number. Microscopically, the particle
system is described ...

#### Gradient Flows, Metastability and Interacting Particle Systems

(2020-06-19)

Many stochastic models exhibit a phenomenon called metastability. The first goal of this thesis is to study this phenomenon for certain classes of interacting particle systems. The second goal of this thesis is the following. ...

#### Geometry of random 3-manifolds

(2020-01-31)

We study random 3-manifolds, as introduced by Dunfield and Thurston, from a geometric point of view. Within this framework, work of Maher allows us to equip a typical random 3-manifold with a canonical geometric structure, ...

#### Algebraic Multigrid for Meshfree Methods

(2020-02-11)

This thesis deals with the development of a new Algebraic Multigrid method (AMG) for the solution of linear systems arising from Generalized Finite Difference Methods (GFDM). In particular, we consider the Finite Pointset ...

#### Mapping Properties of Bäcklund Transformations and the Asymptotic Stability of Soliton Solutions for the Nonlinear Schrödinger and Modified Korteweg-de-Vries Equation

(2020-02-17)

We consider the cubic Nonlinear Schrödinger Equation (NLS) and the Modified Korteweg-de-Vries Equation (mKdV) in the one-dimensional, focusing case. For the mKdV, we also restrict ourselves to the case of real-valued ...

#### Pointwise convergence, maximal functions and regularity issues in harmonic analysis

(2020-03-05)

This cumulative thesis is dedicated to the study of different maximal operators related to pointwise convergence in Fourier Analysis and is divided in three main parts.<br /> The first part is dedicated to regularity results ...

#### Diffusions on Wasserstein Spaces

(2020-05-05)

We construct a canonical diffusion process on the space of probability measures over a closed Riemannian manifold, with invariant measure the Dirichlet–Ferguson measure. Together with a brief survey of the relevant literature, ...

#### Probabilistic and differential geometric methods for relativistic and Euclidean Dirac and radiation fields

(2020-07-09)

The main objective of the thesis is to study relativistic and Euclidean Fermionic quantum fields from a geometrical and probabilistic point of view as opposed to the standard treatment which is more algebraic in nature.<br ...

#### An Easton-like Theorem for all Cardinals in ZF

(2020-07-01)

We show that in Zermelo-Fraenkel Set Theory without the Axiom of Choice (<strong>ZF</strong>), a surjectively modified Continuum Function θ(κ) can take almost arbitrary values on all cardinals. This is in sharp contrast to the situation in <strong>ZFC</strong>, where on the one hand, Easton's Theorem states that the Continuum Function on the class of all

**regular**cardinals is essentially undetermined, but on the other hand, various results show that the value of 2<sup>κ</sup> for**singular**cardinals κ is strongly influenced by the behavior of the Continuum Function below. <br /> Without the Axiom of Choice (<strong>AC</strong>), the powerset of a cardinal is generally not well-orderable, and there are different ways how "largeness" can now be expressed. The θ-function maps any cardinal κ to the least cardinal α for which there is no surjective function from the powerset of κ onto α, thus measuring the**surjective size**of the powersets. <br /> Our first theorem answers a question of Saharon Shelah, who asked whether there are any bounds on the θ-function in the theory <strong>ZF + DC + AX<sub>4</sub></strong>. Here, the axiom <strong>AX<sub>4</sub></strong> is the assertion that for every cardinal λ, the set of all countable subsets of λ can be well-ordered. Together with the Axiom of Dependent Choice (<strong>DC</strong>), the theory <strong>ZF + DC + AX<sub>4</sub> </strong> provides a rich framework for combinatorial set theory in the <strong>¬ AC</strong>-context, in which set theory is "not so far from normal" (Shelah). Nevertheless, we prove that the answer to Shelah's question is**no**: Given any "reasonable" behavior of the θ-function on a set of uncountable cardinals, we construct a model <strong>N</strong> of <strong>ZF + DC + AX<sub>4</sub> </strong> where this behavior is realized. <br /> Our forcing notion is based on ideas from the paper**"Violating the Singular Cardinals Hypothesis without Large Cardinals"**(2012) by Moti Gitik and Peter Koepke. We modify and generalize their construction in order to treat the θ-values of many cardinals simultaneously. Our second theorem deals with the question whether also any "reasonable" behavior of the θ-function on a**class**of infinite cardinals can be realized in <strong>ZF</strong>. (The construction mentioned above can not be straightforwardly generalized to a class-sized forcing notion and is therefore only suitable for treating**set**many θ-values at the same time.) <br /> Given a ground model <strong>V</strong> with a function F on the class of infinite cardinals such that F is weakly monotone and F(κ) ≥ κ<sup>++</sup> holds for all κ, is there a <strong>ZF</strong>-model <strong>N</strong> such that <strong>N</strong>⊇<strong>V</strong> is a cardinal-preserving extension with θ<sup><strong>N</strong></sup>(κ) = F(κ) for all cardinals κ? <br /> We introduce a new notion of class forcing ℙ, consisting of functions on trees with finitely many maximal points. Our eventual model <strong>N</strong> is a symmetric extension by this class forcing ℙ. <br /> We conclude that indeed, any "reasonable" behavior of the θ-function can be realized in <strong>ZF</strong> -- the only restrictions are the obvious ones. In other words: An analogue of Easton's theorem holds for all cardinals....