On Topological Minors in Random Simplicial Complexes

For random graphs, the containment problem considers the probability that a binomial random graph $G(n,p)$ contains a given graph as a substructure. When asking for the graph as a topological minor, i.e., for a copy of a subdivision of the given graph, it is well-known that the (sharp) threshold is at $p=1/n$. We consider a natural analogue of this question for higher-dimensional random complexes $X^k(n,p)$, first studied by Cohen, Costa, Farber and Kappeler for $k=2$. Improving previous results, we show that $p=\Theta(1/\sqrt{n})$ is the (coarse) threshold for containing a subdivision of any fixed complete $2$-complex. For higher dimensions $k>2$, we get that $p=O(n^{-1/k})$ is an upper bound for the threshold probability of containing a subdivision of a fixed $k$-dimensional complex.Comment: 15 page

Higher Dimensional Discrete Cheeger Inequalities

For graphs there exists a strong connection between spectral and combinatorial expansion properties. This is expressed, e.g., by the discrete Cheeger inequality, the lower bound of which states that $\lambda(G) \leq h(G)$, where $\lambda(G)$ is the second smallest eigenvalue of the Laplacian of a graph $G$ and $h(G)$ is the Cheeger constant measuring the edge expansion of $G$. We are interested in generalizations of expansion properties to finite simplicial complexes of higher dimension (or uniform hypergraphs). Whereas higher dimensional Laplacians were introduced already in 1945 by Eckmann, the generalization of edge expansion to simplicial complexes is not straightforward. Recently, a topologically motivated notion analogous to edge expansion that is based on $\mathbb{Z}_2$-cohomology was introduced by Gromov and independently by Linial, Meshulam and Wallach. It is known that for this generalization there is no higher dimensional analogue of the lower bound of the Cheeger inequality. A different, combinatorially motivated generalization of the Cheeger constant, denoted by $h(X)$, was studied by Parzanchevski, Rosenthal and Tessler. They showed that indeed $\lambda(X) \leq h(X)$, where $\lambda(X)$ is the smallest non-trivial eigenvalue of the ($(k-1)$-dimensional upper) Laplacian, for the case of $k$-dimensional simplicial complexes $X$ with complete $(k-1)$-skeleton. Whether this inequality also holds for $k$-dimensional complexes with non-complete $(k-1)$-skeleton has been an open question. We give two proofs of the inequality for arbitrary complexes. The proofs differ strongly in the methods and structures employed, and each allows for a different kind of additional strengthening of the original result.Comment: 14 pages, 2 figure

Not All Saturated 3-Forests Are Tight

A basic statement in graph theory is that every inclusion-maximal forest is connected, i.e. a tree. Using a definiton for higher dimensional forests by Graham and Lovasz and the connectivity-related notion of tightness for hypergraphs introduced by Arocha, Bracho and Neumann-Lara in, we provide an example of a saturated, i.e. inclusion-maximal 3-forest that is not tight. This resolves an open problem posed by Strausz

Improving Cardiovascular Stent Design Using Patient-Specific Models and Shape Optimization

Stent geometry influences local hemodynamic alterations (i.e. the forces moving blood through the cardiovascular system) associated with adverse clinical outcomes. Computational fluid dynamics (CFD) is frequently used to quantify stent-induced hemodynamic disturbances, but previous CFD studies have relied on simplified device or vascular representations. Additionally, efforts to minimize stent-induced hemodynamic disturbances using CFD models often only compare a small number of possible stent geometries. This thesis describes methods for modeling commercial stents in patient-specific vessels along with computational techniques for determining optimal stent geometries that address the limitations of previous studies. An efficient and robust method was developed for virtually implanting stent models into patient-specific vascular geometries derived from medical imaging data. Models of commercial stent designs were parameterized to allow easy control over design features. Stent models were then virtually implanted into vessel geometries using a series of Boolean operations. This approach allowed stented vessel models to be automatically regenerated for rapid analysis of the contribution of design features to resulting hemodynamic alterations. The applicability of the method was demonstrated with patient-specific models of a stented coronary artery bifurcation and basilar trunk aneurysm to reveal how it can be used to investigate differences in hemodynamic performance in complex vascular beds for a variety of clinical scenarios. To identify hemodynamically optimal stents designs, a computational framework was constructed to couple CFD with a derivative-free optimization algorithm. The optimization algorithm was fully-automated such that solid model construction, mesh generation, CFD simulation and time-averaged wall shear stress (TAWSS) quantification did not require user intervention. The method was applied to determine the optimal number of circumferentially repeating stent cells (NC) for a slotted-tube stents and various commercial stents. Optimal stent designs were defined as those minimizing the area of low TAWSS. It was determined the optimal value of NC is dependent on the intrastrut angle with respect to the primary flow direction. Additionally, the geometries of current commercial stents were found to generally incorporate a greater NC than is hemodynamically optimal. The application of the virtual stent implantation and optimization methods may lead to stents with superior hemodynamic performance and the potential for improved clinical outcomes. Future in vivo studies are needed to validate the findings of the computational results obtained from the methods developed in this thesis

Leiharbeit und befristete Beschäftigung: Soziale Teilhabe ist eine Frage von stabilen Jobs

Die Integration in den Arbeitsmarkt gilt als zentral für soziale Teilhabe und gesellschaftliche Integration. Dass die Zahl der Arbeitslosen Ende 2010 auf unter drei Millionen gesunken ist, wäre demnach ein gutes Zeichen für eine Verbesserung des Zusammenhalts in der Gesellschaft. Allerdings wird Personal immer häufiger über befristete Arbeitsverträge eingestellt oder über Zeitarbeitsfirmen ausgeliehen. Hier wird untersucht, wie sich temporäre Beschäftigung auf die subjektive Wahrnehmung der Betroffenen auswirkt: Fühlen sie sich sozial integriert oder von der Gesellschaft ausgeschlossen

Changes in Income Poverty Risks at the Transition from Unemployment to Employment : Comparing the Short-Term and Medium-Term Effects of Fixed-Term and Permanent Jobs

Unemployment is a major risk factor of poverty and employment is regarded key to overcoming it. The present study examines how the income poverty risk of unemployed individuals changes in the short and medium term, when they take up work, and whether the effects differ according to the type of employment. The focus is on permanent and fixed-term job contracts, as the political promotion of fixed-term employment has often been framed as an effort to reduce long-term unemployment and poverty. Drawing on longitudinal data from the German panel study ‘Labour Market and Social Security’ (PASS) 2010–18, we apply a first difference estimator with asymmetric effects to examine the effect of starting a job out of unemployment on income poverty risks in the subsequent four years. Strikingly, starting in a fixed-term and permanent contract have similarly strong and lasting poverty-reducing effects in the short and medium term. Thus, with regard to risks of income poverty, starting a permanent job does not appear more beneficial than starting a fixed-term job for unemployed persons. We discuss the reasons for this finding and also explore how the poverty-reducing effects of transitions from unemployment to fixed-term versus permanent employment vary by household type, occupation, working time and firm size