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

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

Bezahlter Urlaub und Lohnfortzahlung im Krankheitsfall: In der Praxis besteht Nachholbedarf bei Minijobbern

Die gesetzlichen Regelungen in Deutschland sehen die arbeitsrechtliche Gleichstellung von Beschäftigten mit verschiedenen Erwerbsformen vor. In der Praxis zeigen sich jedoch teilweise deutliche Unterschiede. Auf Basis einer Beschäftigten- sowie einer Betriebsbefragung wird die Gewährung von bezahltem Urlaub und von Lohnfortzahlung im Krankheitsfall bei 'atypisch' beschäftigten Arbeitnehmerinnen und Arbeitnehmern im Vergleich zu Beschäftigten in 'Normalarbeitsverhältnissen' untersucht

Identification of Hemodynamically Optimal Coronary Stent Designs Based on Vessel Caliber

Coronary stent design influences local patterns of wall shear stress (WSS) that are associated with neointimal growth, restenosis, and the endothelialization of stent struts. The number of circumferentially repeating crowns NC for a given stent de- sign is often modified depending on the target vessel caliber, but the hemodynamic implications of altering NC have not previously been studied. In this investigation, we analyzed the relationship between vessel diameter and the hemodynamically optimal NC using a derivative-free optimization algorithm coupled with computational fluid dynamics. The algorithm computed the optimal vessel diameter, defined as minimizing the area of stent-induced low WSS, for various configurations (i.e., NC) of a generic slotted-tube design and designs that resemble commercially available stents. Stents were modeled in idealized coronary arteries with a vessel diameter that was allowed to vary between 2 and 5 mm. The results indicate that the optimal vessel diameter increases for stent configurations with greater NC, and the designs of current commercial stents incorporate a greater NC than hemodynamically optimal stent designs. This finding suggests that reducing the NC of current stents may improve the hemodynamic environment within stented arteries and reduce the likelihood of excessive neointimal growth and thrombus formation

A Rapid and Computationally Inexpensive Method to Virtually Implant Current and Next-Generation Stents into Subject-Specific Computational Fluid Dynamics Models

Computational modeling is often used to quantify hemodynamic alterations induced by stenting, but frequently uses simplified device or vascular representations. Based on a series of Boolean operations, we developed an efficient and robust method for assessing the influence of current and next-generation stents on local hemodynamics and vascular biomechanics quantified by computational fluid dynamics. Stent designs were parameterized to allow easy control over design features including the number, width and circumferential or longitudinal spacing of struts, as well as the implantation diameter and overall length. The approach allowed stents 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 constructed from medical imaging data. In the coronary bifurcation, we analyzed the hemodynamic difference between closed-cell and open-cell stent geometries. We investigated the impact of decreased strut size in stents with a constant porosity for increasing flow stasis within the stented basilar aneurysm model. These examples demonstrate the current method can be used to investigate differences in stent performance in complex vascular beds for a variety of stenting procedures and clinical scenarios