35 research outputs found
On Topological Minors in Random Simplicial Complexes
For random graphs, the containment problem considers the probability that a
binomial random graph 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 .
We consider a natural analogue of this question for higher-dimensional random
complexes , first studied by Cohen, Costa, Farber and Kappeler for
.
Improving previous results, we show that is the
(coarse) threshold for containing a subdivision of any fixed complete
-complex. For higher dimensions , we get that is an
upper bound for the threshold probability of containing a subdivision of a
fixed -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 , where is the second smallest eigenvalue of the Laplacian of
a graph and is the Cheeger constant measuring the edge expansion of
. 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 -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 , was studied by Parzanchevski,
Rosenthal and Tessler. They showed that indeed , where
is the smallest non-trivial eigenvalue of the (-dimensional
upper) Laplacian, for the case of -dimensional simplicial complexes with
complete -skeleton.
Whether this inequality also holds for -dimensional complexes with
non-complete -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
Trendreport ambulante soziale Unterstützungsdienstleistungen im Alter - arbeitswissenschaftliche Perspektiven
Der ‚Trendreport ambulante soziale Unterstützungsdienstleistungen im Alter‘ bezieht
sich auf ein Feld, das im Kern durch die Altenpflege und darüber hinaus bzw. darum
herum durch mannigfaltige weitere Unterstützungsangebote gekennzeichnet ist. Der
Trendreport wird der Frage nachgehen, welches Produktivitätsverständnis diesem Feld
personaler Unterstützungstätigkeiten angesichts seiner dynamischen Entwicklung in
den vergangenen zwei Dekaden angemessen ist. Unsere These lautet, dass hier allein
ein erweitertes Produktivitätsverständnis passend ist, das neben der Kosteneffizienz
auch die Unterstützungsqualität sowie die Arbeitsqualität mit einbezieht.17
Opinion on the follow-up of the re-evaluation of sorbic acid (E200) and potassium sorbate (E202) as food additives
Acknowledgements: The FAF Panel wishes to thank EFSA staff member(s): Jose Cortinas Abrahantes, Dimitrios Chrysafidis and Petra Gelgelova for the support provided to this scientific outputPublisher PD
On Eigenvalues of Random Complexes
We consider higher-dimensional generalizations of the normalized Laplacian
and the adjacency matrix of graphs and study their eigenvalues for the
Linial-Meshulam model of random -dimensional simplicial complexes
on vertices. We show that for , the eigenvalues of
these matrices are a.a.s. concentrated around two values. The main tool, which
goes back to the work of Garland, are arguments that relate the eigenvalues of
these matrices to those of graphs that arise as links of -dimensional
faces. Garland's result concerns the Laplacian; we develop an analogous result
for the adjacency matrix. The same arguments apply to other models of random
complexes which allow for dependencies between the choices of -dimensional
simplices. In the second part of the paper, we apply this to the question of
possible higher-dimensional analogues of the discrete Cheeger inequality, which
in the classical case of graphs relates the eigenvalues of a graph and its edge
expansion. It is very natural to ask whether this generalizes to higher
dimensions and, in particular, whether the higher-dimensional Laplacian spectra
capture the notion of coboundary expansion - a generalization of edge expansion
that arose in recent work of Linial and Meshulam and of Gromov. We show that
this most straightforward version of a higher-dimensional discrete Cheeger
inequality fails, in quite a strong way: For every and , there is a -dimensional complex on vertices that
has strong spectral expansion properties (all nontrivial eigenvalues of the
normalised -dimensional Laplacian lie in the interval
) but whose coboundary expansion is bounded
from above by and so tends to zero as ;
moreover, can be taken to have vanishing integer homology in dimension
less than .Comment: Extended full version of an extended abstract that appeared at SoCG
2012, to appear in Israel Journal of Mathematic
Guidance on the use of the Threshold of Toxicological Concern approach in food safety assessment
Peer reviewedPublisher PD