Isoperimetric Inequalities in Simplicial Complexes

In graph theory there are intimate connections between the expansion properties of a graph and the spectrum of its Laplacian. In this paper we define a notion of combinatorial expansion for simplicial complexes of general dimension, and prove that similar connections exist between the combinatorial expansion of a complex, and the spectrum of the high dimensional Laplacian defined by Eckmann. In particular, we present a Cheeger-type inequality, and a high-dimensional Expander Mixing Lemma. As a corollary, using the work of Pach, we obtain a connection between spectral properties of complexes and Gromov's notion of geometric overlap. Using the work of Gunder and Wagner, we give an estimate for the combinatorial expansion and geometric overlap of random Linial-Meshulam complexes

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 $X^k(n,p)$ of random $k$-dimensional simplicial complexes on $n$ vertices. We show that for $p=\Omega(\log n/n)$, 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 $(k-2)$-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 $k$-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 $k\geq 2$ and $n\in \mathbb{N}$, there is a $k$-dimensional complex $Y^k_n$ on $n$ vertices that has strong spectral expansion properties (all nontrivial eigenvalues of the normalised $k$-dimensional Laplacian lie in the interval $[1-O(1/\sqrt{n}),1+O(1/\sqrt{n})]$) but whose coboundary expansion is bounded from above by $O(\log n/n)$ and so tends to zero as $n\rightarrow \infty$; moreover, $Y^k_n$ can be taken to have vanishing integer homology in dimension less than $k$.Comment: Extended full version of an extended abstract that appeared at SoCG 2012, to appear in Israel Journal of Mathematic

Quality of Verbal communication of health care team with hospitalized children under 3

Background and aimFor different reasons, children may experience hospitalization. With respect to the importance of human relationships and verbal communication in this group for creating confidence and a sense of security with healthcare team, this descriptive study was conducted to identify the quality of verbal communication of the team with hospitalized children under 3 in 2008.  Materials and methods253 nurses and 87 physicians working in pediatric wards of hospitals affiliated to Tehran University of Medical Sciences took part. Data collection tools included a demographic questionnaire and a verbal communication checklist in 3 levels: good, moderate and poor. The checklist was designed for 2 age groups (0-1 and 1-3) to evaluate age differences and communication needs more accurately. Chi-square and exact Fisher tests were used for data analysis. Data were analyzed by SPSS 11. Findings%41.9 and %60.8 of nurses had poor communication with children 0-1 and 1-3 respectively. %42.3 of head nurses had good communication with children 0-1 and 1-3. %38.4 and %48.9 of physicians had good communication with children 0-1 and 1-3 respectively. A significant correlation was found between the verbal communication quality of nurses and such demographic variables as having children, educational level and shift time as well as of head nurses and educational level (

The role of working memory in attentional capture

Much previous research has demonstrated that visual search is typically disrupted by the presence of a unique “singleton” distractor in the search display. Here we show that attentional capture by an irrelevant color singleton during shape search critically depends on availability of working memory to the search task: When working memory is loaded in a concurrent yet unrelated verbal short-term memory task, capture increases. These findings converge with previous demonstrations that increasing working memory load results in greater distractor interference in Stroop-like tasks (de Fockert, Rees, Frith, & Lavie, 2001; Lavie, Hirst, de Fockert, & Viding, 2004), which support the hypothesis that working memory provides goal-directed control of visual selective attention allowing to minimize interference by goal-irrelevant distractors

Smallest C2l+1C_{2l+1}-Critical Graphs of Odd-Girth 2k+12k+1

International audienc

Fast FAST

Abstract. We present a randomized subexponential time, polynomial space parameterized algorithm for the k-Weighted Feedback Arc Set in Tournaments (k-FAST) problem. We also show that our algorithm can be derandomized by slightly increasing the running time. To derandomize our algorithm we construct a new kind of universal hash functions, that we coin universal coloring families. For integers m, k and r, a family F of functions from [m] to [r] is called a universal (m, k, r)-coloring family if for any graph G on the set of vertices [m] with at most k edges, there exists an f ∈ F which is a proper vertex coloring of G. Our algorithm is the first non-trivial subexponential time parameterized algorithm outside the framework of bidimensionality.