25 research outputs found
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
Human epicardial cell-conditioned medium contains HGF/IgG complexes that phosphorylate RYK and protect against vascular injury
Complejo universitario en Madrid. Convocatoria Abril. Plan 1996. Proyecto fin de carrera. Universidad Politécnica de Madrid. Escuela Técnica Superior de Arquitectur
Internalized FGF-2-Loaded Nanoparticles Increase Nuclear ERK1/2 Content and Result in Lung Cancer Cell Death
Innovative cancer treatments, which improve adjuvant therapy and reduce adverse events, are desperately needed. Nanoparticles provide controlled intracellular biomolecule delivery in the absence of activating external cell surface receptors. Prior reports suggest that intracrine signaling, following overexpression of basic fibroblast growth factor (FGF-2) after viral transduction, has a toxic effect on diseased cells. Herein, the research goals were to (1) encapsulate recombinant FGF-2 within stable, alginate-based nanoparticles (ABNs) for non-specific cellular uptake, and (2) determine the effects of ABN-mediated intracellular delivery of FGF-2 on cancer cell proliferation/survival. In culture, human alveolar adenocarcinoma basal epithelial cell line (A549s) and immortalized human bronchial epithelial cell line (HBE1s) internalized ABNs through non-selective endocytosis. Compared to A549s exposed to empty (i.e., blank) ABNs, the intracellular delivery of FGF-2 via ABNs significantly increased the levels of lactate dehydrogenase, indicating that FGF-2-ABN treatment decreased the transformed cell integrity. Noticeably, the nontransformed cells were not significantly affected by FGF-2-loaded ABN treatment. Furthermore, FGF-2-loaded ABNs significantly increased nuclear levels of activated-extracellular signal-regulated kinase ½ (ERK1/2) in A549s but had no significant effect on HBE1 nuclear ERK1/2 expression. Our novel intracellular delivery method of FGF-2 via nanoparticles resulted in increased cancer cell death via increased nuclear ERK1/2 activation
Random Chain Complexes
We study random, finite-dimensional, ungraded chain complexes over a finite
field and show that for a uniformly distributed differential a complex has the
smallest possible homology with the highest probability: either zero or
one-dimensional homology depending on the parity of the dimension of the
complex. We prove that as the order of the field goes to infinity the
probability distribution concentrates in the smallest possible dimension of the
homology. On the other hand, the limit probability distribution, as the
dimension of the complex goes to infinity, is a super-exponentially decreasing,
but strictly positive, function of the dimension of the homology