Sparse partition universal graphs for graphs of bounded degree

In 1983, Chvatal, Trotter and the two senior authors proved that for any Delta there exists a constant B such that, for any n, any 2-colouring of the edges of the complete graph K(N) with N >= Bn vertices yields a monochromatic copy of any graph H that has n vertices and maximum degree Delta. We prove that the complete graph may be replaced by a sparser graph G that has N vertices and O(N(2-1/Delta)log(1/Delta)N) edges, with N = [B`n] for some constant B` that depends only on Delta. Consequently, the so-called size-Ramsey number of any H with n vertices and maximum degree Delta is O(n(2-1/Delta)log(1/Delta)n) Our approach is based on random graphs; in fact, we show that the classical Erdos-Renyi random graph with the numerical parameters above satisfies a stronger partition property with high probability, namely, that any 2-colouring of its edges contains a monochromatic universal graph for the class of graphs on n vertices and maximum degree Delta. The main tool in our proof is the regularity method, adapted to a suitable sparse setting. The novel ingredient developed here is an embedding strategy that allows one to embed bounded degree graphs of linear order in certain pseudorandom graphs. Crucial to our proof is the fact that regularity is typically inherited at a scale that is much finer than the scale at which it is assumed. (C) 2011 Elsevier Inc. All rights reserved.

Desingularization of singular Riemannian foliation

Let F be a singular Riemannian foliation on a compact Riemannian manifold M. By successive blow-ups along the strata of F we construct a regular Riemannian foliation (F) over cap on a compact Riemannian manifold (M) over cap and a desingularization map (rho) over cap : (M) over cap -> M that projects leaves of (F) over cap into leaves of F. This result generalizes a previous result due to Molino for the particular case of a singular Riemannian foliation whose leaves were the closure of leaves of a regular Riemannian foliation. We also prove that, if the leaves of F are compact, then, for each small epsilon > 0, we can find (M) over cap and (F) over cap so that the desingularization map induces an epsilon-isometry between M/F and (M) over cap/(F) over cap. This implies in particular that the space of leaves M/F is a Gromov-Hausdorff limit of a sequence of Riemannian orbifolds {((M) over cap (n)/(F) over cap (n))}.

Stability of periodic travelling shallow-water waves determined by Newton`s equation

We study the existence and stability of periodic travelling-wave solutions for generalized Benjamin-Bona-Mahony and Camassa-Holm equations. To prove orbital stability, we use the abstract results of Grillakis-Shatah-Strauss and the Floquet theory for periodic eigenvalue problems.

Retinal removal up-regulates cannabinoid CB1 receptors in the chick optic tectum

The endocannabinoid system has been implicated in several neurobiological processes, including neurodegeneration and neuro protection. The aim of this study was to evaluate the effects of unilateral retinal ablation on the expression of the cannabinoid receptor subtype 1 (CB1) at both protein and mRNA levels in the optic tectum of the adult chick brain. After different survival times postlesion (2-30 days), the chick brains were subjected to immunohistochemical, immunoblotting, and real-time PCR procedures to evaluate CB1 expression. TUNEL and Fluoro-Jade B were used to verify the possible occurrence of cell death, and immunostaining for the microtubule-associated protein MAP-2 was performed to verify possible dendritic remodeling after lesions. No cell death could be observed in the deafferented tectum, at least up to 30 days postlesion, although Fluoro-Jade B could reveal degenerating axons and terminals. Retinal ablation seems to generate an increase of CB1 protein in the optic tectum and other retinorecipient visual areas, which paralleled an increase in MAP-2 staining. On the other hand, CB, mRNA levels were not changed after retinal ablation. Our results reveal that CB, expression in visual structures of the adult chick brain may be negatively regulated by the retinal innervation. The increase of CB1 receptor expression observed after retinal removal indicates that these receptors are not presynaptic in retinal axons projecting to the tectum and suggests a role of the cannabinoid system in plasticity processes ensuing after lesions. (c) 2008 Wiley-Liss, Inc.

A Bayesian model for estimating the malaria transition probabilities considering individuals lost to follow-up

It is known that patients may cease participating in a longitudinal study and become lost to follow-up. The objective of this article is to present a Bayesian model to estimate the malaria transition probabilities considering individuals lost to follow-up. We consider a homogeneous population, and it is assumed that the considered period of time is small enough to avoid two or more transitions from one state of health to another. The proposed model is based on a Gibbs sampling algorithm that uses information of lost to follow-up at the end of the longitudinal study. To simulate the unknown number of individuals with positive and negative states of malaria at the end of the study and lost to follow-up, two latent variables were introduced in the model. We used a real data set and a simulated data to illustrate the application of the methodology. The proposed model showed a good fit to these data sets, and the algorithm did not show problems of convergence or lack of identifiability. We conclude that the proposed model is a good alternative to estimate probabilities of transitions from one state of health to the other in studies with low adherence to follow-up.

Photo-assisted electrochemical degradation of real textile wastewater

In the present study, photo-assisted electrochemical degradation of real textile wastewater was performed. Degradation assays were performed at constant current (40 mA cm(-2)) in a combined electro/photochemical flow-cell using a Ti/Ru(0.3)Ti(0.7)O(2) DSA(R) type electrode. The results show that the method is capable of removing color and chemical oxygen demand (COD) from the effluent. Additionally, the effect of initial pH and type of supporting electrolyte (Na(2)SO(4) or NaCl) was investigated. The principal figures of merit used in this study were COD removal and color removal (605 nm). The results show that up to 72% color and up to 59% COD removal in 120 min is possible under the operating conditions employed. Studies of the phytotoxicity of the wastewater before and after the photo-assisted degradation assays are also presented and the results demonstrate that the toxicity of the effluent is dependent on the length of electrolysis time and the treatment procedure employed.