Isometric immersions into a homogeneous Lorentzian Heisenberg group and rigidity

In this paper we prove an existence result for local and global isometric immersions of semi-Riemannian surfaces into the three dimensional Heisenberg group endowed with a homogeneous left-invariant Lorentzian metric. As a corollary, we prove a rigidity result for such immersions.

SIZE AS A LINE OF LEAST RESISTANCE II: DIRECT SELECTION ON SIZE OR CORRELATED RESPONSE DUE TO CONSTRAINTS?

Evolutionary change in New World Monkey (NWM) skulls occurred primarily along the line of least resistance defined by size (including allometric) variation (g(max)). Although the direction of evolution was aligned with this axis, it was not clear whether this macroevolutionary pattern results from the conservation of within population genetic covariance patterns (long-term constraint) or long-term selection along a size dimension, or whether both, constraints and selection, were inextricably involved. Furthermore, G-matrix stability can also be a consequence of selection, which implies that both, constraints embodied in g(max) and evolutionary changes observed on the trait averages, would be influenced by selection Here, we describe a combination of approaches that allows one to test whether any particular instance of size evolution is a correlated by-product due to constraints (g(max)) or is due to direct selection on size and apply it to NWM lineages as a case study. The approach is based on comparing the direction and amount of evolutionary change produced by two different simulated sets of net-selection gradients (beta), a size (isometric and allometric size) and a nonsize set. Using this approach it is possible to distinguish between the two hypotheses (indirect size evolution due to constraints or direct selection on size), because although both may produce an evolutionary response aligned with g(max), the amount of change produced by random selection operating through the variance/covariance patterns (constraints hypothesis) will be much smaller than that produced by selection on size (selection hypothesis). Furthermore, the alignment of simulated evolutionary changes with g(max) when selection is not on size is not as tight as when selection is actually on size, allowing a statistical test of whether a particular observed case of evolution along the line of least resistance is the result of selection along it or not. Also, with matrix diagonalization (principal components [PC]) it is possible to calculate directly the net-selection gradient on size alone (first PC [PC1]) by dividing the amount of phenotypic difference between any two populations by the amount of variation in PC1, which allows one to benchmark whether selection was on size or not

Melatonin inhibits LPS-induced NO production in rat endothelial cells

Endothelial cells produce NO by activation of constitutive nitric oxide synthase (NOS) and transcription of inducible NOS (iNOS). We have previously shown that melatonin, in the nanomolar range, inhibits activation of constitutive NOS, and in the present paper, we evaluated whether it could interfere with the expression of iNOS, which is activated by lipopolysaccharide (LPS), a major component of gram-negative bacteria cell walls. Primary cultures of rat endothelial cells were loaded with fluorescent probe for NO detection. Nuclear factor kappa B (NF-kappa B) translocation in endothelial cells elicited by LPS was measured by electromobility shift assay, and the vasodilation of aortic rings was accessed by recording isometric contraction. Melatonin in a micromolar but not in a nanomolar range inhibits the NO production induced by LPS. This effect is not dependent on the activation of G protein-coupled melatonin receptors. The nuclear NF-kappa B translocation is a process necessary for iNOS transcription, and melatonin also inhibits its translocation. LPS induced vasodilation only in endothelium-intact aortic rings, and melatonin (10 mu m) inhibits the vasodilation. Here, we show that concentrations compatible with nocturnal melatonin surge (nm) did not interfere with the activity of iNOS. Considering that micromolar melatonin concentrations could be locally achieved through production by activated immune competent cells, extra-pineal melatonin could have a protective effect against tissue injury. We propose that melatonin blocked the LPS-induced vasodilation by inhibiting the NF-kappa B pathway. Finally, we propose that the effect of melatonin on vascular reactivity is one of the mechanisms that underlies the protective effect of this indolamine against LPS.

An existence theorem for G-structure preserving affine immersions

We prove an existence result for local and global G-structure preserving affine immersions between affine manifolds. Several examples are discussed in the context of Riemannian and semi-Riemannian geometry, including the case of isometric immersions into Lie groups endowed with a left-invariant metric, and the case of isometric immersions into products of space forms.

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))}.