Dense packing on uniform lattices

We study the Hard Core Model on the graphs ${\rm {\bf \scriptstyle G}}$ obtained from Archimedean tilings i.e. configurations in $\scriptstyle \{0,1\}^{{\rm {\bf G}}}$ with the nearest neighbor 1's forbidden. Our particular aim in choosing these graphs is to obtain insight to the geometry of the densest packings in a uniform discrete set-up. We establish density bounds, optimal configurations reaching them in all cases, and introduce a probabilistic cellular automaton that generates the legal configurations. Its rule involves a parameter which can be naturally characterized as packing pressure. It can have a critical value but from packing point of view just as interesting are the noncritical cases. These phenomena are related to the exponential size of the set of densest packings and more specifically whether these packings are maximally symmetric, simple laminated or essentially random packings.Comment: 18 page

The moduli space of isometry classes of globally hyperbolic spacetimes

This is the last article in a series of three initiated by the second author. We elaborate on the concepts and theorems constructed in the previous articles. In particular, we prove that the GH and the GGH uniformities previously introduced on the moduli space of isometry classes of globally hyperbolic spacetimes are different, but the Cauchy sequences which give rise to well-defined limit spaces coincide. We then examine properties of the strong metric introduced earlier on each spacetime, and answer some questions concerning causality of limit spaces. Progress is made towards a general definition of causality, and it is proven that the GGH limit of a Cauchy sequence of $\mathcal{C}^{\pm}_{\alpha}$, path metric Lorentz spaces is again a $\mathcal{C}^{\pm}_{\alpha}$, path metric Lorentz space. Finally, we give a necessary and sufficient condition, similar to the one of Gromov for the Riemannian case, for a class of Lorentz spaces to be precompact.Comment: 29 pages, 9 figures, submitted to Class. Quant. Gra

An update on the Hirsch conjecture

The Hirsch conjecture was posed in 1957 in a letter from Warren M. Hirsch to George Dantzig. It states that the graph of a d-dimensional polytope with n facets cannot have diameter greater than n - d. Despite being one of the most fundamental, basic and old problems in polytope theory, what we know is quite scarce. Most notably, no polynomial upper bound is known for the diameters that are conjectured to be linear. In contrast, very few polytopes are known where the bound $n-d$ is attained. This paper collects known results and remarks both on the positive and on the negative side of the conjecture. Some proofs are included, but only those that we hope are accessible to a general mathematical audience without introducing too many technicalities.Comment: 28 pages, 6 figures. Many proofs have been taken out from version 2 and put into the appendix arXiv:0912.423

Causal structures and causal boundaries

We give an up-to-date perspective with a general overview of the theory of causal properties, the derived causal structures, their classification and applications, and the definition and construction of causal boundaries and of causal symmetries, mostly for Lorentzian manifolds but also in more abstract settings.Comment: Final version. To appear in Classical and Quantum Gravit

The repulsive lattice gas, the independent-set polynomial, and the Lov\'asz local lemma

We elucidate the close connection between the repulsive lattice gas in equilibrium statistical mechanics and the Lovasz local lemma in probabilistic combinatorics. We show that the conclusion of the Lovasz local lemma holds for dependency graph G and probabilities {p_x} if and only if the independent-set polynomial for G is nonvanishing in the polydisc of radii {p_x}. Furthermore, we show that the usual proof of the Lovasz local lemma -- which provides a sufficient condition for this to occur -- corresponds to a simple inductive argument for the nonvanishing of the independent-set polynomial in a polydisc, which was discovered implicitly by Shearer and explicitly by Dobrushin. We also present some refinements and extensions of both arguments, including a generalization of the Lovasz local lemma that allows for "soft" dependencies. In addition, we prove some general properties of the partition function of a repulsive lattice gas, most of which are consequences of the alternating-sign property for the Mayer coefficients. We conclude with a brief discussion of the repulsive lattice gas on countably infinite graphs.Comment: LaTex2e, 97 pages. Version 2 makes slight changes to improve clarity. To be published in J. Stat. Phy

Inducible cAMP Early Repressor (ICER) and Brain Functions

The inducible cAMP early repressor (ICER) is an endogenous repressor of cAMP-responsive element (CRE)-mediated gene transcription and belongs to the CRE-binding protein (CREB)/CRE modulator (CREM)/activating transcription factor 1 (ATF-1) gene family. ICER plays an important role in regulating the neuroendocrine system and the circadian rhythm. Other aspects of ICER function have recently attracted heightened attention. Being a natural inducible CREB antagonist, and more broadly, an inducible repressor of CRE-mediated gene transcription, ICER regulates long-lasting plastic changes that occur in the brain in response to incoming stimulation. This review will bring together data on ICER and its functions in the brain, with a special emphasis on recent findings highlighting the involvement of ICER in the regulation of long-term plasticity underlying learning and memory

Successful Cognitive Aging in Rats: A Role for mGluR5 Glutamate Receptors, Homer 1 Proteins and Downstream Signaling Pathways

Normal aging is associated with impairments in cognition, especially learning and memory. However, major individual differences are known to exist. Using the classical Morris Water Maze (MWM) task, we discriminated a population of 24-months old Long Evans aged rats in two groups - memory-impaired (AI) and memory-unimpaired (AU) in comparison with 6-months old adult animals. AI rats presented deficits in learning, reverse memory and retention. At the molecular level, an increase in metabotropic glutamate receptors 5 (mGluR5) was observed in post-synaptic densities (PSD) in the hippocampus of AU rats after training. Scaffolding Homer 1b/c proteins binding to group 1 mGluR facilitate coupling with its signaling effectors while Homer 1a reduces it. Both Homer 1a and 1b/c levels were up-regulated in the hippocampus PSD of AU animals following MWM task. Using immunohistochemistry we further demonstrated that mGluR5 as well as Homer 1b/c stainings were enhanced in the CA1 hippocampus sub-field of AU animals. In fact mGluR5 and Homer 1 isoforms were more abundant and co-localized in the hippocampal dendrites in AU rats. However, the ratio of Homer 1a/Homer 1b/c bound to mGluR5 in the PSD was four times lower for AU animals compared to AI rats. Consequently, AU animals presented higher PKCγ, ERK, p70S6K, mTOR and CREB activation. Finally the expression of immediate early gene Arc/Arg3.1 was shown to be higher in AU rats in accordance with its role in spatial memory consolidation. On the basis of these results, a model of successful cognitive aging with a critical role for mGluR5, Homer 1 proteins and downstream signalling pathways is proposed here