Representation Theory of Twisted Group Double

This text collects useful results concerning the quasi-Hopf algebra \D . We give a review of issues related to its use in conformal theories and physical mathematics. Existence of such algebras based on 3-cocycles with values in ${R} / {Z}$ which mimic for finite groups Chern-Simons terms of gauge theories, open wide perspectives in the so called "classification program". The modularisation theorem proved for quasi-Hopf algebras by two authors some years ago makes the computation of topological invariants possible. An updated, although partial, bibliography of recent developments is provided.Comment: 15 pages, no figur

Comments on the Links between su(3) Modular Invariants, Simple Factors in the Jacobian of Fermat Curves, and Rational Triangular Billiards

We examine the proposal made recently that the su(3) modular invariant partition functions could be related to the geometry of the complex Fermat curves. Although a number of coincidences and similarities emerge between them and certain algebraic curves related to triangular billiards, their meaning remains obscure. In an attempt to go beyond the su(3) case, we show that any rational conformal field theory determines canonically a Riemann surface.Comment: 56 pages, 4 eps figures, LaTeX, uses eps

Generalised Fermat Hypermaps and Galois Orbits

We consider families of quasiplatonic Riemann surfaces characterised by the fact that -- as in the case of Fermat curves of exponent $n$ -- their underlying regular (Walsh) hypermap is the complete bipartite graph $K_{n,n}$, where $n$ is an odd prime power. We will show that all these surfaces, regarded as algebraic curves, are defined over abelian number fields. We will determine the orbits under the action of the absolute Galois group, their minimal fields of definition, and in some easier cases also their defining equations. The paper relies on group-- and graph--theoretic results by G. A. Jones, R. Nedela and M.\v{S}koviera about regular embeddings of the graphs $K_{n,n}$ [JN\v{S}] and generalises the analogous question for maps treated in [JStW], partly using different methods.Comment: 14 pages, new version with extended introduction, minor corrections and updated reference

Single File Diffusion of particles with long ranged interactions: damping and finite size effects

We study the Single File Diffusion (SFD) of a cyclic chain of particles that cannot cross each other, in a thermal bath, with long ranged interactions, and arbitrary damping. We present simulations that exhibit new behaviors specifically associated to systems of small number of particles and to small damping. In order to understand those results, we present an original analysis based on the decomposition of the particles motion in the normal modes of the chain. Our model explains all dynamic regimes observed in our simulations, and provides convincing estimates of the crossover times between those regimes.Comment: 30 pages, 9 figure

Spatial Mixing and Non-local Markov chains

We consider spin systems with nearest-neighbor interactions on an $n$-vertex $d$-dimensional cube of the integer lattice graph $\mathbb{Z}^d$. We study the effects that exponential decay with distance of spin correlations, specifically the strong spatial mixing condition (SSM), has on the rate of convergence to equilibrium distribution of non-local Markov chains. We prove that SSM implies $O(\log n)$ mixing of a block dynamics whose steps can be implemented efficiently. We then develop a methodology, consisting of several new comparison inequalities concerning various block dynamics, that allow us to extend this result to other non-local dynamics. As a first application of our method we prove that, if SSM holds, then the relaxation time (i.e., the inverse spectral gap) of general block dynamics is $O(r)$, where $r$ is the number of blocks. A second application of our technology concerns the Swendsen-Wang dynamics for the ferromagnetic Ising and Potts models. We show that SSM implies an $O(1)$ bound for the relaxation time. As a by-product of this implication we observe that the relaxation time of the Swendsen-Wang dynamics in square boxes of $\mathbb{Z}^2$ is $O(1)$ throughout the subcritical regime of the $q$-state Potts model, for all $q \ge 2$. We also prove that for monotone spin systems SSM implies that the mixing time of systematic scan dynamics is $O(\log n (\log \log n)^2)$. Systematic scan dynamics are widely employed in practice but have proved hard to analyze. Our proofs use a variety of techniques for the analysis of Markov chains including coupling, functional analysis and linear algebra

Unexpected Transcripts in Tn7 orf19.2646 C. albicans Mutant Lead to Low Fungal Burden Phenotype In vivo.

The commensal fungus Candida albicans is the major cause of fungal systemic infection in immuno-compromised patients, with a mortality rate approaching 50% in the case of bloodstream infections. There is therefore a clear need to better understand fungal biology during infection to improve treatment. One of the particularities of C. albicans is its capacity to adapt to drastically diverse environments such as brain, bloodstream or gut. Adaptations to environmental change are mediated by transcription factors (TF) that modulate the expression of their target genes. Previous screening of a collection of Tn7 C. albicans TF mutants in vivo identified orf19.2646 as playing a crucial role in the ability of the fungus to survive within its host. Indeed, the orf19.2646 Tn7 interruption mutant strain displayed a reduced fungal burden compared to the wild-type strain. Surprisingly, an independent deletion mutant did not recapitulate the phenotype of the Tn7 interruption mutant. In the present study, we therefore investigated the difference between these two mutants and determined by performing a RACE analysis whether unexpected transcripts of the Tn7 mutant occurred. We found that two such transcripts upstream and downstream of the Tn7 insertion site were produced. The two transcripts were expressed in an orf19.2646 deletion mutant which displayed a significantly reduced fungal burden level compared to the wild-type in G. mellonella. When the regions corresponding to these transcripts were deleted in the Tn7 mutants, the strains lacking both regions displayed a fungal burden similar to that of the wild-type strain. This study shows for the first time that mRNA transcription may occur downstream of a Tn7 sequence. In addition, these results demonstrated that the low fungal burden phenotype observed in the orf19.2646 Tn7 mutant is due to the presence of these two transcripts together participating to an unidentified virulence mechanism to be further elucidated

IL-1α and TNF-α Down-Regulate CRH Receptor-2 mRNA Expression in the Mouse Heart

Two receptors (CRH receptor type 1 and CRH receptor type 2) have been identified for the stress-induced neuropeptide, CRH and related peptides, urocortin, and urocortin II. We previously found marked down-regulation of cardiac CRH receptor type 2 expression following administration of bacterial endotoxin, lipopolysaccharide, a model of systemic immune activation, and inflammation. We postulated that inflammatory cytokines may regulate CRH receptor type 2. We show that systemic IL-1α administration significantly down-regulates CRH receptor type 2 mRNA in mouse heart. In addition, TNFα treatment also reduces CRH receptor type 2 mRNA expression, although the effect was not as marked as with IL-1α. However, CRH receptor type 2 mRNA expression is not altered in adult mouse ventricular cardiomyocytes stimulated in vitro with TNFα or IL-1α. Thus, cytokine regulation may be indirect. Exogenous administration of corticosterone in vivo or acute restraint stress also reduces cardiac CRH receptor type 2 mRNA expression, but like cytokines, in vitro corticosterone treatment does not modulate expression in cardiomyocytes. Interestingly, treatment with urocortin significantly decreases CRH receptor type 2 mRNA in cultured cardiomyocytes. We speculate that in vivo, inflammatory mediators such as lipopolysaccharide and/or cytokines may increase urocortin, which in turn down-regulates CRH receptor type 2 expression in the heart. Because CRH and urocortin increase cardiac contractility and coronary blood flow, impaired CRH receptor type 2 function during systemic inflammation may ultimately diminish the adaptive cardiac response to adverse conditions