Quicksort with unreliable comparisons: a probabilistic analysis

We provide a probabilistic analysis of the output of Quicksort when comparisons can err.Comment: 29 pages, 3 figure

Statistics of reduced words in locally free and braid groups: Abstract studies and application to ballistic growth model

We study numerically and analytically the average length of reduced (primitive) words in so-called locally free and braid groups. We consider the situations when the letters in the initial words are drawn either without or with correlations. In the latter case we show that the average length of the reduced word can be increased or lowered depending on the type of correlation. The ideas developed are used for analytical computation of the average number of peaks of the surface appearing in some specific ballistic growth modelComment: 29 pages, LaTeX, 7 separated Postscript figures (available on request), submitted to J. Phys. (A): Math. Ge

The topological structure of scaling limits of large planar maps

We discuss scaling limits of large bipartite planar maps. If p is a fixed integer strictly greater than 1, we consider a random planar map M(n) which is uniformly distributed over the set of all 2p-angulations with n faces. Then, at least along a suitable subsequence, the metric space M(n) equipped with the graph distance rescaled by the factor n to the power -1/4 converges in distribution as n tends to infinity towards a limiting random compact metric space, in the sense of the Gromov-Hausdorff distance. We prove that the topology of the limiting space is uniquely determined independently of p, and that this space can be obtained as the quotient of the Continuum Random Tree for an equivalence relation which is defined from Brownian labels attached to the vertices. We also verify that the Hausdorff dimension of the limit is almost surely equal to 4.Comment: 45 pages Second version with minor modification

Random trees between two walls: Exact partition function

We derive the exact partition function for a discrete model of random trees embedded in a one-dimensional space. These trees have vertices labeled by integers representing their position in the target space, with the SOS constraint that adjacent vertices have labels differing by +1 or -1. A non-trivial partition function is obtained whenever the target space is bounded by walls. We concentrate on the two cases where the target space is (i) the half-line bounded by a wall at the origin or (ii) a segment bounded by two walls at a finite distance. The general solution has a soliton-like structure involving elliptic functions. We derive the corresponding continuum scaling limit which takes the remarkable form of the Weierstrass p-function with constrained periods. These results are used to analyze the probability for an evolving population spreading in one dimension to attain the boundary of a given domain with the geometry of the target (i) or (ii). They also translate, via suitable bijections, into generating functions for bounded planar graphs.Comment: 25 pages, 7 figures, tex, harvmac, epsf; accepted version; main modifications in Sect. 5-6 and conclusio

Tangling clustering of inertial particles in stably stratified turbulence

We have predicted theoretically and detected in laboratory experiments a new type of particle clustering (tangling clustering of inertial particles) in a stably stratified turbulence with imposed mean vertical temperature gradient. In this stratified turbulence a spatial distribution of the mean particle number density is nonuniform due to the phenomenon of turbulent thermal diffusion, that results in formation of a gradient of the mean particle number density, \nabla N, and generation of fluctuations of the particle number density by tangling of the gradient, \nabla N, by velocity fluctuations. The mean temperature gradient, \nabla T, produces the temperature fluctuations by tangling of the gradient, \nabla T, by velocity fluctuations. These fluctuations increase the rate of formation of the particle clusters in small scales. In the laboratory stratified turbulence this tangling clustering is much more effective than a pure inertial clustering that has been observed in isothermal turbulence. In particular, in our experiments in oscillating grid isothermal turbulence in air without imposed mean temperature gradient, the inertial clustering is very weak for solid particles with the diameter 10 microns and Reynolds numbers Re =250. Our theoretical predictions are in a good agreement with the obtained experimental results.Comment: 16 pages, 4 figures, REVTEX4, revised versio

Interactions with M cells and macrophages as key steps in the pathogenesis of enterohemorrhagic Escherichia coli infections

Enterohemorrhagic Escherichia coli (EHEC) are food-borne pathogens that can cause serious infections ranging from diarrhea to hemorrhagic colitis (HC) and hemolytic-uremic syndrome (HUS). Translocation of Shiga-toxins (Stx) from the gut lumen to underlying tissues is a decisive step in the development of the infection, but the mechanisms involved remain unclear. Many bacterial pathogens target the follicle-associated epithelium, which overlies Peyer's patches (PPs), cross the intestinal barrier through M cells and are captured by mucosal macrophages. Here, translocation across M cells, as well as survival and proliferation of EHEC strains within THP-1 macrophages were investigated using EHEC O157:H7 reference strains, isogenic mutants, and 15 EHEC strains isolated from HC/HUS patients. We showed for the first time that E. coli O157:H7 strains are able to interact in vivo with murine PPs, to translocate ex vivo through murine ileal mucosa with PPs and across an in vitro human M cell model. EHEC strains are also able to survive and to produce Stx in macrophages, which induce cell apoptosis and Stx release. In conclusion, our results suggest that the uptake of EHEC by M cells and underlying macrophages in the PP may be a critical step in Stx translocation and release in vivo. A new model for EHEC infection in humans is proposed that could help in a fuller understanding of EHEC-associated diseases

Back pressure effects on variable geometry turbine performances

Paper presented at the 6th International Conference on Heat Transfer, Fluid Mechanics and Thermodynamics, South Africa, 30 June - 2 July, 2008.Turbochargers are widely used in applications to increase specific power and decrease fuel consumption. However, recent anti-pollution regulations have became stricter and pressed automotive engineers to find new solutions to reduce Nox emissions. Two of these solutions are the catalytic converter and the intercooler system. All these modifications will change the initial matching of the turbocharger performance characteristics to the engine requirements. In this paper, several compressor wheel sizes are investigated to evaluate the turbine/compressor matching. The intercooler and catalytic converter back pressure induced are respectively modeled by a lower duct section downstream the compressor stage and a variable valve downstream the turbine stage. The influences of the different modifications are identified through the loading and the flow coefficients and also on classical turbine performance maps. First, an analogy between compressor wheel size and back pressure effects is underlined. Second, it is shown that initial control settings of turbine nozzle vanes are no longer appropriate with a catalytic converter.vk201

Universality classes in Burgers turbulence

We establish necessary and sufficient conditions for the shock statistics to approach self-similar form in Burgers turbulence with L\'{e}vy process initial data. The proof relies upon an elegant closure theorem of Bertoin and Carraro and Duchon that reduces the study of shock statistics to Smoluchowski's coagulation equation with additive kernel, and upon our previous characterization of the domains of attraction of self-similar solutions for this equation

Random Operator Approach for Word Enumeration in Braid Groups

We investigate analytically the problem of enumeration of nonequivalent primitive words in the braid group B_n for n >> 1 by analysing the random word statistics and the target space on the basis of the locally free group approximation. We develop a "symbolic dynamics" method for exact word enumeration in locally free groups and bring arguments in support of the conjecture that the number of very long primitive words in the braid group is not sensitive to the precise local commutation relations. We consider the connection of these problems with the conventional random operator theory, localization phenomena and statistics of systems with quenched disorder. Also we discuss the relation of the particular problems of random operator theory to the theory of modular functionsComment: 36 pages, LaTeX, 4 separated Postscript figures, submitted to Nucl. Phys. B [PM