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 scaling attractor and ultimate dynamics for Smoluchowski's coagulation equations

We describe a basic framework for studying dynamic scaling that has roots in dynamical systems and probability theory. Within this framework, we study Smoluchowski's coagulation equation for the three simplest rate kernels $K(x,y)=2$, $x+y$ and $xy$. In another work, we classified all self-similar solutions and all universality classes (domains of attraction) for scaling limits under weak convergence (Comm. Pure Appl. Math 57 (2004)1197-1232). Here we add to this a complete description of the set of all limit points of solutions modulo scaling (the scaling attractor) and the dynamics on this limit set (the ultimate dynamics). The main tool is Bertoin's L\'{e}vy-Khintchine representation formula for eternal solutions of Smoluchowski's equation (Adv. Appl. Prob. 12 (2002) 547--64). This representation linearizes the dynamics on the scaling attractor, revealing these dynamics to be conjugate to a continuous dilation, and chaotic in a classical sense. Furthermore, our study of scaling limits explains how Smoluchowski dynamics ``compactifies'' in a natural way that accounts for clusters of zero and infinite size (dust and gel)

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

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

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

The effect of cigarette smoke exposure on the development of inflammation in lungs, gut and joints of TNFΔARE mice

The inflammatory cytokine TNF-alpha is a central mediator in many immune-mediated diseases, such as Crohn's disease (CD), spondyloarthritis (SpA) and chronic obstructive pulmonary disease (COPD). Epidemiologic studies have shown that cigarette smoking (CS) is a prominent common risk factor in these TNF-dependent diseases. We exposed TNF Delta ARE mice; in which a systemic TNF-alpha overexpression leads to the development of inflammation; to 2 or 4 weeks of air or CS. We investigated the effect of deregulated TNF expression on CS-induced pulmonary inflammation and the effect of CS exposure on the initiation and progression of gut and joint inflammation. Upon 2 weeks of CS exposure, inflammation in lungs of TNF Delta ARE mice was significantly aggravated. However, upon 4 weeks of CS-exposure, this aggravation was no longer observed. TNF Delta ARE mice have no increases in CD4+ and CD8+ T cells and a diminished neutrophil response in the lungs after 4 weeks of CS exposure. In the gut and joints of TNF Delta ARE mice, 2 or 4 weeks of CS exposure did not modulate the development of inflammation. In conclusion, CS exposure does not modulate gut and joint inflammation in TNF Delta ARE mice. The lung responses towards CS in TNF Delta ARE mice however depend on the duration of CS exposure

Mouse models for pseudoxanthoma elasticum: Genetic and dietary modulation of the ectopic mineralization phenotypes

Pseudoxanthoma elasticum (PXE), a heritable ectopic mineralization disorder, is caused by mutations in the ABCC6 gene. Null mice ( Abcc6 -/-) recapitulate the genetic, histopathologic and ultrastructural features of PXE, and they demonstrate early and progressive mineralization of vibrissae dermal sheath, which serves as a biomarker of the overall mineralization process. Recently, as part of a mouse aging study at The Jackson Laboratory, 31 inbred mouse strains were necropsied, and two of them, KK/HlJ and 129S1/SvImJ, were noted to have vibrissae dermal mineralization similar to Abcc6-/- mice. These two strains were shown to harbor a single nucleotide polymorphism (rs32756904) in the Abcc6 gene, which resulted in out-of-frame splicing and marked reduction in ABCC6 protein expression in the liver of these mice. The same polymorphism is present in two additional mouse strains, DBA/2J and C3H/HeJ, with similar reduction in Abcc6 protein levels, yet these mice did not demonstrate tissue mineralization when kept on standard rodent diet. However, all four mouse strains, when placed on experimental diet enriched in phosphate and low in magnesium, developed extensive ectopic mineralization. These results indicate that the genetic background of mice and the mineral composition of their diet can profoundly modulate the ectopic mineralization process predicated on mutations in the Abcc6 gene. These mice provide novel model systems to study the pathomechanisms and the reasons for strain background on phenotypic variability of PXE. © 2014 Li et al

Considerations for best practices in studies of fiber or other dietary components and the intestinal microbiome

Considerations for best practices in studies of fiber or other dietary components and the intestinal microbiome. Am J Physiol Endocrinol Metab 315: E1087–E1097, 2018. First published August 21, 2018; doi:10.1152/ajpendo.00058.2018.—A 2-day workshop organized by the National Institutes of Health and U.S. Department of Agriculture included 16 presentations focused on the role of diet in alterations of the gastrointestinal microbiome, primarily that of the colon. Although thousands of research projects have been funded by U.S. federal agencies to study the intestinal microbiome of humans and a variety of animal models, only a minority addresses dietary effects, and a small subset is described in sufficient detail to allow reproduction of a study. Whereas there are standards being developed for many aspects of microbiome studies, such as sample collection, nucleic acid extraction, data handling, etc., none has been proposed for the dietary component; thus this workshop focused on the latter specific point. It is important to foster rigor in design and reproducibility of published studies to maintain high quality and enable designs that can be compared in systematic reviews. Speakers addressed the influence of the structure of the fermentable carbohydrate on the microbiota and the variables to consider in design of studies using animals, in vitro models, and human subjects. For all types of studies, strengths and weaknesses of various designs were highlighted, and for human studies, comparisons between controlled feeding and observational designs were discussed. Because of the lack of published, best-diet formulations for specific research questions, the main recommendation is to describe dietary ingredients and treatments in as much detail as possible to allow reproduction by other scientists

Random Convex Hulls and Extreme Value Statistics

In this paper we study the statistical properties of convex hulls of $N$ random points in a plane chosen according to a given distribution. The points may be chosen independently or they may be correlated. After a non-exhaustive survey of the somewhat sporadic literature and diverse methods used in the random convex hull problem, we present a unifying approach, based on the notion of support function of a closed curve and the associated Cauchy's formulae, that allows us to compute exactly the mean perimeter and the mean area enclosed by the convex polygon both in case of independent as well as correlated points. Our method demonstrates a beautiful link between the random convex hull problem and the subject of extreme value statistics. As an example of correlated points, we study here in detail the case when the points represent the vertices of $n$ independent random walks. In the continuum time limit this reduces to $n$ independent planar Brownian trajectories for which we compute exactly, for all $n$, the mean perimeter and the mean area of their global convex hull. Our results have relevant applications in ecology in estimating the home range of a herd of animals. Some of these results were announced recently in a short communication [Phys. Rev. Lett. {\bf 103}, 140602 (2009)].Comment: 61 pages (pedagogical review); invited contribution to the special issue of J. Stat. Phys. celebrating the 50 years of Yeshiba/Rutgers meeting