Super congruences and Euler numbers

Let $p>3$ be a prime. We prove that $$\sum_{k=0}^{p-1}\binom{2k}{k}/2^k=(-1)^{(p-1)/2}-p^2E_{p-3} (mod p^3),$$ $$\sum_{k=1}^{(p-1)/2}\binom{2k}{k}/k=(-1)^{(p+1)/2}8/3*pE_{p-3} (mod p^2),$$ $$\sum_{k=0}^{(p-1)/2}\binom{2k}{k}^2/16^k=(-1)^{(p-1)/2}+p^2E_{p-3} (mod p^3)$$, where E_0,E_1,E_2,... are Euler numbers. Our new approach is of combinatorial nature. We also formulate many conjectures concerning super congruences and relate most of them to Euler numbers or Bernoulli numbers. Motivated by our investigation of super congruences, we also raise a conjecture on 7 new series for $\pi^2$, $\pi^{-2}$ and the constant $K:=\sum_{k>0}(k/3)/k^2$ (with (-) the Jacobi symbol), two of which are $$\sum_{k=1}^\infty(10k-3)8^k/(k^3\binom{2k}{k}^2\binom{3k}{k})=\pi^2/2$$ and $$\sum_{k>0}(15k-4)(-27)^{k-1}/(k^3\binom{2k}{k}^2\binom{3k}k)=K.$

Statistical properties of the Burgers equation with Brownian initial velocity

We study the one-dimensional Burgers equation in the inviscid limit for Brownian initial velocity (i.e. the initial velocity is a two-sided Brownian motion that starts from the origin x=0). We obtain the one-point distribution of the velocity field in closed analytical form. In the limit where we are far from the origin, we also obtain the two-point and higher-order distributions. We show how they factorize and recover the statistical invariance through translations for the distributions of velocity increments and Lagrangian increments. We also derive the velocity structure functions and we recover the bifractality of the inverse Lagrangian map. Then, for the case where the initial density is uniform, we obtain the distribution of the density field and its $n$-point correlations. In the same limit, we derive the $n-$point distributions of the Lagrangian displacement field and the properties of shocks. We note that both the stable-clustering ansatz and the Press-Schechter mass function, that are widely used in the cosmological context, happen to be exact for this one-dimensional version of the adhesion model.Comment: 42 pages, published in J. Stat. Phy

The number of flags in finite vector spaces: Asymptotic normality and Mahonian statistics

We study the generalized Galois numbers which count flags of length r in N-dimensional vector spaces over finite fields. We prove that the coefficients of those polynomials are asymptotically Gaussian normally distributed as N becomes large. Furthermore, we interpret the generalized Galois numbers as weighted inversion statistics on the descent classes of the symmetric group on N elements and identify their asymptotic limit as the Mahonian inversion statistic when r approaches infinity. Finally, we apply our statements to derive further statistical aspects of generalized Rogers-Szegoe polynomials, re-interpret the asymptotic behavior of linear q-ary codes and characters of the symmetric group acting on subspaces over finite fields, and discuss implications for affine Demazure modules and joint probability generating functions of descent-inversion statistics.Comment: 19 pages. Corrected proof of asymptotic normality (Theorem 3.5). Previous Proposition 3.3 is fals

Topologies Refining the Cantor Topology on X ω

International audienceThe space of one-sided infinite words plays a crucial rôle in several parts of Theoretical Computer Science. Usually, it is convenient to regard this space as a metric space, the Cantor-space. It turned out that for several purposes topologies other than the one of the Cantor-space are useful, e.g. for studying fragments of first-order logic over infinite words or for a topological characterisation of random infinite words. It is shown that both of these topologies refine the topology of the Cantor-space. Moreover, from common features of these topologies we extract properties which characterise a large class of topologies. It turns out that, for this general class of topologies, the corresponding closure and interior operators respect the shift operations and also, to some respect, the definability of sets of infinite words by finite automata

Mycobacterium tuberculosis complex genetic diversity: mining the fourth international spoligotyping database (SpolDB4) for classification, population genetics and epidemiology

BACKGROUND: The Direct Repeat locus of the Mycobacterium tuberculosis complex (MTC) is a member of the CRISPR (Clustered regularly interspaced short palindromic repeats) sequences family. Spoligotyping is the widely used PCR-based reverse-hybridization blotting technique that assays the genetic diversity of this locus and is useful both for clinical laboratory, molecular epidemiology, evolutionary and population genetics. It is easy, robust, cheap, and produces highly diverse portable numerical results, as the result of the combination of (1) Unique Events Polymorphism (UEP) (2) Insertion-Sequence-mediated genetic recombination. Genetic convergence, although rare, was also previously demonstrated. Three previous international spoligotype databases had partly revealed the global and local geographical structures of MTC bacilli populations, however, there was a need for the release of a new, more representative and extended, international spoligotyping database. RESULTS: The fourth international spoligotyping database, SpolDB4, describes 1939 shared-types (STs) representative of a total of 39,295 strains from 122 countries, which are tentatively classified into 62 clades/lineages using a mixed expert-based and bioinformatical approach. The SpolDB4 update adds 26 new potentially phylogeographically-specific MTC genotype families. It provides a clearer picture of the current MTC genomes diversity as well as on the relationships between the genetic attributes investigated (spoligotypes) and the infra-species classification and evolutionary history of the species. Indeed, an independent Naïve-Bayes mixture-model analysis has validated main of the previous supervised SpolDB3 classification results, confirming the usefulness of both supervised and unsupervised models as an approach to understand MTC population structure. Updated results on the epidemiological status of spoligotypes, as well as genetic prevalence maps on six main lineages are also shown. Our results suggests the existence of fine geographical genetic clines within MTC populations, that could mirror the passed and present Homo sapiens sapiens demographical and mycobacterial co-evolutionary history whose structure could be further reconstructed and modelled, thereby providing a large-scale conceptual framework of the global TB Epidemiologic Network. CONCLUSION: Our results broaden the knowledge of the global phylogeography of the MTC complex. SpolDB4 should be a very useful tool to better define the identity of a given MTC clinical isolate, and to better analyze the links between its current spreading and previous evolutionary history. The building and mining of extended MTC polymorphic genetic databases is in progress

Epidermólisis ampollosa, reporte de un caso

Introduction: epidermolysis bullosa refers to a heterogeneous group of chronic hereditary pimple-like diseases affecting the skin and mucosae with blisters and vesicles after minimal injury, with variable involvement of other organs.Case report: 3-year-old female patient with exulcerated skin lesions and some erythema crust-plaques, with scaling at exposure sites such as hands, knees, feet, back of the neck and genitals. She was admitted to “Hermanos Cordové” Pediatric Teaching Hospital in Manzanillo. The medical care was based on maintaining the integrity of the skin avoiding trauma, temperature control, nutrition and prevention of secondary infections. A simple epidermolysis bullosa is diagnosed by skin biopsy.Conclusions: the disease is scarcely known with low incidence and prevalence. It is a major problem in the family and social environment, as parents are dealing with a rare disease of genetic origin and poor prognosis. Medical-social support helps to minimize the problems by means of information and coordination. Treatment requires the care of a multidisciplinary and specialized team.Introducción: la epidermólisis bulosa se refiere a un grupo heterogéneo de enfermedades hereditarias ampulosas crónicas, que afectan a la piel y las mucosas con formación de ampollas y vesículas tras mínimos traumatismos, con afectación variable de otros órganos.Presentación de caso: paciente femenina de 3 años de edad con lesiones de piel exulceradas y algunas eritematocostrosas en placas, con descamación en sitios de exposición como manos, rodillas, pies, parte posterior del cuello y genitales. Se encontraba ingresada en el Hospital Pediátrico Docente “Hermanos Cordové", de Manzanillo. Los cuidados médicos se basaron en mantener la entereza de la piel evitando traumatismo, control de la temperatura, nutrición y prevención de infecciones secundarias. Se diagnostica por biopsia de piel una epidermólisis bulosa simple. Conclusiones: la enfermedad es poco conocida con baja incidencia y prevalencia. Supone un problema de gran magnitud en el entorno familiar y social, al enfrentarse los padres a una enfermedad rara con origen genético y de mal pronóstico. El apoyo médico-social ayuda a minimizar los problemas, a través de la información y coordinación. Para su tratamiento es necesaria la atención de un equipo multidisciplinario y especializado.