Generalized Riemann sums

The primary aim of this chapter is, commemorating the 150th anniversary of Riemann's death, to explain how the idea of {\it Riemann sum} is linked to other branches of mathematics. The materials I treat are more or less classical and elementary, thus available to the "common mathematician in the streets." However one may still see here interesting inter-connection and cohesiveness in mathematics

Equidistribution Rates, Closed String Amplitudes, and the Riemann Hypothesis

We study asymptotic relations connecting unipotent averages of $Sp(2g,\mathbb{Z})$ automorphic forms to their integrals over the moduli space of principally polarized abelian varieties. We obtain reformulations of the Riemann hypothesis as a class of problems concerning the computation of the equidistribution convergence rate in those asymptotic relations. We discuss applications of our results to closed string amplitudes. Remarkably, the Riemann hypothesis can be rephrased in terms of ultraviolet relations occurring in perturbative closed string theory.Comment: 15 page

Open complete dislocation of trapezium with a vertically split fracture: a case report

Open complete dislocation of the trapezium is an extraordinarily rare injury with only a few cases reported so far in literature. The association of a vertically split fracture makes this injury even rare and hence worth reporting. A 14 year old Kashmiri boy presented to us with a history of massive trauma to the non dominant left hand sustained as a result of a blow from a heavy hammer. The thenar area was burst out and the trapezium was vertically split apart into two halves which were dislocated from the articular surfaces of the scaphoid as well as the first metacarpal. The mechanism of injury as in other such reported cases was a massive direct force localized over the carpal bone which causes its enucleation and fracture. Although some authors have recommended excision of the dislocated trapezium, open reduction of the fracture dislocation and fixation with K wires was carried out under General anesthesia. At the end of one year although there was some functional deficit in the affected thumb, especially in opposition, the patient was quite satisfied with the outcome as this was the non dominant hand

Spherical averages in the space of marked lattices

A marked lattice is a $d$-dimensional Euclidean lattice, where each lattice point is assigned a mark via a given random field on ${\mathbb Z}^d$. We prove that, if the field is strongly mixing with a faster-than-logarithmic rate, then for every given lattice and almost every marking, large spheres become equidistributed in the space of marked lattices. A key aspect of our study is that the space of marked lattices is not a homogeneous space, but rather a non-trivial fiber bundle over such a space. As an application, we prove that the free path length in a crystal with random defects has a limiting distribution in the Boltzmann-Grad limit

Recent Results on the Periodic Lorentz Gas

The Drude-Lorentz model for the motion of electrons in a solid is a classical model in statistical mechanics, where electrons are represented as point particles bouncing on a fixed system of obstacles (the atoms in the solid). Under some appropriate scaling assumption -- known as the Boltzmann-Grad scaling by analogy with the kinetic theory of rarefied gases -- this system can be described in some limit by a linear Boltzmann equation, assuming that the configuration of obstacles is random [G. Gallavotti, [Phys. Rev. (2) vol. 185 (1969), 308]). The case of a periodic configuration of obstacles (like atoms in a crystal) leads to a completely different limiting dynamics. These lecture notes review several results on this problem obtained in the past decade as joint work with J. Bourgain, E. Caglioti and B. Wennberg.Comment: 62 pages. Course at the conference "Topics in PDEs and applications 2008" held in Granada, April 7-11 2008; figure 13 and a misprint in Theorem 4.6 corrected in the new versio

Pediatric Cushing disease: disparities in disease severity and outcomes in the Hispanic and African-American populations.

BackgroundLittle is known about the contribution of racial and socioeconomic disparities to severity and outcomes in children with Cushing disease (CD).MethodsA total of 129 children with CD, 45 Hispanic/Latino or African-American (HI/AA) and 84 non-Hispanic White (non-HW), were included in this study. A 10-point index for rating severity (CD severity) incorporated the degree of hypercortisolemia, glucose tolerance, hypertension, anthropomorphic measurements, disease duration, and tumor characteristics. Race, ethnicity, age, gender, local obesity prevalence, estimated median income, and access to care were assessed in regression analyses of CD severity.ResultsThe mean CD severity in the HI/AA group was worse than that in the non-HW group (4.9±2.0 vs. 4.1±1.9, P=0.023); driving factors included higher cortisol levels and larger tumor size. Multiple regression models confirmed that race (P=0.027) and older age (P=0.014) were the most important predictors of worse CD severity. When followed up a median of 2.3 years after surgery, the relative risk for persistent CD combined with recurrence was 2.8 times higher in the HI/AA group compared with that in the non-HW group (95% confidence interval: 1.2-6.5).ConclusionOur data show that the driving forces for the discrepancy in severity of CD are older age and race/ethnicity. Importantly, the risk for persistent and recurrent CD was higher in minority children

Risk assessment for the spread of Serratia marcescens within dental-unit waterline systems using Vermamoeba vermiformis

Vermamoeba vermiformis is associated with the biofilm ecology of dental-unit waterlines (DUWLs). This study investigated whether V. vermiformis is able to act as a vector for potentially pathogenic bacteria and so aid their dispersal within DUWL systems. Clinical dental water was initially examined for Legionella species by inoculating it onto Legionella selective-medium plates. The molecular identity/profile of the glassy colonies obtained indicated none of these isolates were Legionella species. During this work bacterial colonies were identified as a non-pigmented Serratia marcescens. As the water was from a clinical DUWL which had been treated with Alpron™ this prompted the question as to whether S. marcescens had developed resistance to the biocide. Exposure to Alpron™ indicated that this dental biocide was effective, under laboratory conditions, against S. marcescens at up to 1x108 colony forming units/millilitre (cfu/ml). V. vermiformis was cultured for eight weeks on cells of S. marcescens and Escherichia coli. Subsequent electron microscopy showed that V. vermiformis grew equally well on S. marcescens and E. coli (p = 0.0001). Failure to detect the presence of S. marcescens within the encysted amoebae suggests that V. vermiformis is unlikely to act as a vector supporting the growth of this newly isolated, nosocomial bacterium

The development of path integration: combining estimations of distance and heading

Efficient daily navigation is underpinned by path integration, the mechanism by which we use self-movement information to update our position in space. This process is well-understood in adulthood, but there has been relatively little study of path integration in childhood, leading to an underrepresentation in accounts of navigational development. Previous research has shown that calculation of distance and heading both tend to be less accurate in children as they are in adults, although there have been no studies of the combined calculation of distance and heading that typifies naturalistic path integration. In the present study 5-year-olds and 7-year-olds took part in a triangle-completion task, where they were required to return to the startpoint of a multi-element path using only idiothetic information. Performance was compared to a sample of adult participants, who were found to be more accurate than children on measures of landing error, heading error, and distance error. 7-year-olds were significantly more accurate than 5-year-olds on measures of landing error and heading error, although the difference between groups was much smaller for distance error. All measures were reliably correlated with age, demonstrating a clear development of path integration abilities within the age range tested. Taken together, these data make a strong case for the inclusion of path integration within developmental models of spatial navigational processing

Rapture of renal angiomyolipoma during pregnancy: a case report

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