We begin with the same setup as before: Let x be a Heegner point on X0(N), let c be the divisor (x)− (∞), and let d be the divisor (x)− (0). We need to compute the Néron height pairing 〈c, Tmc〉 = 〈c, Tmd〉. We have seen that this global height pairing decomposes into a sum of local height pairings. Andrew computed the local height pairings at the archimedean places two weeks ago; our goal will b...

In this paper, we give a refinement of the work of Gross and Zagier on singular moduli. Let K1 and K2 be two imaginary quadratic fields with relatively prime discriminants d1 and d2, and let F = Q( √ d1d2). Hecke constructed a Hilbert modular Eisenstein series over F of weight 1 whose functional equation forces it to vanish at s = 0. For CM elliptic curves E1 and E2 with complex multiplication ...

Although a few cases of sinonasal carcinoma with focal sarcomatous differentiation have been reported, pure sarcomatoid carcinoma has not been reported in the English literature. Imaging studies and gross inspection in a 60-year-old man with left-sided face pain revealed a mass in the left maxillary sinus and nasal cavity. A large incisional biopsy specimen from the nasal cavity revealed prolif...

Assessing children’s perceptions of their movement abilities (i.e., perceived competence) is traditionally done using picture scales—Pictorial Scale of Perceived Competence and Acceptance for Young Children or Pictorial Scale of Perceived Movement Skill Competence. Pictures fail to capture the temporal components of movement. To address this limitation, we created a digital-based instrument to ...

Recently it has been shown that the gross structure of the bottomonium spectrum is reproduced reasonably well within the non–relativistic boundstate theory based on perturba-tive QCD. In that calculation, however, the fine splittings and the S–P level splittings are predicted to be considerably narrower than the corresponding experimental values. We investigate the bottomonium spectrum within a...

We apply the projected Gross-Pitaevskii equation (PGPE) formalism to the experimental problem of the shift in critical temperature Tc of a harmonically confined Bose gas as reported in Gerbier et al., Phys. Rev. Lett. 92, 030405 (2004). The PGPE method includes critical fluctuations and we find the results differ from various mean-field theories, and are in best agreement with experimental data...

One typically observes differing trends in the distribution of gross incomes across countries. Where do these inequalities originate from? We consider a simple model where individuals have identical preferences but different abilities, and we investigate the impact on the distribution of gross income of changes in the way abilities are distributed among the individuals. Journal of Economic Lite...

Strongly-interacting bosons in two-dimensional harmonic traps are described through breaking of rotational symmetry at the Hartree-Fock level and subsequent symmetry restoration via projection techniques, thus incorporating correlations beyond the Gross-Pitaevskii (GP) solution. The bosons localize and form polygonal-ringlike crystalline patterns, both for a repulsive contact potential and a Co...

Prenatal exposure to an inflammatory stimulus has been shown to cause renal damage in offspring. Our present study explored the role of intra-renal NF-κB activation in the development of progressive renal fibrosis in offspring that underwent prenatal exposure to an inflammatory stimulus. Time-dated pregnant rats were treated with saline (control group) or 0.79 mg/kg lipopolysaccharide (LPS) thr...