An explicit height bound for the classical modular polynomial

For a prime m, let Phi_m be the classical modular polynomial, and let h(Phi_m) denote its logarithmic height. By specializing a theorem of Cohen, we prove that h(Phi_m) <= 6 m log m + 16 m + 14 sqrt m log m. As a corollary, we find that h(Phi_m) <= 6 m log m + 18 m also holds. A table of h(Phi_m) values is provided for m <= 3607.Comment: Minor correction to the constants in Theorem 1 and Corollary 9. To appear in the Ramanujan Journal. 17 pages

Shipping in the north-east Atlantic : identifying spatial and temporal patterns of change

This work was supported by a faculty PhD bursary from the University of Portsmouth. Work was supported by the Marine Ecosystems Research Programme.Maritime traffic is increasing globally, with a four-fold increase in commercial vessel movements between 1992 and 2012. Vessels contribute to noise and air pollution, provide pathways for non-native species, and collide with marine wildlife. While knowledge of shipping trends and potential environmental impacts exists at both local and global levels, key information on vessel density for regional-scale management is lacking. This study presents the first in-depth spatio-temporal analysis of shipping in the north-east Atlantic region, over three years in a five-year period. Densities increased by 34%, including in 73% of Marine Protected Areas. Western Scotland and the Bay of Biscay experienced the largest increases in vessel density, predominantly from small and slow vessels. Given well-documented impacts that shipping can have on the marine environment, it is crucial that this situation continues to be monitored – particularly in areas designated to protect vulnerable species and ecosystems which may already be under pressure.Publisher PDFPeer reviewe

Growth, microstructure, and failure of crazes in glassy polymers

We report on an extensive study of craze formation in glassy polymers. Molecular dynamics simulations of a coarse-grained bead-spring model were employed to investigate the molecular level processes during craze nucleation, widening, and breakdown for a wide range of temperature, polymer chain length $N$, entanglement length $N_e$ and strength of adhesive interactions between polymer chains. Craze widening proceeds via a fibril-drawing process at constant drawing stress. The extension ratio is determined by the entanglement length, and the characteristic length of stretched chain segments in the polymer craze is $N_e/3$. In the craze, tension is mostly carried by the covalent backbone bonds, and the force distribution develops an exponential tail at large tensile forces. The failure mode of crazes changes from disentanglement to scission for $N/N_e\sim 10$, and breakdown through scission is governed by large stress fluctuations. The simulations also reveal inconsistencies with previous theoretical models of craze widening that were based on continuum level hydrodynamics

Factors Associated with Revision Surgery after Internal Fixation of Hip Fractures

Background: Femoral neck fractures are associated with high rates of revision surgery after management with internal fixation. Using data from the Fixation using Alternative Implants for the Treatment of Hip fractures (FAITH) trial evaluating methods of internal fixation in patients with femoral neck fractures, we investigated associations between baseline and surgical factors and the need for revision surgery to promote healing, relieve pain, treat infection or improve function over 24 months postsurgery. Additionally, we investigated factors associated with (1) hardware removal and (2) implant exchange from cancellous screws (CS) or sliding hip screw (SHS) to total hip arthroplasty, hemiarthroplasty, or another internal fixation device. Methods: We identified 15 potential factors a priori that may be associated with revision surgery, 7 with hardware removal, and 14 with implant exchange. We used multivariable Cox proportional hazards analyses in our investigation. Results: Factors associated with increased risk of revision surgery included: female sex, [hazard ratio (HR) 1.79, 95% confidence interval (CI) 1.25-2.50; P = 0.001], higher body mass index (fo

Behavioural and neurochemical effects of early social deprivation in the rat

The behavioural and neurochemical sequelae of rearing in early social isolation are reviewed. Rats isolated from their peers at ~ 20 days of age are hyperactive, exhibit ambivalent responses to novelty, altered response to stressors, and cognitive impairments when tested as adults. They also overrespond to stimuli predictive of reinforcement, as shown by their elevated levels of conditioned activity when the presentation of food reliably follows daily activity tests in photocell cages. The dose-effect curve for D-amphetamine is shifted to the left in this situation, confirming the enhanced locomotor and stereotyped behaviours seen following systemic treatment with a variety of other psychomotor stimulant drugs. In vivo measurement of amine activity in isolates using microdialysis shows elevations in dopamine concentration in both the nucleus accumbens (ventral striatum) and the caudate-putamen. There are also changes in central 5-HT metabolism. These results are considered in the light of recent studies in isolates using the latent inhibition or pre-pulse inhibition procedures, two putative models of the attentional dysfunction in schizophrenia. Finally, the implications of these findings are considered and future problems requiring resolution are raised

Shipping in the north-east Atlantic:identifying spatial and temporal patterns of change

Maritime traffic is increasing globally, with a four-fold increase in commercial vessel movements between 1992 and 2012. Vessels contribute to noise and air pollution, provide pathways for non-native species, and collide with marine wildlife. While knowledge of shipping trends and potential environmental impacts exists at both local and global levels, key information on vessel density for regional-scale management is lacking. This study presents the first in-depth spatio-temporal analysis of shipping in the north-east Atlantic region, over three years in a five-year period. Densities increased by 34%, including in 73% of Marine Protected Areas. Western Scotland and the Bay of Biscay experienced the largest increases in vessel density, predominantly from small and slow vessels. Given well-documented impacts that shipping can have on the marine environment, it is crucial that this situation continues to be monitored – particularly in areas designated to protect vulnerable species and ecosystems which may already be under pressure