Multivariate Diophantine equations with many solutions

We show that for each n-tuple of positive rational integers (a_1,..,a_n) there are sets of primes S of arbitrarily large cardinality s such that the solutions of the equation a_1x_1+...+a_nx_n=1 with the x_i all S-units are not contained in fewer than exp((4+o(1))s^{1/2}(log s)^{-1/2}) proper linear subspaces of C^n. This generalizes a result of Erdos, Stewart and Tijdeman for m=2 [Compositio 36 (1988), 37-56]. Furthermore we prove that for any algebraic number field K of degree n, any integer m with 1<=m<n, and any sufficiently large s there are integers b_0,...,b_m in a number field which are linearly independent over the rationals, and prime numbers p_1,...,p_s, such that the norm polynomial equation |N_{K/Q}(b_0+b_1x_1+...+b_mx_m)|=p_1^{z_1}...p_s^{z_s} has at least exp{(1+o(1)){n/m}s^{m/n}(log s)^{-1+m/n}) solutions in integers x_1,..,x_m,z_1,..,z_s. This generalizes a result of Moree and Stewart [Indag. Math. 1 (1990), 465-472]. Our main tool, also established in this paper, is an effective lower bound for the number of ideals in a number field K of norm <=X composed of prime ideals which lie outside a given finite set of prime ideals T and which have norm <=Y. This generalizes a result of Canfield, Erdos and Pomerance [J. Number Th. 17 (1983), 1-28], and of Moree and Stewart (see above).Comment: 29 page

Jorgensen's inequality for non-Archimedean metric spaces.

Jørgensen’s inequality gives a necessary condition for a non-elementary group of Möbius transformations to be discrete. In this paper we generalise this to the case of groups of Möbius transformations of a non-Archimedean metric space. As an application, we give a version of Jørgensen’s inequality for SL(2, ℚ p )

Principal forms X^2 + nY^2 representing many integers

In 1966, Shanks and Schmid investigated the asymptotic behavior of the number of positive integers less than or equal to x which are represented by the quadratic form X^2+nY^2. Based on some numerical computations, they observed that the constant occurring in the main term appears to be the largest for n=2. In this paper, we prove that in fact this constant is unbounded as n runs through positive integers with a fixed number of prime divisors.Comment: 10 pages, title has been changed, Sections 2 and 3 are new, to appear in Abh. Math. Sem. Univ. Hambur

Mauritius on fire: Tracking historical human impacts on biodiversity loss

Fire was rare on Mauritius prior to human arrival (AD 1598); subsequently three phases of elevated fire activity occurred: ca 1630–1747, 1787–1833, and 1950–modern. Elevated fire frequency coincided with periods of high human impact evidenced from the historical record, and is linked to the extinction of island endemics

The effectiveness of psychosocial interventions for anxiety in children and adolescents with autism spectrum disorder:a systematic review and meta-analysis

Anxiety is a common problem in children and adolescents with autism spectrum disorder (ASD). This meta-analysis aimed to systematically evaluate the evidence for the use of psychosocial interventions to manage anxiety in this population. Cognitive behavioural therapy (CBT) was the primary intervention modality studied. A comprehensive systematic search and study selection process was conducted. Separate statistical analyses were carried out for clinician-, parent-, and self-reported outcome measures. Sensitivity analyses were conducted by removing any outlying studies and any studies that did not use a CBT intervention. A subgroup analysis was performed to compare individual and group delivery of treatment. Ten randomised control trials involving a total of 470 participants were included. The overall SMD was d = 1.05 (95 % CI 0.45, 1.65; z = 3.45, p = 0.0006) for clinician- reported outcome measures; d = 1.00 (95%CI 0.21, 1.80; z = 2.47, p = 0.01) for parent-reported outcome measures; and d = 0.65 (95%CI -0.10, 1.07; z = 1.63, p = 0.10) for self-reported outcome measures. Clinician- and parent-reported outcome measures showed that psychosocial interventions were superior to waitlist and treatment-as-usual control conditions at post-treatment. However, the results of self-reported outcome measures failed to reach significance. The sensitivity analyses did not significantly change these results and the subgroup analysis indicated that individual treatment was more effective than group treatment. The main limitations of this review were the small number of included studies as well as the clinical and methodological variability between studies

Evolution of metabolic divergence in <i>Pseudomonas aeruginosa</i> during long-term infection facilitates a proto-cooperative interspecies interaction

The effect of polymicrobial interactions on pathogen physiology and how it can act either to limit pathogen colonization or to potentiate pathogen expansion and virulence are not well understood. Pseudomonas aeruginosa and Staphylococcus aureus are opportunistic pathogens commonly found together in polymicrobial human infections. However, we have previously shown that the interactions between these two bacterial species are strain dependent. Whereas P. aeruginosa PAO1, a commonly used laboratory strain, effectively suppressed S. aureus growth, we observed a commensal-like interaction between the human host-adapted strain, DK2-P2M24-2003, and S. aureus. In this study, characterization by matrix-assisted laser desorption ionization-time of flight (MALDI-TOF) imaging mass spectrometry (IMS) and mass spectral (MS) molecular networking revealed a significant metabolic divergence between P. aeruginosa PAO1 and DK2-P2M24-2003, which comprised several virulence factors and signaling 4-hydroxy-2-alkylquinoline (HAQ) molecules. Strikingly, a further modulation of the HAQ profile was observed in DK2-P2M24-2003 during interaction with S. aureus, resulting in an area with thickened colony morphology at the P. aeruginosa–S. aureus interface. In addition, we found an HAQ-mediated protection of S. aureus by DK2-P2M24-2003 from the killing effect of tobramycin. Our findings suggest a model where the metabolic divergence manifested in human host-adapted P. aeruginosa is further modulated during interaction with S. aureus and facilitate a proto-cooperative P. aeruginosa–S. aureus relationship

Structural Insights from Binding Poses of CCR2 and CCR5 with Clinically Important Antagonists: A Combined In Silico Study

Chemokine receptors are G protein-coupled receptors that contain seven transmembrane domains. In particular, CCR2 and CCR5 and their ligands have been implicated in the pathophysiology of a number of diseases, including rheumatoid arthritis and multiple sclerosis. Based on their roles in disease, they have been attractive targets for the pharmaceutical industry, and furthermore, targeting both CCR2 and CCR5 can be a useful strategy. Owing to the importance of these receptors, information regarding the binding site is of prime importance. Structural studies have been hampered due to the lack of X-ray crystal structures, and templates with close homologs for comparative modeling. Most of the previous models were based on the bovine rhodopsin and β2-adrenergic receptor. In this study, based on a closer homolog with higher resolution (CXCR4, PDB code: 3ODU 2.5 Å), we constructed three-dimensional models. The main aim of this study was to provide relevant information on binding sites of these receptors. Molecular dynamics simulation was done to refine the homology models and PROCHECK results indicated that the models were reasonable. Here, binding poses were checked with some established inhibitors of high pharmaceutical importance against the modeled receptors. Analysis of interaction modes gave an integrated interpretation with detailed structural information. The binding poses confirmed that the acidic residues Glu291 (CCR2) and Glu283 (CCR5) are important, and we also found some additional residues. Comparisons of binding sites of CCR2/CCR5 were done sequentially and also by docking a potent dual antagonist. Our results can be a starting point for further structure-based drug design