Let K be a number field, let ϕ(x) ∈ K(x) be a rational function of degree d > 1, and let α ∈ K be a wandering point such that ϕ n (α) = 0 for all n > 0. We prove that if the abc-conjecture holds for K, then for all but finitely many positive integers n, there is a prime p of K such that v p (ϕ n (α)) > 0 and vp(ϕ m (α)) 0 for all positive integers m < n. Under appropriate ramification hypotheses, we can replace the condition v p (ϕ n (α)) > 0 with the stronger condition vp(ϕ n (α)) = 1. We prove the same result unconditionally for function fields of characteristic 0 when ϕ is not isotrivial.
Because the insulin-responsive glucose transporter, GLUT4, is expressed in renal vascular and glomerular cells, we determined the effects of experimental diabetes mellitus on GLUT4 expression and glucose uptake by these tissues. Quantitative reverse-transcription polymerase chain reaction studies of microdissected afferent microvessels and renal glomeruli showed that, after 1 wk of diabetes, GLUT4 mRNA was decreased to 26 and 34% of control values, respectively. GLUT4 immunoblots of renal glomerular and microvessel samples showed that GLUT4 polypeptide was decreased to 51% of control values. These results were confirmed by indirect immunofluorescence, which showed decreased GLUT4 expression in glomerular cells and in vascular smooth muscle cells of the afferent microvasculature of diabetic animals. Uptake of the glucose analogue, 2-deoxyglucose, was also depressed in microvessels of diabetic rats to 57% of control values, supporting the conclusion that fewer total glucose transporters were available for glucose uptake into diabetic renal glomerular and microvascular cells. Thus both GLUT4 expression and glucose uptake by glomerular and microvascular cells are decreased in diabetic animals. These results have led us to suggest a mechanism by which decreased renal GLUT4 expression could contribute to glomerular hyperfiltration and hypertension seen in early diabetes.
We prove the Dynamical Bogomolov Conjecture for endomorphisms Φ : P 1 ×P 1 −→ P 1 ×P 1 , where Φ(x, y) := (f (x), g(y)) for any rational functions f and g defined overQ. We use the equidistribution theorem for points of small height with respect to an algebraic dynamical system, combined with a theorem of Levin regarding symmetries of the Julia set. Using a specialization theorem of Yuan and Zhang, we can prove the Dynamical Manin-Mumford Conjecture for endomorhisms Φ = (f, g) of P 1 × P 1 , where f and g are rational functions defined over an arbitrary field of characteristic 0.2010 Mathematics Subject Classification. Primary: 37P05. Secondary: 37P30. Key words and phrases. Dynamical Manin-Mumford Conjecture, equidistribution of points of small height, symmetries of the Julia set of a rational function.
Summaryobjectives To determine the effectiveness of hand hygiene in a developing healthcare setting in reducing nosocomial infections (NIs).method Prospective study measuring NI rates in a urology ward in Ho Chi Minh City, Vietnam, before and after implementation of a hand hygiene programme with an alcohol-based decontaminant, and compliance rates of medical staff and carers with hand hygiene using standardised observation sheets.
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