A note on weighted badly approximable linear forms

We prove a result in the area of twisted Diophantine approximation related to the theory of Schmidt games. In particular, under certain restrictions we give an affirmative answer to the analogue in this setting of a famous conjecture of Schmidt from Diophantine approximation

A problem in non-linear Diophantine approximation

In this paper we obtain the Lebesgue and Hausdorff measure results for the set of vectors satisfying infinitely many fully non-linear Diophantine inequalities. The set is associated with a class of linear inhomogeneous partial differential equations whose solubility depends on a certain Diophantine condition. The failure of the Diophantine condition guarantees the existence of a smooth solution

Presentations of major peripheral arterial disease and risk of major outcomes in patients with type 2 diabetes: results from the ADVANCE-ON study.

BACKGROUND: Peripheral arterial disease (PAD) is known to be associated with high cardiovascular risk, but the individual impact of PAD presentations on risk of macrovascular and microvascular events has not been reliably compared in patients with type 2 diabetes. We aimed to evaluate the impact of major PAD, and its different presentations, on the 10-year risk of death, major macrovascular events, and major clinical microvascular events in these patients. METHODS: Participants in the action in diabetes and vascular disease: PreterAx and DiamicroN modified-release controlled evaluation (ADVANCE) trial and the ADVANCE-ON post-trial study were followed for a median of 5.0 (in-trial), 5.4 (post-trial), and 9.9 (overall) years. Major PAD at baseline was subdivided into lower-extremity chronic ulceration or amputation secondary to vascular disease and history of peripheral revascularization by angioplasty or surgery. RESULTS: Among 11,140 participants, 516 (4.6 %) had major PAD at baseline: 300 (2.7 %) had lower-extremity ulceration or amputation alone, 190 (1.7 %) had peripheral revascularization alone, and 26 (0.2 %) had both presentations. All-cause mortality, major macrovascular events, and major clinical microvascular events occurred in 2265 (20.3 %), 2166 (19.4 %), and 807 (7.2 %) participants, respectively. Compared to those without PAD, patients with major PAD had increased rates of all-cause mortality (HR 1.35, 95 % CI 1.15-1.60, p = 0.0004), and major macrovascular events (1.47 [1.23-1.75], p < 0.0001), after multiple adjustments for region of origin, cardiovascular risk factors and treatments, peripheral neuropathy markers, and randomized treatments. We have also observed a trend toward an association of baseline PAD with risk of major clinical microvascular events [1.31 (0.96-1.78), p = 0.09]. These associations were comparable for patients with a lower-extremity ulceration or amputation and for those with a history of peripheral revascularization. Furthermore, the risk of retinal photocoagulation or blindness, but not renal events, increased in patients with lower-extremity ulceration or amputation [1.53 (1.01-2.30), p = 0.04]. CONCLUSIONS: Lower-extremity ulceration or amputation, and peripheral revascularization both increased the risks of death and cardiovascular events, but only lower-extremity ulceration or amputation increased the risk of severe retinopathy in patients with type 2 diabetes. Screening for major PAD and its management remain crucial for cardiovascular prevention in patients with type 2 diabetes (ClinicalTrials.gov number, NCT00949286)

Estimation of individual beneficial and adverse effects of intensive glucose control for patients with type 2 diabetes

AIMS/HYPOTHESIS: Intensive glucose control reduces the risk of vascular complications while increasing the risk of severe hypoglycaemia at a group level. We sought to estimate individual beneficial and adverse effects of intensive glucose control in patients with type 2 diabetes. METHODS: We performed a post hoc analysis of the Action in Diabetes and Vascular Disease: Preterax and Diamicron MR Controlled Evaluation (ADVANCE) trial, a randomised controlled trial evaluating standard vs intensive glucose control (HbA1c target ≤6.5% [48 mmol/mol]). In 11,140 participants, we estimated the individual 5 year absolute risk reduction (ARR) for the composite outcome of major micro- and macrovascular events and absolute risk increase (ARI) for severe hypoglycaemia for intensive vs standard glucose control. Predictions were based on competing risks models including clinical characteristics and randomised treatment. RESULTS: Based on these models, 76% of patients had a substantial estimated 5 year ARR for major vascular events (>1%, 5 year number-needed-to-benefit [NNTB5] 200). Similarly, 36% of patients had a substantial estimated ARI for severe hypoglycaemia (5 year number-needed-to-harm [NNTH5] 200). When assigning similar or half the weight to severe hypoglycaemia compared with a major vascular event, net benefit was positive in 85% or 99% of patients, respectively. Limiting intensive treatment to the 85% patient subgroup had no significant effect on the overall incidence of major vascular events and severe hypoglycaemia compared with treating all patients. CONCLUSIONS/INTERPRETATION: Taking account of the effects of intensive glucose control on major micro- and macrovascular events and severe hypoglycaemia for individual patients, the estimated net benefit was positive in the majority of the participants in the ADVANCE trial. The estimated individual effects can inform treatment decisions once individual weights assigned to positive and adverse effects have been specified. TRIAL REGISTRATION: ClinicalTrials.gov NCT00145925

A loss-of-function polymorphism in the human P2X4 receptor is associated with increased pulse pressure

The P2X4 receptor is involved in endothelium-dependent changes in large arterial tone in response to shear stress and is, therefore, potentially relevant to arterial compliance and pulse pressure. Four identified nonsynonymous polymorphisms in P2RX4 were reproduced in recombinantly expressed human P2X4. Electrophysiological studies showed that one of these, the Tyr315&gt;Cys mutation (rs28360472), significantly reduced the peak amplitude of the ATP-induced inward current to 10.9% of wild-type P2X4 receptors in transfected HEK-293 cells (10 µmol/L of ATP; n=4-8 cells; P&lt;0.001). Concentration-response curves for ATP and the partial agonist BzATP demonstrate that the 315Cys-P2X4 mutant had an increased EC50 value for both ligands. Mutation of Tyr315&gt;Cys likely disrupts the agonist binding site of P2X4 receptors, a finding supported by molecular modeling based on the zebrafish P2X4 receptor crystal structure. We tested inheritance of rs28360472 encoding the Tyr315&gt;Cys mutation in P2RX4 against pulse pressure in 2874 subjects from the Victorian Family Heart Study. The minor allele frequency was 0.014 (1.4%). In a variance components analysis we found significant association with pulse pressure (P=0.023 for total association) where 1 minor allele increased pulse pressure by 2.84 mm Hg (95% CI: 0.41-5.27). This increase in pulse pressure associated with inheritance of 315Cys-P2X4 receptors might reflect reduced large arterial compliance as a result of impaired endothelium-dependent vasodilation in large arteries

Clinical Utility of Short-Term Blood Pressure Measures to Inform Long-Term Blood Pressure Management

Background: Decisions about hypertension management are substantially influenced by blood pressure (BP) levels measured before and soon after starting BP lowering drugs. We aimed to assess the utility of short-term BP changes in individuals in terms of long-term treatment response. Methods: Post hoc analyses of 2 randomized trials with 4-to-6 weeks active run-in for all participants, followed by randomization to active BP lowering treatment (combination perindopril±indapamide) or placebo. We categorized individuals by degree of systolic BP (SBP) change during active run-in treatment and assessed associations with subsequent postrandomization placebo-corrected BP reduction, cardiovascular events, and tolerability. We included individuals with baseline BP ≥140/90 mm Hg from the PROGRESS trial (Perindopril Protection Against Recurrent Stroke Study; 4275 individuals with cerebrovascular disease) and ADVANCE trial (The Action in Diabetes and Vascular Disease: Preterax and Diamicron-MR Controlled Evaluation; 6610 individuals with diabetes). Results: During the active run-in period, the proportion of participants with initial SBP changes in 4 categories (SBP increase, 0-9.9 mm Hg decrease, 10-19.9 mm Hg decrease, and ≥20 mm Hg decrease) were 17%, 27%, 28%, and 28% in PROGRESS and 21%, 22%, 24%, and 33% in ADVANCE. Randomization to active therapy achieved similar placebo-corrected long-term BP reductions across the 4 initial SBP change groups in both trials (P-values for heterogeneity >0.1). There was no significant difference in achieving BP 0.1). Conclusions: An individual's apparent BP change immediately after commencing therapy has limited clinical utility. Therefore, more emphasis should be given to use of evidence-based regimens and measures over the long-term to ensure sustained BP control. Registration: URL: https://www.clinicaltrials.gov; Unique identifier: NCT00145925

Cumulative Systolic Blood Pressure Load and Cardiovascular Risk in Patients With Diabetes

Background: Standard measures of blood pressure (BP) do not account for both the magnitude and duration of exposure to elevated BP over time. Objectives: The purpose of this study was to assess the association between cumulative systolic blood pressure (SBP) load and risk of cardiovascular events in patients with type 2 diabetes. Methods: A post hoc analysis of patients with type 2 diabetes followed by the ADVANCE-ON (Action in Diabetes and Vascular Disease: Preterax and Diamicron Modified Release Controlled Evaluation - Observational Study). Cumulative SBP load was defined as the area under curve for SBP values ≥130 mm Hg divided by the area under curve for all measured SBP values over a 24-month exposure period. HRs for the association between cumulative SBP load with major cardiovascular events and death were estimated using Cox models. Results: Over a median 7.6 years of follow-up, 1,469 major cardiovascular events, 1,615 deaths, and 660 cardiovascular deaths were observed in 9,338 participants. Each 1-SD increase in cumulative SBP load was associated with a 14% increase in major cardiovascular events (HR: 1.14; 95% CI: 1.09-1.20), 13% increase in all-cause mortality (HR: 1.13; 95% CI: 1.13-1.18), and 21% increase in cardiovascular death (HR: 1.21; 95% CI: 1.13-1.29). For the prediction of cardiovascular events and death, cumulative SBP load outperformed mean SBP, time-below-target SBP, and visit-to-visit SBP variability in terms of Akaike information criterion and net reclassification indexes. Conclusions: Cumulative SBP load may provide better prediction of major cardiovascular events compared with traditional BP measures among patients with type 2 diabetes. These findings reinforce the importance of both the magnitude and duration of exposure to elevated SBP in assessing cardiovascular risk. (Action in Diabetes and Vascular Disease Preterax and Diamicron MR Controlled Evaluation Post Trial Observational Study [ADVANCE-ON]; NCT00949286

Follow-up of blood-pressure lowering and glucose control in type 2 diabetes.

BACKGROUND In the Action in Diabetes and Vascular Disease: Preterax and Diamicron Modified Release Controlled Evaluation (ADVANCE) factorial trial, the combination of perindopril and indapamide reduced mortality among patients with type 2 diabetes, but intensive glucose control, targeting a glycated hemoglobin level of less than 6.5%, did not. We now report results of the 6-year post-trial follow-up. METHODS We invited surviving participants, who had previously been assigned to perindopril–indapamide or placebo and to intensive or standard glucose control (with the glucose-control comparison extending for an additional 6 months), to participate in a post-trial follow-up evaluation. The primary end points were death from any cause and major macrovascular events. RESULTS The baseline characteristics were similar among the 11,140 patients who originally underwent randomization and the 8494 patients who participated in the post-trial follow-up for a median of 5.9 years (blood-pressure–lowering comparison) or 5.4 years (glucose-control comparison). Between-group differences in blood pressure and glycated hemoglobin levels during the trial were no longer evident by the first post-trial visit. The reductions in the risk of death from any cause and of death from cardiovascular causes that had been observed in the group receiving active blood-pressure–lowering treatment during the trial were attenuated but significant at the end of the post-trial follow-up; the hazard ratios were 0.91 (95% confidence interval [CI], 0.84 to 0.99; P=0.03) and 0.88 (95% CI, 0.77 to 0.99; P=0.04), respectively. No differences were observed during follow-up in the risk of death from any cause or major macrovascular events between the intensive-glucose-control group and the standard-glucose-control group; the hazard ratios were 1.00 (95% CI, 0.92 to 1.08) and 1.00 (95% CI, 0.92 to 1.08), respectively. CONCLUSIONS The benefits with respect to mortality that had been observed among patients originally assigned to blood-pressure–lowering therapy were attenuated but still evident at the end of follow-up. There was no evidence that intensive glucose control during the trial led to long-term benefits with respect to mortality or macrovascular events