Lattice-point enumerators of ellipsoids

Minkowski's second theorem on successive minima asserts that the volume of a 0-symmetric convex body K over the covolume of a lattice \Lambda can be bounded above by a quantity involving all the successive minima of K with respect to \Lambda. We will prove here that the number of lattice points inside K can also accept an upper bound of roughly the same size, in the special case where K is an ellipsoid. Whether this is also true for all K unconditionally is an open problem, but there is reasonable hope that the inductive approach used for ellipsoids could be extended to all cases.Comment: 9 page

Songs by Ethelbert Nevin: summer longings

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Global patient outcomes after elective surgery: prospective cohort study in 27 low-, middle- and high-income countries.

BACKGROUND: As global initiatives increase patient access to surgical treatments, there remains a need to understand the adverse effects of surgery and define appropriate levels of perioperative care. METHODS: We designed a prospective international 7-day cohort study of outcomes following elective adult inpatient surgery in 27 countries. The primary outcome was in-hospital complications. Secondary outcomes were death following a complication (failure to rescue) and death in hospital. Process measures were admission to critical care immediately after surgery or to treat a complication and duration of hospital stay. A single definition of critical care was used for all countries. RESULTS: A total of 474 hospitals in 19 high-, 7 middle- and 1 low-income country were included in the primary analysis. Data included 44 814 patients with a median hospital stay of 4 (range 2-7) days. A total of 7508 patients (16.8%) developed one or more postoperative complication and 207 died (0.5%). The overall mortality among patients who developed complications was 2.8%. Mortality following complications ranged from 2.4% for pulmonary embolism to 43.9% for cardiac arrest. A total of 4360 (9.7%) patients were admitted to a critical care unit as routine immediately after surgery, of whom 2198 (50.4%) developed a complication, with 105 (2.4%) deaths. A total of 1233 patients (16.4%) were admitted to a critical care unit to treat complications, with 119 (9.7%) deaths. Despite lower baseline risk, outcomes were similar in low- and middle-income compared with high-income countries. CONCLUSIONS: Poor patient outcomes are common after inpatient surgery. Global initiatives to increase access to surgical treatments should also address the need for safe perioperative care. STUDY REGISTRATION: ISRCTN5181700

Endovascular strategy or open repair for ruptured abdominal aortic aneurysm: one-year outcomes from the IMPROVE randomized trial.

AIMS: To report the longer term outcomes following either a strategy of endovascular repair first or open repair of ruptured abdominal aortic aneurysm, which are necessary for both patient and clinical decision-making. METHODS AND RESULTS: This pragmatic multicentre (29 UK and 1 Canada) trial randomized 613 patients with a clinical diagnosis of ruptured aneurysm; 316 to an endovascular first strategy (if aortic morphology is suitable, open repair if not) and 297 to open repair. The principal 1-year outcome was mortality; secondary outcomes were re-interventions, hospital discharge, health-related quality-of-life (QoL) (EQ-5D), costs, Quality-Adjusted-Life-Years (QALYs), and cost-effectiveness [incremental net benefit (INB)]. At 1 year, all-cause mortality was 41.1% for the endovascular strategy group and 45.1% for the open repair group, odds ratio 0.85 [95% confidence interval (CI) 0.62, 1.17], P = 0.325, with similar re-intervention rates in each group. The endovascular strategy group and open repair groups had average total hospital stays of 17 and 26 days, respectively, P < 0.001. Patients surviving rupture had higher average EQ-5D utility scores in the endovascular strategy vs. open repair groups, mean differences 0.087 (95% CI 0.017, 0.158), 0.068 (95% CI -0.004, 0.140) at 3 and 12 months, respectively. There were indications that QALYs were higher and costs lower for the endovascular first strategy, combining to give an INB of £3877 (95% CI £253, £7408) or €4356 (95% CI €284, €8323). CONCLUSION: An endovascular first strategy for management of ruptured aneurysms does not offer a survival benefit over 1 year but offers patients faster discharge with better QoL and is cost-effective. CLINICAL TRIAL REGISTRATION: ISRCTN 48334791

The effect of aortic morphology on peri-operative mortality of ruptured abdominal aortic aneurysm

Aims To investigate whether aneurysm shape and extent, which indicate whether a patient with ruptured abdominal aortic aneurysm (rAAA) is eligible for endovascular repair (EVAR), influence the outcome of both EVAR and open surgical repair. Methods and results The influence of six morphological parameters (maximum aortic diameter, aneurysm neck diameter, length and conicality, proximal neck angle, and maximum common iliac diameter) on mortality and reinterventions within 30 days was investigated in rAAA patients randomized before morphological assessment in the Immediate Management of the Patient with Rupture: Open Versus Endovascular strategies (IMPROVE) trial. Patients with a proven diagnosis of rAAA, who underwent repair and had their admission computerized tomography scan submitted to the core laboratory, were included. Among 458 patients (364 men, mean age 76 years), who had either EVAR (n = 177) or open repair (n = 281) started, there were 155 deaths and 88 re-interventions within 30 days of randomization analysed according to a pre-specified plan. The mean maximum aortic diameter was 8.6 cm. There were no substantial correlations between the six morphological variables. Aneurysm neck length was shorter in those undergoing open repair (vs. EVAR). Aneurysm neck length (mean 23.3, SD 16.1 mm) was inversely associated with mortality for open repair and overall: adjusted OR 0.72 (95% CI 0.57, 0.92) for each 16 mm (SD) increase in length. There were no convincing associations of morphological parameters with reinterventions. Conclusion Short aneurysm necks adversely influence mortality after open repair of rAAA and preclude conventional EVAR. This may help explain why observational studies, but not randomized trials, have shown an early survival benefit for EVAR. Clinical trial registration: ISRCTN 48334791

The Number of Solutions of Bounded Height to a System of Linear Equations

We Count the number of solutions with height less than or equal to B to a system of linear equations over a number field. We give explicit asymptotic estimates for the number of such solutions as B goes to infinity, where the constants involved depend on the classical invariants of the number field (degree, discriminant, class number. etc.). The problem is reformulated as an estimate for the number of lattice points in a certain bounded domain.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/30992/1/0000667.pd