Effective computability of solutions of ordinary differential equations: the thousand monkeys approach

In this note we consider the computability of the solution of the initial- value problem for ordinary di erential equations with continuous right- hand side. We present algorithms for the computation of the solution using the \thousand monkeys" approach, in which we generate all possi- ble solution tubes, and then check which are valid. In this way, we show that the solution of a di erential equation de ned by a locally Lipschitz function is computable even if the function is not e ectively locally Lips- chitz. We also recover a result of Ruohonen, in which it is shown that if the solution is unique, then it is computable, even if the right-hand side is not locally Lipschitz. We also prove that the maximal interval of existence for the solution must be e ectively enumerable open, and give an example of a computable locally Lipschitz function which is not e ectively locally Lipschitz

Two-Dimensional Critical Percolation: The Full Scaling Limit

We use SLE(6) paths to construct a process of continuum nonsimple loops in the plane and prove that this process coincides with the full continuum scaling limit of 2D critical site percolation on the triangular lattice -- that is, the scaling limit of the set of all interfaces between different clusters. Some properties of the loop process, including conformal invariance, are also proved.Comment: 45 pages, 12 figures. This is a revised version of math.PR/0504036 without the appendice

Area minimizing discs in metric spaces

We solve the classical problem of Plateau in the setting of proper metric spaces. Precisely, we prove that among all disc-type surfaces with prescribed Jordan boundary in a proper metric space there exists an area minimizing disc which moreover has a quasi-conformal parametrization. If the space supports a local quadratic isoperimetric inequality for curves we prove that such a solution is locally Hölder continuous in the interior and continuous up to the boundary. Our results generalize corresponding results of Douglas Radò and Morrey from the setting of Euclidean space and Riemannian manifolds to that of proper metric spaces

Feel4Diabetes healthy diet score: Development and evaluation of clinical validity

Background: The aim of this paper is to present the development of the Feel4Diabetes Healthy Diet Score and to evaluate its clinical validity. Methods: Study population consisted of 3268 adults (63% women) from high diabetes risk families living in 6 European countries. Participants filled in questionnaires at baseline and after 1 year, reflecting the dietary goals of the Feel4Diabetes intervention. Based on these questions the Healthy Diet Score was constructed, consisting of the following components: breakfast, vegetables, fruit and berries, sugary drinks, whole-grain cereals, nuts and seeds, low-fat dairy products, oils and fats, red meat, sweet snacks, salty snacks, and family meals. Maximum score for each component was set based on its estimated relative importance regarding T2DM risk, higher score indicating better quality of diet. Clinical measurements included height, weight, waist circumference, heart rate, blood pressure, and fasting blood sampling, with analyses of glucose, total cholesterol, HDL-cholesterol, LDL-cholesterol, and triglycerides. Analysis of (co) variance was used to compare the Healthy Diet Score and its components between countries and sexes using baseline data, and to test differences in clinical characteristics between score categories, adjusted for age, sex and country. Pearson''s correlations were used to study the association between changes from baseline to year 1 in the Healthy Diet Score and clinical markers. To estimate reproducibility, Pearson''s correlations were studied between baseline and 1 year score, within the control group only. Results: The mean total score was 52.8 ± 12.8 among women and 46.6 ± 12.8 among men (p < 0.001). The total score and its components differed between countries. The change in the Healthy Diet Score was significantly correlated with changes in BMI, waist circumference, and total and LDL cholesterol. The Healthy Diet Score as well as its components at baseline were significantly correlated with the values at year 1, in the control group participants. Conclusion: The Feel4Diabetes Healthy Diet Score is a reproducible method to capture the dietary information collected with the Feel4Diabetes questionnaire and measure the level of and changes in the adherence to the dietary goals of the intervention. It gives a simple parameter that associates with clinical risk factors in a meaningful manner

Des cartes contre l'impérialisme. Les atlas de Frank Horrabin et Alexandre Radó pendant l'entre-deux-guerres

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