We prove that if the outer billiard map around a plane oval is algebraically integrable in a certain non-degenerate sense then the oval is an ellipse. In this note, an outer billiard table is a compact convex domain in the plane bounded by an oval (closed smooth strictly convex curve) C. Pick a point x outside of C. There are two tangent lines from x to C; choose one of them, say, the right one...

OBJECTIVE Our objective was to draw growth curves and assess obesity prevalence in adolescents of public schools, and to explore selected food consumption frequency and physical activity. METHODS A cross-sectional study was conducted in Lebanese public schools. From the list of schools provided by the Ministry of Education, a random sample of 20 schools was chosen, distributed in all Lebanese...

Receiver Operating Characteristic (ROC) curves provide a powerful tool for visualizing and comparing classification results. Regression Error Characteristic (REC) curves generalize ROC curves to regression. REC curves plot the error tolerance on the xaxis versus the percentage of points predicted within the tolerance on the y-axis. The resulting curve estimates the cumulative distribution funct...

This paper presents efficient formulas for computing cryptographic pairings on the curve y = cx + 1 over fields of large characteristic. We provide examples of pairing-friendly elliptic curves of this form which are of interest for efficient pairing implementations.

by the implicit function theorem. This second form is far more convenient than the first (for example one can find approximate solutions using the relationship Sy = f(x, y)dx (<5x small), or in the analytic case find a series solution by repeated differentiation). Let us assume that 0 is a regular value of the function F, so F~ (0) is a smooth surface in IR. The locus F = Fp — 0 is the set of c...

We give a detailed account of the classical Van Kampen method for computing presentations of fundamental groups of complements of complex algebraic curves, and of a variant of this method, working with arbitrary projections (even with vertical asymptotes). Introduction In the 1930’s, Van Kampen described a general technique for computing presentations of fundamental groups of complements of com...

Individual pig growth curves were derived on 445 Yorkshire, 323 Hampshire, and 316 Landrace boars and gilts. The range in age was from 1 to 35 d and these growth curves were linear. The regression parameters (x- and y-intercepts, and slopes) were different among the three breeds, and differences between boars and gilts were found in the Yorkshire breed for the y-intercept and slope and in the L...

We show that the Diophantine equation of the title has, for n > 1, no solution in coprime nonzero integers x, y and z. Our proof relies upon Frey curves and related results on the modularity of Galois representations.

1. Geometry of the geodesic curvature vector. 2. Geometry of the constant velocity projection. 3. A simple planar example. The nal geodesic curve is as straight line as expected. (a) Top view (the ^ C curve) of the process. (b) The nal geodesic curve is presented on the 3D surface. 4. Example with an \egg" box surface. The initial curve (^ C 0) is given by a sin function on the (x; y){plane. 5....

BACKGROUND MUC4 is a high molecular weight membrane protein that is overexpressed in pancreatic cancer (PC) and is associated with the development and progression of this disease. However, the exact mechanisms through which MUC4 domains promote these biological processes have rarely been studied, partly because of its high molecular weight, difficulty to overexpress it. Here, we use MUC4/Y, one...