We define the notion of Bernstein measures and Bernstein approximations over general convex polytopes. This generalizes well-known Bernstein polynomials which are used to prove the Weierstrass approximation theorem on one dimensional intervals. We discuss some properties of Bernstein measures and approximations, and prove an asymptotic expansion of the Bernstein approximations for smooth functi...

which are now called Bernstein polynomials, in order to present a short proof of the Weierstrass Approximation Theorem. The subsequent history is well documented, see, e.g., [29] for the period up to 1955, the monograph [18] published in 1953, and the survey article [9] which appeared on the occasion of the hundredth anniversary of the above paper by Bernstein. Since the latter publication prov...

For a compact metric space X , consider a linear subspace A of C{X) containing the constant functions. One version of the Stone-Weierstrass Theorem states that, if A separates points, then the closure of A under both minima and maxima is dense in C{X). By the Hahn-Banach Theorem, if A separates probability measures, A is dense in C{X). It is shown that if A separates points from probability mea...

1. What is an elliptic curve? 2 2. Mordell-Weil Groups 5 2.1. The Group Law on a Smooth, Plane Cubic Curve 5 2.2. Reminders on Commutative Groups 8 2.3. Some Elementary Results on Mordell-Weil Groups 9 2.4. The Mordell-Weil Theorem 11 2.5. K-Analytic Lie Groups 13 3. Background on Algebraic Varieties 15 3.1. Affine Varieties 15 3.2. Projective Varieties 18 3.3. Homogeneous Nullstellensätze 20 3...

We characterize ideals of subsets of natural numbers for which some versions of Ramsey’s theorem and Schur’s theorem hold. These are similar to generalizations shown by Frankl, Graham and Rödl in [6]. We show that, in some sense, Ramsey’s theorem, theorem of Bolzano-Weierstrass and “monotone subsequence” theorem characterize the same class of ideals.