On the number of polynomials with small discriminants in the Euclidean and <Emphasis Type="Italic">p</Emphasis>-adic metrics

In this article it is proved that there exist a large number of polynomials which have small discriminant in terms of the Euclidean and p-adic metrics simultaneously. The measure of the set of points which satisfy certain polynomial and derivative conditions is also determined.     

Development of Electrospun Three‐arm Star Poly(ε‐caprolactone) Meshes for Tissue Engineering Applications

We have developed three‐dimensional electrospun microfibrous meshes of a novel star branched three‐arm poly(ε‐caprolactone) (*PCL) as potential scaffolds for tissue engineering applications. The processing conditions required to obtain uniform fibers were optimized by studying their influence on fiber morphology and size. Polymer molecular weight and solution feed rate influenced both the mesh microstructure and the tensile properties of the developed mats. Electrospun samples were also tested for their mechanical properties in wet conditions, showing higher yield strength and strain in comparison to that observed in dry conditions. Cell culture experiments employing MC3T3‐E1 osteoblast like cells showed good cell viability adhesion and collagen production on the *PCL scaffolds.

     

Sets of exact ‘logarithmic’ order in the theory of Diophantine approximation

For each real number , let denote the set of real numbers with exact order . A theorem of Güting states that for the Hausdorff dimension of is equal to . In this note we introduce the notion of exact t–logarithmic order which refines the usual definition of exact order. Our main result for the associated refined sets generalizes Güting's result to linear forms and moreover determines the Hausdorff measure at the critical exponent. In fact, the sets are shown to satisfy delicate zero-infinity laws with respect to Lebesgue and Hausdorff measures. These laws are reminiscent of those satisfied by the classical set of well approximable real numbers, for example as demonstrated by Khintchine's theorem. Received: 12 December 2000 / Published online: 25 June 2001     

Diophantine approximation on non-degenerate curves with non-monotonic error function

It is shown that a non-degenerate curve in ⁿ satisfies a convergentGroshev theorem with a non-monotonic error function. In otherwords it is shown that if a volume sum converges the set ofpoints lying on the curve which satisfy a Diophantine conditionhas Lebesgue measure zero.