Single-site beta-diiminate zinc catalysts for the ring-opening polymerization of beta-butyrolactone and beta-valerolactone to poly(3-hydroxyalkanoates)

Polymerization of beta-butyrolactone (BBL) and beta-valerolactone (BVL) using the zinc alkoxide initiator (BDI-1)ZnO(i)()Pr [(BDI-1) = 2-((2,6-diisopropylphenyl)amido)-4-((2,6-diisopropylphenyl)imino)-2-pentene] proceeds very rapidly under mild conditions to produce poly(3-hydroxybutyrate) (PHB) and poly(3-hydroxyvalerate) (PHV), respectively. For the monomer-to-initiator ratio 200:1, PHB number-average molecular weights (M(n)) are proportional to conversion throughout the reaction and polydispersity indices (PDIs) are narrow, consistent with a living polymerization. Higher monomer-to-initiator ratios can be used to achieve high molecular weight PHB (M(n) > 100 000). End-group analysis verifies that the polymerization of BBL follows a coordination-insertion mechanism, where complexes of the form (BDI-1)ZnOCH(Me)CH(2)CO(2)R are the active species. Variable temperature NMR experiments show that (BDI-1)ZnO(i)()Pr is monomeric in benzene-d(6) solution. In contrast, (BDI-2)ZnO(i)()Pr [(BDI-2) = 2-((2,6-diethylphenyl)amido)-4-((2,6-diethylphenyl)imino)-2-pentene] is a poor initiator at room temperature because it prefers to form a bis-mu-isopropoxide dimer in solution. According to kinetic studies, propagation by (BDI-1)ZnO(i)()Pr is first order in both monomer as well as (BDI-1)ZnO(i)()Pr concentration. These results lead us to propose a monometallic active species. Several results suggest that elimination side reactions are slowly catalyzed by zinc alkoxides, leading to degradation of the polymer.     

On G 2 Continuous Spline Interpolation of Curves in ℝ d

In this paper the problem of G ² continuous interpolation of curves in ^d by polynomial splines of degree n is studied. The interpolation of the data points and two tangent directions at the boundary is considered. The case n = r + 2 = d, where r is the number of interior points interpolated by each segment of the spline curve, is studied in detail. It is shown that the problem is uniquely solvable asymptotically, e., when the data points are sampled regularly and sufficiently dense, and lie on a regular, convex parametric curve in ^d . In this case the optimal approximation order is also determined.     

Structure of V(2)O(5)*nH(2)O xerogel solved by the atomic pair distribution function technique

A long-standing issue regarding the local and long-range structure of V(2)O(5)*nH(2)O xerogel has been successfully addressed. The full three-dimensional structure of the lamellar turbostratic V(2)O(5)*nH(2)O xerogel was determined by the atomic pair distribution function technique. We show that on the atomic scale the slabs of the xerogel can be described well as almost perfect pairs (i.e., bilayers) of single V(2)O(5) layers made of square pyramidal VO(5) units. These slabs are separated by water molecules and stack along the z-axis of a monoclinic unit cell (space group C2/m) with parameters a = 11.722(3) A, b = 3.570(3) A, c = 11.520(3) A, and beta = 88.65 degrees. The stacking sequence shows signatures of turbostratic disorder and a structural coherence limited to 50 A.     

The Bergman kernel function: Explicit formulas and zeroes

We show how to compute the Bergman kernel functions of some special domains in a simple way. As an application of the explicit formulas, we show that the Bergman kernel functions of some convex domains, for instance the domain in defined by the inequality , have zeroes.

     

Focal shift in focused truncated gaussian beams

It is known that under certain circumstances the point of maximum intensity in a focused aberration-free wave is not at the geometrical focus, but is closer to the focusing lens. In the present note such a focal shift is analyzed for the case of a gaussian beam, focused by a thin lens, which fills a circular aperture of any prescribed radius, the waist of the beam being assumed to be located in the aperture plane. Many previously known results follow as limiting cases of our analysis.     

Generalized stokes parameters of random electromagnetic beams

A generalization of the Stokes parameters of a random electromagnetic beam is introduced. Unlike the usual Stokes parameters, which depend on one spatial variable, the generalized Stokes parameters, depend on two spatial variables. They obey precise laws of propagation, both in free space and in any linear medium, whether deterministic or random. With the help of the generalized Stokes parameters, the changes in the ordinary Stokes parameters upon propagation can be determined. Numerical examples of such changes are presented. The generalized Stokes parameters contain information not only about the polarization properties of the beam but also about its coherence properties. We illustrate this fact by expressing the degree of coherence of the electromagnetic beam in terms of one of the generalized Stokes parameters.