An analysis of cubic approximation schemes for conic sections

In this paper a piece of a conic section is approximated by a cubic or piecewise cubic polynomial. The main tool is to define the two inner control points of the cubic as an affine combination, defined by [0, 1], of two control points of the conic. If is taken to depend on the weightw of the latter, a function (w) results which is used to distinguish between different algorithms and to analyze their properties. One of the approximations is a piecewise cubic havingG ⁴ continuity at the break points.     

Some Inequalities for Derivatives of Trigonometric and Algebraic Polynomials

We investigate inequalities for derivatives of trigonometric and algebraic polynomials in weighted L ^P spaces with weights satisfying the Muckenhoupt A _p condition. The proofs are based on an identity of Balázs and Kilgore [1] for derivatives of trigonometric polynomials. Also an inequality of Brudnyi in terms of rth order moduli of continuity ω_r will be given. We are able to give values to the constants in the inequalities.     

Curvature and finite domination

Upper bounds obtained by Gromov on the Betti numbers of certain closed Riemannian manifolds are shown to be upper bounds on the minimum number of cells in --spaces dominating such manifolds.

     

Spectrally determined growth is generic

Let be the infinitesimal generator of a -semigroup of operators in a Hilbert space. We consider the class of operators , where is bounded. It is proved that the spectrum of determines the growth of the associated semigroup for ``most" operators (in the sense of Baire category).

     

Eclipsed Isopropyl Conformations of Two Dimeric Hydrazines

The X-ray structures of 4,10-di-tert-butyl-5,9-diisopropyl-4,5,9,10-tetraazatetracyclo[6.2.2.2(3,6)]tetradecane (s4iPr) and its 4,9-di-tert-butyl-5,10-diisopropyl isomer (a4iPr) are reported. Both compounds are in conformations having their in-N-alkyl groups (directed toward the central CH-CH bond of the molecule) anti to each other, as expected from previous work. The principal feature of interest is that one in-isopropyl group in each compound is in an eclipsed conformation, NN,C(alpha)Me twist angle -0.5(5) degrees for s4iPr and -6.4(4) degrees for a4iPr. Low energy (somewhat less) eclipsed in-isopropyl conformations are predicted by both molecular mechanics (MM2) and semiempirical quantum mechanical (AM1) calculations. The asymmetry of the potentially C(2) symmetric a4iPr because the two in-isopropyl groups are in different rotamers is apparently not a result of crystal packing forces, because a conformation with different isopropyl rotamers is the more stable one by at least 1.0 kcal/mol in solution, determined by (13)C-NMR spectroscopy. This result is not predicted by either calculation method. The "monomer", 2-tert-butyl-3-isopropyl-2,3-diazabicyclo[2.2.2]octane (3), proves to be a poor model for the conformations of 4iPr.     

Approximate minimum-cost multicommodity flows in\tilde O(ɛ −2 KNM) timetime

We show that an ε-approximate solution of the cost-constrainedK-commodity flow problem on anN-nodeM-arc network,G can be computed by sequentially solving O(K(? ^?2+logGK) logGM log (G? ^?1 GK)) single-commodity minimum-cost flow problems on the same network. In particular, an approximate minimum-cost multicommodity flow can be computed in $\tilde O$ (G? ^?2 GKNM) running time, where the notation Õ(·) means “up to logarithmic factors”. This result improves the time bound mentioned by Grigoriadis and Khachiyan [4] by a factor ofM/N and that developed more recently by Karger and Plotkin [8] by a factor of? ^?1. We also provide a simple $\tilde O$ (NM)-time algorithm for single-commodity budget-constrained minimum-cost flows which is $\tilde O$ (? ^?3) times faster than the algorithm developed in the latter paper.     

Distance distributions and dynamics of a zinc finger peptide from fluorescence resonance energy transfer measurements

Time-resolved fluorescence resonance energy transfer (FRET) measurements were used to measure distance distributions and intramolecular dynamics (site-to-site diffusion) of a 28-residue single-domain zinc finger peptide in the absence and presence of zinc ion. Energy transfer was measured between TRP₁₄ and a N-terminal DNS group. As expected, the TRP-to-DNS distance distribution for zinc-bound peptide is shorter and narrower (R _av=11.2 Å,hw=2.8 Å) than the metal-free peptide (R _av=20.1 Å,hw=14.5 Å). The degree of mutual donor-to-acceptor diffusion (D) was also determined for these distributions. For zinc-bound peptide there is no detectible diffusion (D0.2 Ų/ns), whereas for metal-free peptide a considerable amount of motion is occurring between the donor and the acceptor (D=12 Ų/ns). These results indicate that the zinc-bound peptide folds into a unique, well-defined conformation, whereas the metal-free conformation is flexible and rapidly changing. The absence of detectible mutual site-to-site diffusion between the donor and the acceptor in the metal-bound zinc finger peptide indicates that intramolecular motion is essentially frozen out, on the FRET time scale, as a consequence of zinc coordination.Dedicated to the memory of Barbara D. Wells.