Solid-phase synthesis of dihydrovirginiamycin S1, a streptogramin B antibiotic

We describe the first solid-phase synthesis of dihydrovirginiamycin S(1), a member of the streptogramin B family of antibiotics, which are nonribosomal-peptide natural products produced by Streptomyces. These compounds, along with the synergistic group A components, are "last line of defense" antimicrobial agents for the treatment of life-threatening infections such as vancomycin-resistant enterococci. The synthesis features an on-resin cyclization and is designed to allow production of streptogramin B analogues with diversification at positions 1', 1, 2, 3, 4, and 6. Several synthetic challenges known to hinder the synthesis of this class of compounds were solved, including sensitivity to acids and bases, and epimerization and rearrangements, through the judicious choice of deprotection conditions, coupling conditions, and synthetic strategy. This work should enable a better understanding of structure-activity relationships in the streptogramin B compounds, possible identification of analogues that bypass known resistance mechanisms, and perhaps the identification of analogues with novel biological activities.     

QUADRATURE METHODS FOR HIGHLY OSCILLATORY SINGULAR INTEGRALS

We address the evaluation of highly oscillatory integrals,with power-law and logarithmic singularities.Such problems arise in numerical methods in engineering.Notably,the evaluation of oscillatory integrals dominates the run-time for wave-enriched boundary integral formulations for wave scattering,and many of these exhibit singularities.We show that the asymptotic behaviour of the integral depends on the integrand and its derivatives at the singular point of the integrand,the stationary points and the endpoints of the integral.A truncated asymptotic expansion achieves an error that decays faster for increasing frequency.Based on the asymptotic analysis,a Filon-type method is constructed to approximate the integral.Unlike an asymptotic expansion,the Filon method achieves high accuracy for both small and large frequency.Complex-valued quadrature involves interpolation at the zeros of polynomials orthogonal to a complex weight function.Numerical results indicate that the complex-valued Gaussian quadrature achieves the highest accuracy when the three methods are compared.However,while it achieves higher accuracy for the same number of function evaluations,it requires signi cant additional cost of computation of orthogonal polynomials and their zeros.     

Laser generation of narrow-band and directed ultrasound

An aluminium hemicylindrical sample has been irradiated with an array of laser lines, with each line acting as a source of acoustic waves. Detection of the generated ultrasonic waves was performed using both a wide-band stabilized Michelson interferometer and a 20 MHz piezoelectric transducer. Experimental and theoretical results are presented which reveal that the use of a spatially modulated laser source produces significant narrow-banding of the detected ultrasound, compared with a single point or single line source case. Additionally, for a given line spacing, ultrasound of a particular frequency can be directed. Owing to the nature of the acoustic signals generated by each individual array element, superposition of several signals does not result in any energy directivity similar to that encountered in phase electromagnetic array antennas. While time or frequency feature enhancement may be obtained in a desired direction, in most cases the far field energy directivity pattern is simply the incoherent sum of the energy directivity of each array element.     

The Oshima-Sekiguchi and Liouville theorems on Heintze groups

Let L be an elliptic operator on a Riemannian manifold M. A function F annihilated by L is said to be L-harmonic. F is said to have moderate growth if and only if F grows at most exponentially in the Riemannian distance. If M is a rank-one symmetric space and L is the Laplace-Beltrami operator for M, the Oshima-Sekiguchi theorem [T. Oshima, J. Sekiguchi, Eigenspaces of invariant differential operators on an affine symmetric space, Invent. Math. 57 (1980) 1-81] states that a L-harmonic function F has moderate growth if and only if F is the Poisson integral of a distribution on the Furstenberg boundary. In this work we prove that this result generalizes to a very large class of homogeneous Riemannian manifolds of negative curvature. We also (i) prove a Liouville type theorem that characterizes the “polynomial-like” harmonic functions which vanish on the boundary in terms of their growth properties, (ii) describe all “polynomial-like” harmonic functions, and (iii) give asymptotic expansions for the Poisson kernel. One consequence of this work is that every Schwartz distribution on the boundary is the boundary value for a L-harmonic function F which is uniquely determined modulo “polynomial-like” harmonic functions.     

On a class of relatively prime sequences

For each natural number n, let a₀(n) = n, and if a₀(n),…,a_i(n) have already been defined, let a_i+1(n) > a_i(n) be minimal with (a_i+1(n), a₀(n) … a_i(n)) = 1. Let g(n) be the largest a_i(n) not a prime or the square of a prime. We show that g(n) ～ n and that $\text{g(n) > n + cn}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{log}\text{(n)}$ for some c > 0. The true order of magnitude of g(n) ? n seems to be connected with the fine distribution of prime numbers. We also show that “most” a_i(n) that are not primes or squares of primes are products of two distinct primes. A result of independent interest comes of one of our proofs: For every sufficiently large n there is a prime $\text{p < n}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ with [$\text{n}\text{p}$] composite.     

Relationship of crystal growth parameters to the stoichiometry of EuO as determined by I.R. and conductivity measurements

Single crystals of EuO have been grown from melts containing excess Eu-metal. i.r. absorption and conductivity measurements were made on these crystals and the results correlated with the crystal growth parameters. It is seen that as the concentration of europium in the initial melt is increased, the growth temperature is lowered and the crystals become richer in europium. The variations in the i.r. absorption and conductivity were studied and are shown to be strongly related to composition. According to the i.r. and conductivity behavior, the EuO crystals can be classified into five types, with compositions ranging from oxygen-rich, to stoichiometric, to europium-rich. The oxygen-rich EuO crystals are insulators and exhibit i.r. absorption which is attributed to Eu₃O₄ and to Eu³⁺. Only weak i.r. absorption is seen in the stoichiometric crystals. Those with a deficiency of oxygen show a new i.r. absorption and have either metallic conduction or show large insulator-metal transitions which are attributed to oxygen vacancies.