Separating the solution sets of analytical and polynomial systems

A linear inequality system with infinitely many constraints is polynomial (analytical) if its index set is a compact interval of the real line and all its coefficients are polynomial (analytical, respectively) functions of the index on this interval. This paper provides an example of analytical system whose solution set cannot be the solution set of any polynomial system. Research supported by DGES of Spain and FEDER of UE, Grant BFM2002-04114-C02-01. Research supported by CONACyT of Mexico, Grant 130036. Research partially supported by CONACyT of Mexico, Grant 44003.     

New results for isotropic point scatterers: Foldy revisited

Since 1945, Foldy’s method has been used to predict velocity and attenuation for various types of scatterers. In this paper, it is shown that Foldy’s method also yields predictions of reflection and transmission of scalar waves by a random distribution of point or line scatterers contained in a slab. Results are given in two and three dimensions, and for normal and oblique incidences. Formulae are also obtained for the reflection and transmission of longitudinal waves by point or line scatterers distributed in an elastic (non-viscous) fluid. Energy equations are derived, and expressions are obtained for the energy dissipated in the slab on average over one period. Curves for the reflection and transmission coefficients are presented in the case of solid cylindrical bars immersed in a fluid. The results obtained in this paper are expected to be valid for a low density of scatterers. Potential applications of this work occur in ultrasonic evaluation of materials, seismic exploration and medical ultrasonics, where reflected (or backscattered) data are used to construct maps or images of the materials (metals, composites, earth subsurface, tissue). The formulae of this work are expected to provide useful tools for better and more efficient mapping or imaging.     

Long-Range proton-proton coupling constants. I. Propanic coupling involving a methyl group 4JMeH

An equation is formulated on the basis of theoretical INDO/FPT calculations which describes the angular dependence of the propanic long-range coupling constant ⁴J_MeH in substituted HCCCH₃ fragments. This equation is a truncated Fourier series in the torsion angle ?, HCCMe, which takes into account the dependence of the Fourier coefficients on the bond angle θ, CCMe. The substituent effects are assumed to be additive. Some parameters in the equation may be obtained from the ⁴J_MeH couplings in propane and neopentane derivatives. The calculated effect upon ⁴J_MeH of changes in the bond angle θ is significant and it seems to be in part the cause of some effects which have been attributed to conformational dependence.     

Conjugate points and Maslov index in locally symmetric semi-Riemannian manifolds

We study the singularities of the exponential map in semi Riemannian locally symmetric manifolds. Conjugate points along geodesics depend only on real negative eigenvalues of the curvature tensor, and their contribution to the Maslov index of the geodesic is computed explicitly. We prove that degeneracy of conjugate points, which is a phenomenon that can only occur in semi-Riemannian geometry, is caused in the locally symmetric case by the lack of diagonalizability of the curvature tensor. The case of Lie groups endowed with a bi-invariant metric is studied in some detail, and conditions are given for the lack of local injectivity of the exponential map around its singularities.     

The preparation of 4-oxo-N-aryl-4H-1-benzopyran-2-acetamides by the condensation/cyclization of trilithiated acetoacetanilides with lithiated methyl salicylates

Several acetoacetanilides were trilithiated in excess lithium diisopropylamide, and the resulting polylithiated intermediates were regioselectively condensed with lithiated methyl salicylates followed by acid cyclization to substituted 4-oxo-N-aryl-4H-1-benzopyran-2-acetamides (benzopyranone-2-acetamides).     

Determination of propazine by differential pulse polarography in micellar and emulsified media

An electroanalytical study of the herbicide propazine's reduction process in micellar solutions and oil-in-water emulsions is reported. The anionic surfactant sodium pentanesulphonate was chosen as the most suitable. The differential pulse polarograms of micellar solutions had two reduction peaks below pH 2.0, whereas only one peak was obtained above pH 2.O. Ethyl acetate was chosen as the organic solvent to form propazine emulsions. Unlike in micellar solutions, the DPP polarograms of propazine emulsions showed only one peak even at pH < 2.0, suggesting that propazine hydrolysis was hindered in the emulsified medium. The limiting current is diffusion-controlled and the electrode process is irreversible. Propazine can be determined by differential pulse polarography over the 1.0 × 10^–1 – 1.0 × 10^–1moll^–1 and 1.0 × 10^–15 – 4.0 × 10^–1 moll^–1 concentration ranges and the limit of detection was 2.8 × 10^–1 moll^–1. Of the potential interferents simazine, methoprotryne and terbutryn (alls-triazines), thiram (a dithiocarbamate), dinoseb (nitrophenolic), and heptachlor (chlorinated cyclo-diene herbicide), only the first two were significant (10% error for equimolar concentrations). The method was applied to the determination of propazine in spiked drinking water. At a concentration level of 2.0 × 10^–1 moll^–1 a recovery of 94 ± 6% was obtained, after tenfold concentration on Sep-Pak.