Spline fitting discontinuous functions given just a few Fourier coefficients

This paper considers the use of polynomial splines to approximate periodic functions with jump discontinuities of themselves and their derivatives when the information consists only of the first few Fourier coefficients and the location of the discontinuities. Spaces of splines are introduced which include, members with discontinuities at those locations. The main results deal with the orthogonal projection of such a spline space on spaces of trigonometric polynomials corresponding to the known coefficients. An approximation is defined based on inverting this projection. It is shown that when discontinuities are sufficiently far apart, the projection is invertible, its inverse has norm close to 1, and the approximation is nearly as good as directL ₂ approximation by members of the spline space. An example is given which illustrates the results and which is extended to indicate how the approximation technique may be used to provide smoothing which which accurately represents discontinuities.     

Vanadium-Based Organic-Inorganic Hybrid Materials Prepared by a Sol-Gel Method

Synthesis and characterization of hybrid organic-inorganic nanocomposites based on reaction of vanadium i-propoxide with an organically functionalized poly(ethylene glycol) having di-carbonyl groups is reported. The bulk gels were characterized by thermal analyses (DSC, TGA and TMA) and FT-IR spectroscopy. The results revealed that the gels consist of vanadium-oxygen-vanadium groupings which are cross-linked with the functionalized polymer through the di-carbonyl groups. Gels doped with lithium salt show significant electrical conductivity.     

Theory and simulation of supercontinuum generation in transparent bulk media

Comprehensive model and computer simulations of the supercontinuum generation in bulk media are presented. Using water and air as representative condensed and gaseous media, the standard model for white-light generation is put to test. It is shown that besides the intensity clamping and plasma-induced frequency blue shift, the linear chromatic dispersion plays an important role. It is the major factor that determines the achievable spectral extent of the supercontinuum. PACS 42.65.-k; 42.65.Hw; 42.65.Ky; 42.65.Sf     

A neutron diffraction investigation of the structure of vitreous boron trioxide

An extended review is given of previous structural work on vitreous B₂O₃ which comes to the conclusion that the best structural model comprises a mixed random network of boroxol groups and independent BO₃ triangles. The neutron scattering length for ¹¹B is slightly greater than that for oxygen so that neutrons should be more sensitive than X-rays to the presence of boroxol groups. Results are presented from a neutron diffraction investigation of vitreous ¹¹B₂O₃, using the Harwell linac pulsed neutron source, in which data have been obtained to high scattering vector values (Q_max=40 Å⁻¹) yielding good real space resolution. It is concluded that these data are consistent with a mixed boroxol group - independent BO₃ triangle network having 60% of the boron atoms in boroxol rings.     

Singularities of the solution of a mixed problem for a general second order elliptic equation and boundary stabilizationof the wave equation

We deal here with a second order elliptic mixed problem which is posed in a regular open bounded domain of . We study the regularity of its solution. We apply our results to the boundary stabilization of the wave equation.     

Solid-state (67)zn NMR spectroscopy in bioinorganic chemistry. Spectra of four- and six-coordinate zinc pyrazolylborate complexes obtained by management of proton relaxation rates with a paramagnetic dopant

Solid-state (67)Zn NMR spectra of model compounds for metalloproteins, such as [H(2)B(3,5-Me(2)pz)(2)](2)Zn (pz denotes pyrazolyl ring), have been obtained using low temperatures (10 K) to enhance the Boltzmann factor in combination with cross polarization (CP) from (1)H to (67)Zn. Attempts to observe spectra of other model compounds, such as [H(2)B(pz)(2)](2)Zn, were hindered by long relaxation times of the protons. To decrease the proton relaxation times, the high-spin six-coordinate complex [HB(3,4,5-Me(3)pz)(3)](2)Fe has been investigated as a dopant. NMR and EPR measurements have shown that this Fe(II) dopant effectively reduces the (1)H spin lattice relaxation time, T(1), of the zinc samples in the temperature range 5-10 K with minimal perturbations of the (1)H spin lattice relaxation time in the rotating frame, T(1)(rho). Using this methodology, we have determined the (67)Zn NMR parameters of four- and six-coordinate zinc(II) poly(pyrazolyl)borate complexes that are useful models for systems of biological importance. The (67)Zn NMR parameters are contrasted to the corresponding changes in the (113)Cd NMR parameters for the analogous compounds. Further, these investigations have demonstrated that a temperature-dependent phase transition occurs in the neighborhood of 185 K for [HB(3,5-Me(2)pz)(3)](2)Zn; the other poly(pyrazolyl)borate complexes we investigated did not show this temperature-dependent behavior. This conclusion is confirmed by a combination of room-temperature high-field (18.8 T) solid-state (67)Zn NMR spectroscopy and low-temperature X-ray methods. The utilization of paramagnetic dopants should enable low-temperature cross polarization experiments to be performed on a wide variety of nuclides that are important in bioinorganic chemistry, for example, (25)Mg, (43)Ca, and (67)Zn.     

New chiral building blocks from tetrabromocyclopropene and furan

The cyclocondensation of tetrabromocyclopropene and furan leads directly to a halogenated oxabicyclo[3.2.1]octadiene derivative. Over the past several years, we have utilized these compounds as intermediates for natural product synthesis. Herein, we describe the preparation of nonracemic dibromoenone building blocks from the racemic cycloadduct. Conversion of the adduct to a mixture of tartrate-derived ketals followed by separation of the diastereomers and hydrolysis allows for the formation of novel chiral synthons with either absolute configuration.