Wavelet compression and nonlinearn-widths

It is shown that certain algorithms of compression based on wavelet decompositions are optimal in the sense of nonlinearn-widths.This research was supported by the Office of Naval Research Contract N0014-91-J1343 and a Binational Science Foundation Grant 89-00505.This research was completed while D. Leviatan and V.M. Tikhomirov were visiting scholars at the University of South Carolina.     

Determining factors for high performance silicone rubber microwave absorbing materials

Silicone rubber microwave absorbing materials (RMAMs) based on ferrite as the major absorbent were prepared by the mechanical blending method. The determining factors for the complex permittivity, complex permeability, and reflectivity of RMAM were thoroughly investigated with various samples including different crystal structures of Ba-ferrite (M-type, W-type, and Y-type), the ferrite with doped elements (Ba, Sr), the materials' thickness, the combination ratio of ferrite and carbonyl iron. The effects of surface modification and loading amount of ferrite on the mechanical properties, processing performance, and absorbing property of RMAM were also assessed. The results show that W-type Ba-ferrite based RMAM exhibits better absorbing property at high frequencies (8-18 GHz) than the other two barium ferrites (M-type and Y-type) based ones, and the absorbing property of RMAM based on Sr-ferrite is best. As the thickness of RMAM and the amount of absorbents increase, the absorption peak moves toward low frequency, the absorption frequency bandwidth is narrowed, and the reflectivity first decreases and later increases. The optimum thickness is 1.5-1.7 mm, and the amount of ferrite is 450 parts per hundreds of rubber (phr). Surface modification of the absorbent with silane coupling agent could improve the mechanical properties and processing performance of RMAM. It is concluded that there will be a synergistic effect when carbonyl iron (CI) is used in combination with Sr-ferrite (Sr-W) in an appropriate proportion. When the total volume fraction of absorbents is 51%, the optimum ratio of Cl to Sr-W is 17:34, the absorption frequency bandwidth (<−10 dB) is about 8 GHz, and the absorption area is −99 dB.     

Widths and shape-preserving widths of Sobolev-type classes of s-monotone functions

Let I be a finite interval, , and 1p∞. Given a set M, of functions defined on I, denote by the subset of all functions yM such that the s-difference is nonnegative on I, τ>0. Further, denote by the Sobolev class of functions x on I with the seminorm x^(r)_Lp1. We obtain the exact orders of the Kolmogorov and the linear widths, and of the shape-preserving widths of the classes in L_q for s>r+1 and (r,p,q)≠(1,1,∞). We show that while the widths of the classes depend in an essential way on the parameter s, which characterizes the shape of functions, the shape-preserving widths of these classes remain asymptotically ≈n^-2.     

Convex polynomial and spline approximation inL p,O<p<∞

We prove that a convex functionf ∈ L _p[−1, 1], 0<p<∞, can be approximated by convex polynomials with an error not exceeding Cω ₃ ^ϕ (f,1/n)_p where ω ₃ ^ϕ (f,·) is the Ditzian-Totik modulus of smoothness of order three off. We are thus filling the gap between previously known estimates involving ω ₃ ^ϕ (f,1/n)_p, and the impossibility of having such estimates involving ω₄. We also give similar estimates for the approximation off by convexC ⁰ andC ¹ piecewise quadratics as well as convexC ² piecewise cubic polynomials. Communicated by Dietrich Braess     

Evaluation of multi-element analysis of blood serum by energy-dispersive x-ray spectrometry

Conventional energy-dispersive x-ray fluorescence is applied in the analysis of blood serum to give the concentrations of 7–10 elements simultaneously with minimal manipulation of the samples. Simple spotting onto a Mylar carrier of 250 μl of serum, doped with two internal standards, was chosen as the sample preparation step. Some 200 serum samples, analyzed in replicate (n = 2–6), were used to evaluate this procedure. The detection limits are 4 μg ml^-1 for K and Ca, 0.5–0.2 μg ml^-1 for Fe, Cu, Pb and Zn, and less than 0.1 μg ml^-1 for Se, Rb and Sr. Well above these limits, the standard deviation is around 10%. Comparison with the results of other measurements on the same samples indicates an accuracy of that order. The simplicity and high throughput, and the possibility of automating the x.r.f. measurements, make the proposed procedure suitable for screening large numbers of sera.     

Cannabielsoic acids : Isolation and synthesis by a novel oxidative cyclization

Two tricyclic dihydrobenzofuran cannabinoids, cannabielsoic acids A (4a) and B (5a) were isolated from hashish. Their structures were elucidated by chemical transformations and from spectral data. Cannabielsoic acid A was synthesized from cannabidiolic acid by an oxidative cyclization in the presence of air under irradiation, or with manganese dioxide.     

Positive Results and Counterexamples in Comonotone Approximation

We estimate the degree of comonotone polynomial approximation of continuous functions f, on [?1,1], that change monotonicity s??1 times in the interval, when the degree of unconstrained polynomial approximation E _n (f)??n ^??? , n??1. We ask whether the degree of comonotone approximation is necessarily ??c(??,s)n ^??? , n??1, and if not, what can be said. It turns out that for each s??1, there is an exceptional set A _s of ????s for which the above estimate cannot be achieved.     

On Bivariate Smoothness Spaces Associated with Nonlinear Approximation

In recent years there have been various attempts at the representations of {\mbox multivariate} signals such as images, which outperform wavelets. As is well known, wavelets are not optimal in that they do not take full advantage of the geometrical regularities and singularities of the images. Thus these approaches have been based on tracing curves of singularities and applying bandlets, curvelets, ridgelets, etc., or allocating some weights to curves of singularities like the Mumford–Shah functional and its modifications. In the latter approach a function is approximated on subdomains where it is smoother but there is a penalty in the form of the total length (or other measurement) of the partitioning curves. We introduce a combined measure of smoothness of the function in several dimensions by augmenting its smoothness on subdomains by the smoothness of the partitioning curves. Also, it is known that classical smoothness spaces fail to characterize approximation spaces corresponding to multivariate piecewise polynomial nonlinear approximation. We show how the proposed notion of smoothness can almost characterize these spaces. The question whether the characterization proposed in this work can be further simplified remains open.     

Broken O(5) and O(3) symmetries in a general non-spherical boson basis

We discuss the spontaneous symmetry breaking of O(3) and O(5) rotations occurring in an exact ground state of certain schematic IBM-interactions. The ground state is a condensate built from one of six bosons forming a general non-spherical boson basis whose members depend continuously on the deformation parametersβ andγ. Interpreting some of the basis members as Goldstone bosons allows a convenient separation between bosons representing physical and spurious modes. The schematic interactions are shown to be suitable for forming Hamiltonians whose eigenstates are clustered into bands.