Multivariate approximation in the worst case setting over reproducing kernel Hilbert spaces

We study the worst case setting for approximation of d variate functions from a general reproducing kernel Hilbert space with the error measured in the L_ȡe; norm. We mainly consider algorithms that use n arbitrary continuous linear functionals. We look for algorithms with the minimal worst case errors and for their rates of convergence as n goes to infinity. Algorithms using n function values will be analyzed in a forthcoming paper.We show that the L_ȡe; approximation problem in the worst case setting is related to the weighted L₂ approximation problem in the average case setting with respect to a zero-mean Gaussian stochastic process whose covariance function is the same as the reproducing kernel of the Hilbert space. This relation enables us to find optimal algorithms and their rates of convergence for the weighted Korobov space with an arbitrary smoothness parameter αe;1, and for the weighted Sobolev space whose reproducing kernel corresponds to the Wiener sheet measure. The optimal convergence rates are n^-(α-1)/2 and n^-1/2, respectively.We also study tractability of L_ȡe; approximation for the absolute and normalized error criteria, i.e., how the minimal worst case errors depend on the number of variables, d, especially when d is arbitrarily large. We provide necessary and sufficient conditions on tractability of L_ȡe; approximation in terms of tractability conditions of the weighted L₂ approximation in the average case setting. In particular, tractability holds in weighted Korobov and Sobolev spaces only for weights tending sufficiently fast to zero and does not hold for the classical unweighted spaces.     

Phase-retrieval in hard x-ray diffraction imaging: Can we quantitatively reconstruct a “real” transmission function?

X-ray phase-retrieval algorithms are widely exploited in contemporary hard x-ray diffraction techniques to image at the nanoscale, less than 10-20 nm. Often reconstruction of the sample shape (image) suffices for the purpose of experiment. Identification of specimen composition requires a quantitative profiling of the complex refractive index. This Letter shows that, although such quantitative analysis is possible in many cases, there is a lower limit to variations in optical density, which can be quantitatively reconstructed using the common phase-retrieval methods.     

Probabilistic properties of neuron spiking time-series obtained in vivo

Probabilistic properties of spiking time-series obtained in vivo from singular neurons belonging to Red Nucleus of brain are analyzed for two groups of rats: genetically defined rat model of depression (Flinders Sensitive Rat Line - FSL) and a control (healthy) group. The FSL group shows a distribution of interspike intervals with a much longer tail than that found for normal rats. The former distribution (for the FSL group) indicates a power-law with exponent α = - 1±0.1. A simple thermodynamic (noise) model is elaborated to explain obtained results. Received 13 May 2001 and Received in final form 20 September 2001     

Spin-cast, thin, glassy polymer films: Highly metastable forms of matter

Ultra thin films of glassy polymers such as polystyrene (PS) can show a) anomalously large thickness changes, b) unexpected dewetting properties, c) large shifts in the glass temperature T_g. The present discussion focusses mainly on point a). A certain cascade of metastable states is presented together with (tentative) explanations. Received 1 March 2001 and Received in final form 10 May 2001     

Electron capture into few-electron heavy ions: Independent particle model

We apply the density matrix theory to re-investigate the radiative electron capture into heavy ions with one valence electron. Attention has been paid particularly to the magnetic sublevel population of the residual ions, as described in terms of alignment parameters. Simple method, based on an independent particle model, which takes into account the Pauli principle, is proposed for evaluating the alignment of the excited ionic states. By making use of this method, detailed calculations are performed for electron capture into (initially) hydrogen-like and lithium-like europium, gold and uranium ions, and are compared with the results of the multiconfiguration Dirac-Fock approach.As seen from the calculations and from the comparison with available experimental results, the independent particle model provides a good estimate for the alignment parameters of few-electron heavy ions. Therefore, our simple model may help to understand the basic properties of the X-ray emission from heavy, few-electron ions without the need for invoking sophisticated MCDF calculations.     

Yang's model as triply special relativity and the Snyder's model–de Sitter special relativity duality

We show that if Yang's quantized space–time model is completed at both classical and quantum level, it should contain both Snyder's model, the de Sitter special relativity and their duality.     

Filtering properties of defect-bearing periodic and triadic cantor multilayers

Defect-bearing periodic as well as pre-fractal dielectric multilayers can be designed to operate as narrow-band filters. For the same dielectric constituents and the same (or closest to same) number of layers, pre-fractal structures are remarkably better than the periodic ones, both in terms of filter bandwidth and transmittance peak, and are less sensitive to variations in the defect thickness. For structures having about 30 layers, optical losses do not significantly affect the filtering features of both morphologies if the loss tangent of the constituent materials is less than 10⁻⁵.     

Control of high-k/germanium interface properties through selection of high-k materials and suppression of GeO volatilization

Two approaches to control high-k/Ge interface qualities were investigated. The first approach was using high-k materials that are intimate with Ge. These Ge-intimate high-k materials should have moderate reactivity with Ge to form an amorphized interface that will reduce the interface defects and will suppress the GeO desorption at the interface. The second approach was modifying the annealing processes by using a cap layer to block GeO out-diffusion. We found that Si works as the cap layer very efficiently. By combining those two approaches, we achieved fairly good high-k/Ge metal-insulator-semiconductor (MIS) characteristics with LaYO₃ as the Ge-intimate high-k material, and a NiSi_<em>Xem> electrode as the cap layer. These results provide us an important guide for controlling the high-k/Ge interface properties.     

Second-order Born calculation of coplanar symmetric(e, 2e) process on Mg

The second-order distorted wave Born aPl6roximation （DWBA） method is employed to investigate the triple differen- tial cross sections （TDCS） of coplanar doubly symmetric （e, 2e） collisions for magnesium at excess energies of 6 eV-20 eV. Comparing with the standard first-order DWBA calculations, the inclusion of the second-order Born term in the scattering amplitude improves the degree of agreement with experiments, especially for backward scattering region of TDCS. This indicates that the present second-order Born term is capable to give a reasonable correction to DWBA model in studying coplanar symmetric （e, 2e） problems of two-valence-electron target in low energy range.