Scattering spectra and colour of disperse systems of low-absorptive particles

In this article, the results of calculations of scattering spectra of low-absorptive spherical particles are presented. It was obtained using different approximate and numerical methods. The comparative evaluations of application of single scattering approximation, the average field approximation and of the methods of computer modeling on the basis of Monte-Carlo method on various scattering multiplicity were performed. The comparison demonstrates the good agreement of calculated results with experimental scattering spectra of eye lens. By using the obtained spectral data the colour coordinates are calculated. The regularities of changes in colour characteristics in dependence of disperse system parameters are analyzed.     

Helly-Type Theorems for Line Transversals to Disjoint Unit Balls

We prove Helly-type theorems for line transversals to disjoint unit balls in ℝ ^d . In particular, we show that a family of n≥2d disjoint unit balls in ℝ ^d has a line transversal if, for some ordering ≺ of the balls, any subfamily of 2d balls admits a line transversal consistent with ≺. We also prove that a family of n≥4d−1 disjoint unit balls in ℝ ^d admits a line transversal if any subfamily of size 4d−1 admits a transversal. Andreas Holmsen was supported by the Research Council of Norway, prosjektnummer 166618/V30. Otfried Cheong and Xavier Goaoc acknowledge support from the French-Korean Science and Technology Amicable Relationships program (STAR).     

Pore expansion of highly monodisperse phenylene-bridged organosilica spheres for chromatographic application

Monodisperse phenylene-bridged organosilica spheres show great potential as chromatographic stationary phase. In this paper, the tunable particle size of monodisperse phenylene-bridged organosilica spheres were prepared by co-condensing different proportion of 1,4-bis(triethoxysilyl)benzene (1,4-BTEB) and tetraethylorthosilicate (TEOS), and then pore size was expanded by two-step post-synthesis hydrothermal treatments using N,N-dimethyldecylamine (DMDA)/dodecylamine (DDA) and tris-(hydroxymethyl)-aminomethane (TRIS) in turn. Phenylene-bridged organosilica spheres with particle size of 3.0-3.5 μm and pore size of 85 Å were further surface modified by C₁₈ group and tested in reversed-phase high performance liquid chromatography (RP-HPLC). The primary chromatographic results demonstrated that C₁₈ bonded phenylene-bridged organosilica stationary phase has high retention and good chemical stability in the high pH mobile phase, which indicated that the phenylene-bridged organosilica can be used for HPLC packing supports.     

球胶态颗粒在中等电位条件下双电层相互作用能和力

根据有效表面电位的定义,使用改进的Derjaguin法,推导出在中等电位条件下等同和不等同球形胶态颗粒相互作用能和力的近似表达式.与精确的数值解相比,对等同球颗粒,表面电位≤100mV;对不等同球颗粒,表面电位≤75mV;最大相对误差均小于±10%,表明该近似式是目前较好的.     

Conditionally positive definite kernels on Euclidean domains

This paper characterizes several classes of conditionally positive definite kernels on a domain Ω of either or . Among the classes is that composed of strictly conditionally positive definite kernels. These kernels are known to be useful in the solution of variational interpolation problems on Ω. Our study covers the case in which Ω is the sphere S^l−1 of or a similar manifold. Among other things, our results imply that the characterization of (strict) conditional positive definiteness on Ω can be obtained from a characterization of (strict) positive definiteness on Ω. The bi-zonal strictly conditionally positive definite kernels on S^l−1, l?3, are described.     

Interpolation by Polynomials and Radial Basis Functions on Spheres

The paper obtains error estimates for approximation by radial basis functions on the sphere. The approximations are generated by interpolation at scattered points on the sphere. The estimate is given in terms of the appropriate power of the fill distance for the interpolation points, in a similar manner to the estimates for interpolation in Euclidean space. A fundamental ingredient of our work is an estimate for the Lebesgue constant associated with certain interpolation processes by spherical harmonics. These interpolation processes take place in ``spherical caps' whose size is controlled by the fill distance, and the important aim is to keep the relevant Lebesgue constant bounded. This result seems to us to be of independent interest. March 27, 1997. Dates revised: March 19, 1998; August 5, 1999. Date accepted: December 15, 1999.     

Preparation and properties of uniform praseodymium-doped ceria colloidal particles

 A procedure to prepare submicrometre spherical particles of Pr(III), Ce(III) or Pr(III)-doped C(III) hydroxycarbonates with narrow size distribution is reported. The particles were obtained by aging aqueous solutions of Pr(III) chloride and/or Ce(III) nitrate in the presence of urea at 100 °C for 2 h. The effect of Pr and/or Ce salt concentrations in the starting solutions on the size, shape and composition of the precipitated particles is reported. The thermal behaviour of the basic carbonates up to decomposition into the metal oxides was followed by differential thermal and thermogravimetric analyses, X-ray diffraction, IR spectroscopy and transmission electron microscopy. All the systems were also characterized by their electrokinetic behaviour (determination of isoelectric point) and specific surface areas. The colour of the powders was also evaluated as a function of the Pr content and temperature and was compared with previous results on red pigments of similar composition. Received: 30 May 2001 Revised: 17 September 2001 Accepted: 20 September 2001     

Reproducing Properties of Differentiable Mercer-Like Kernels on the Sphere

We study differentiability of functions in the reproducing kernel Hilbert space (RKHS) associated with a smooth Mercer-like kernel on the sphere. We show that differentiability up to a certain order of the kernel yields both, differentiability up to the same order of the elements in the series representation of the kernel and a series representation for the corresponding derivatives of the kernel. These facts are used to embed the RKHS into spaces of differentiable functions and to deduce reproducing properties for the derivatives of functions in the RKHS. We discuss compactness and boundedness of the embedding and some applications to Gaussian-like kernels.     

The Gaussian limit for high-dimensional spherical means

We show that the limit of integrals along slices of a high dimensional sphere is a Gaussian integral on a corresponding finite-codimension affine subspace in infinite dimensions.     

A generalization of image theory to predict the interaction of multipole fields with plane surfaces

A derivation and computational scheme, based on exact image theory, for the field produced by the interaction of an outgoing vector wave harmonic with an infinite-extent plane surface is presented. The method represents the angular-dependent Fresnel reflection coefficients of the surface as Laplace transforms of a spatially dependent function, which results in the reflected field appearing as a superposition of image sources located at complex points along the normal axis within the surface medium. Exact, analytical formulas are given for the transformed reflection coefficients for arbitrary surface refractive index, and an efficient computation scheme for evaluation of the scattered field coupling between a particle and the surface is presented.