Spline collocation for strongly elliptic equations on the torus

Summary We prove convergence and error estimates in Sobolev spaces for the collocation method with tensor product splines for strongly elliptic pseudodifferential equations on the torus. Examples of applications include elliptic partial differential equations with periodic boundary conditions but also the classical boundary integral operators of potential theory on torus-shaped domains in three or more dimensions. For odd-degree splines, we prove convergence of nodal collocation for any strongly elliptic operator. For even-degree splines and midpoint collocation, we find an additional condition for the convergence which is satisfied for the classical boundary integral operators. Our analysis is a generalization to higher dimensions of the corresponding analysis of Arnold and Wendland [4].     

Ion pair generation rates near planar3H and63Ni electron capture detector foils

The ion-pair generation rate (ionization topography) in plasmas from⁶³Ni and particularly Ti³H₄ foils, as used in electron capture detectors, was measured at room temperature using large, parallel plates of low backscattering ability in nitrogen gas of varying density. For one atmosphere pressure, the fall-off of ion pair formation as calculated from the exponential region equalsN ₀·e ^–0.19d for⁶³Ni andN ₀·e ^–1.4d for³H (whereN ₀ is the initial ionization rate immediately adjacent to the foil andd is the distance from the foil in mm). The experimentally measured half ranges (distances from the foil within which 50% of all possible ion pairs are created) are 2.7 mm for⁶³Ni and 0.27 mm for³H. The half ranges calculated from the exponential region where there is less interference from electron backscattering, are 3.7 and 0.5 mm, respectively. The latter values are considered closer to the true, unimpeded ionization topography near planar⁶³Ni and³H foils.Material taken from doctoral thesis     

~(14)N ENDOR Study on Bis(2-Hydroxylphenyl Ketoxime)-~(63)Cu(Ⅱ) Complex

The ENDOR spectrum of ~(14)N-~(63)Cu-HAP complex in DMSO/EtOH (5:1) freezing solution at 20 K has been studied using orientational selective method in this paper. The anisotropic superhyperfine coupling tensor and qusdrupole coupling tensor of ligand ~(14)N nucleus, and the superhyperfine coupling tensor of various ~1H nuclei have been measured precisely. Comparing the superhyperfine coupling tensor of ~(14)N-nucleus with previous work shows that the analytical method of spectrum is reasonable and the data are reliable in our previous work.     

Quasi-interpolation with translates of a function having noncompact support

We establish a result related to a theorem of de Boor and Jia [1]. Their theorem, in turn, corrected and extended a result of Fix and Strang [5] concerning controlled approximation. In our result, the approximating functions are not required to have compact support, but satisfy instead conditions on their behavior at . Our theorem includes some recent results of Jackson [6] and is closely related to the work of Buhmann [2].Communicated by Carl de Boor     

Fourier analysis of the approximation power of principal shift-invariant spaces

The approximation order provided by a directed set {S _h } _h>0 of spaces, each spanned by thehZ ^d -translates of one function, is analyzed. The nearoptimal approximants of [R2] from eachs _h to the exponential functions are used to establish upper bounds on the approximation order. These approximants are also used on the Fourier transform domain to yield approximations for other smooth functions, and thereby provide lower bounds on the approximation order. As a special case, the classical Strang-Fix conditions are extended to bounded summable generating functions.The second part of the paper consists of a detailed account of various applications of these general results to spline and radial function theory. Emphasis is given to the case when the scale {s _h } is obtained froms ₁ by means other than dilation. This includes the derivation of spectral approximation orders associated with smooth positive definite generating functions.     

Structural Patterns and Dimensionality in Magnesium Borophosphates: The Crystal Structures of Mg2(H2O)[BP3O9(OH)4] and Mg(H2O)2[B2P2O8(OH)2]·H2O

Hydrothermal investigations in the system MgO/B₂O₃/P₂O₅(/H₂O) yielded two new magnesium borophosphates, Mg₂(H₂O)[BP₃O₉(OH)₄] and Mg(H₂O)₂[B₂P₂O₈(OH)₂]·H₂O. The crystal structures were solved by means of single crystal X‐ray diffraction. While the acentric crystal structure of Mg₂(H₂O)[BP₃O₉(OH)₄] (orthorhombic, P2₁2₁2₁ (No. 19), a = 709.44(5) pm, b = 859.70(4) pm, c = 1635.1(1) pm, V = 997.3(3) × 10⁶ pm³, Z = 4) contains 1D infinite chains of magnesium coordination octahedra interconnected by a borophosphate tetramer, Mg(H₂O)₂[B₂P₂O₈(OH)₂]·H₂O (monoclinic, P2₁/c (No. 14), a = 776.04(5) pm, b = 1464.26(9) pm, c = 824.10(4) pm, β = 90.25(1)°, V = 936.44(9) × 10⁶ pm³,Z = 4) represents the first layered borophosphate with 6³ net topology. The structures are discussed and classified in terms of structural systematics.     

Splitting families of sets in ZFC

Miller's 1937 splitting theorem was proved for every finite n>0$n>0$ for all ρ-uniform families of sets in which ρ is infinite. A simple method for proving Miller-type splitting theorems is presented here and an extension of Miller's theorem is proved in ZFC for every cardinal ν for all ρ  -uniform families in which ρ≥_?ω(ν)$\rho \ge {\mathrm{?}}_{\omega }(\nu )$. The main ingredient in the method is an asymptotic infinitary Löwenheim–Skolem theorem for anti-monotone set functions.     

Twisted Poincaré Duality for some Quadratic Poisson Algebras

We exhibit a Poisson module restoring a twisted Poincaré duality between Poisson homology and cohomology for the polynomial algebra endowed with Poisson bracket arising from a uniparametrised quantum affine space. This Poisson module is obtained as the semiclassical limit of the dualising bimodule for Hochschild homology of the corresponding quantum affine space. As a corollary we compute the Poisson cohomology of R, and so retrieve a result obtained by direct methods (so completely different from ours) by Monnier. This research of the first author was supported by a Marie Curie Intra-European Fellowship within the 6th European Community Framework Programme. The second author was supported by EPSRC Grant EP/D034167/1.