Generalized Multiresolution Analyses with Given Multiplicity Functions

Generalized multiresolution analyses are increasing sequences of subspaces of a Hilbert space ℋ that fail to be multiresolution analyses in the sense of wavelet theory because the core subspace does not have an orthonormal basis generated by a fixed scaling function. Previous authors have studied a multiplicity function m which, loosely speaking, measures the failure of the GMRA to be an MRA. When the Hilbert space ℋ is L ²(ℝ ⁿ ), the possible multiplicity functions have been characterized by Baggett and Merrill. Here we start with a function m satisfying a consistency condition, which is known to be necessary, and build a GMRA in an abstract Hilbert space with multiplicity function m.     

NP -Hardness of Largest Contained and Smallest Containing Simplices for V - and H -Polytopes

Abstract. The problem of finding a d -simplex of maximum volume in an arbitrary d -dimensional V -polytope, for arbitrary d , was shown by Gritzmann et al. [GKL] in 1995 to be NP-hard. They conjectured that the corresponding problem for H -polytopes is also NP-hard. This paper presents a unified way of proving the NP-hardness of both these problems. The approach also yields NP-hardness proofs for the problems of finding d -simplices of minimum volume containing d -dimensional V - or H -polytopes. The polytopes that play the key role in the hardness proofs are truncations of simplices. A construction is presented which associates a truncated simplex to a given directed graph, and the hardness results follow from the hardness of detecting whether a directed graph has a partition into directed triangles.     

Iterated snap rounding with bounded drift

Snap Rounding and its variant, Iterated Snap Rounding, are methods for converting arbitrary-precision arrangements of segments into a fixed-precision representation (we call them SR and ISR for short). Both methods approximate each original segment by a polygonal chain, and both may lead, for certain inputs, to rounded arrangements with undesirable properties: in SR the distance between a vertex and a non-incident edge of the rounded arrangement can be extremely small, inducing potential degeneracies. In ISR, a vertex and a non-incident edge are well separated, but the approximating chain may drift far away from the original segment it approximates. We propose a new variant, Iterated Snap Rounding with Bounded Drift, which overcomes these two shortcomings of the earlier methods. The new solution augments ISR with simple and efficient procedures that guarantee the quality of the geometric approximation of the original segments, while still maintaining the property that a vertex and a non-incident edge in the rounded arrangement are well separated. We investigate the properties of the new method and compare it with the earlier variants. We have implemented the new scheme on top of CGAL, the Computational Geometry Algorithms Library, and report on experimental results.     

Diffusion in thin films on the surface of a porous solid.

A water-wet mono-dispersed glass bead system is saturated with two phases, a wetting phase of water and a non-wetting phase of tetrachloroethylene (no 1H signal). Pulsed field gradient NMR measurements of the one-dimensional probability density distribution P(Delta) (X) for the diffusive displacements of water molecules in times, Delta, are presented for the whole accessible water saturation range. At lower water contents the distributions show a distinctive shape, which is attributed to the distribution of the aqueous phase in thin surface wetting films connecting pendular rings where the beads are in contact. The data are reproduced well by a computer simulation of a random walk model based on diffusion of molecules within such a structure, allowing determination of the surface film thickness.     

Complexation-induced chemical shifts--ab initio parameterization of transferable bond anisotropies

Complexation-induced changes in proton chemical shifts provide a potent tool for conformational analysis, being highly dependent on intermolecular orientation. An important contribution to these shifts arises from the molecular magnetisability anisotropy, or more specifically from the anisotropy of certain groups, such as aromatic rings and unsaturated bonds. While the influence of aromatic rings has been well characterised via the ring current effect, unsaturated bonds have received much less attention and prediction of complexation shifts is hampered by the lack of accurate anisotropy parameters for these bonds. We have therefore used ab initio calculations at the HF/aug-cc-pVDZ level to obtain bond anisotropies for C-H, N-H, C=O, C=C, C triple bond N, N=N, C triple bond C, and C triple bond N. Fitting the anisotropies to bond magnetic dipoles (the McConnell equation) gives non-transferable values for C-H and N-H bonds. We have therefore expanded in terms of bond magnetic dipoles, quadrupoles, and octopoles for double and triple bonds only, obtaining highly accurate shielding surfaces in all cases. The transferable nature of the anisotropies is confirmed by comparing with shifts obtained in larger molecules containing unsaturated bonds.     

Effect of crosslinking on mitochondrial structure and function.

Rat liver mitochondria were treated with ethylacetimidate and methylbutyrimidate, monofunctional imidates, and with dimethylsuberimidate, a bifunctional imidate, and the effects on structure and function studied. Mitochondria treated with 5 mM dimethylsuberimidate or greater did not respond osmotically when placed in deionized water. Sodium dodecylsulfate-polyacrylamide gel electrophoresis revealed that at concentrations greater than 5 mM dimethylsuberimidate nearly all mitochondrial polypeptides failed to enter 6% gels, indicating crosslinking of both membrane and soluble proteins. Extensive amidination by ethylacetimidate and methylbutyrimidate had little effect on ascorbate-tetramethylphenylenediamine oxidase while extensive inhibition resulted from dimethylsuberimidate treatment. The possible involvement of molecular motion in electron transport is discussed.     

The effects of morphology on 1H NMR spectra and relaxation in semicrystalline polyolefins

A brief résumé is given of the role of structural heterogeneity, magnetic dipolar couplings, molecular structure, and molecular motion in determining the ¹H NMR spectra and relaxation properties of heterogeneous solids such as semicrystalline polymers. Measurements of ¹H spin-lattice relaxation in laboratory (T₁) and rotating frames (T) are reported for a number of solid polyolefin samples. These include solution-crystallized and melt-crystallized polyethylene, annealed and quenched isotactic polypropene, and isotactic polybut-1-ene. In addition, broad-line ¹H spectra, both normal and partially (T) relaxed, are reported for these materials as well as a number of pulsed NMR experiments having the philosophy of the so-called Goldman–Shen experiment. Spin-lattice relaxation (T₁) for all samples is a single exponential process, whereas rotating-frame relaxation comprises three exponential processes both on-resonance (θ = 90°) and off-resonance at the magic angle (θ = 54.7°), with the latter generally being much slower. The spectra show clearly the existence of components having differing degrees of mobility and, with the exception of the solution-crystallized polyethylene, the partially (T) relaxed spectra indicate a correlation between breadth of resonance line and magnitude of T_1ρ. The Goldman–Shen-type experiments indicate a spin-diffusional transport of magnetization between the different spectral and (T) components. A computer program has been used to simulate the NMR behavior of a three-region system comprising repeating units of infinite lamellae of different widths, each region having different intrinsic relaxation times and spin diffusion coefficients. The results demonstrate that the observed ¹H NMR behavior of these samples can be interpreted in terms of this model and that, inter alia, the long-time T behavior reflects, qualitatively, the time taken for magnetization to diffuse a distance of the order of the dimensions of the region to which it corresponds.     

High-resolution 13C NMR studies of solid 1,4-bis(n-butylamino)anthraquinone

High-resolution ¹³C NMR spectra of solid 1,4-bis(n-butylamino)anthraquinone are presented. In spectra measured at 22.6 MHz the effects of two short-range and two long-range residual (¹³C, ¹⁴N) dipolar couplings have been observed. Only the short-range effects are observable in the 50.3 MHz spectra. These typical asymmetric doublets allow the determination of the ¹⁴N nuclear quadrupole coupling constant (χ=e²Qq/h=?3.0 MHz), the asymmetry parameter (η=0.3) and the orientation of the electric field gradient tensor of the ¹⁴N in the molecular framework.