Approximation by translates of refinable functions

Summary. The functions are refinable if they are combinations of the rescaled and translated functions . This is very common in scientific computing on a regular mesh. The space of approximating functions with meshwidth is a subspace of with meshwidth . These refinable spaces have refinable basis functions. The accuracy of the computations depends on , the order of approximation, which is determined by the degree of polynomials that lie in . Most refinable functions (such as scaling functions in the theory of wavelets) have no simple formulas. The functions are known only through the coefficients in the refinement equation – scalars in the traditional case, matrices for multiwavelets. The scalar "sum rules" that determine are well known. We find the conditions on the matrices that yield approximation of order from . These are equivalent to the Strang–Fix conditions on the Fourier transforms , but for refinable functions they can be explicitly verified from the . Received August 31, 1994 / Revised version received May 2, 1995     

SSZ-53 and SSZ-59: two novel extra-large pore zeolites

The syntheses, structure solutions, and physicochemical and catalytic characterizations of the novel zeolites SSZ-53 and SSZ-59 are described. SSZ-53 and SSZ-59 were synthesized under hydrothermal conditions with the [1-(4-fluorophenyl)cyclopentylmethyl]trimethyl ammonium cation and 1-[1-(4-chlorophenyl)cyclopentylmethyl]-1-methyl azocanium cation, respectively, as structure-directing agents. The framework topology of SSZ-53 was solved with the FOCUS method, and the structure of SSZ-59 was determined by model building. Rietveld refinement of synchrotron X-ray powder diffraction data confirms each proposed model. SSZ-53 and SSZ-59 each possess a one-dimensional channel system delimited by 14-membered rings. Results from transmission electron microscopy, electron diffraction, catalytic experiments (spaciousness index and constraint index tests), and argon and hydrocarbon adsorption experiments are consistent with the proposed structures.     

Styrene emulsion polymerization: Particle-size distributions

Data are presented on the time evolution of particle-size distributions (PSDs) in seeded and ab initio styrene emulsion polymerization systems. Initiation was by chemical reagent (potassium persulfate) or γ-radiation. The unswollen PSDs at various times during interval II of the polymerization were obtained by direct measurement of calibrated electron micrographs. Experimental results were fitted with the equations that describe the time evolution of an initial PSD. Analytic solutions to these equations that allow for entry, exit, and propagation of free radicals were obtained. The values of the rate coefficients for these processes used to fit the experimental data were in excellent agreement with those obtained from dilatometric kinetics experiments.     

Molecular models of lipopeptide detergents: large coiled-coils with hydrocarbon interiors

We have constructed molecular models of octameric micelles formed by a recently developed lipopeptide detergent consisting of a single amphipathic alpha-helix coupled to two acyl chains at either end of the helix. The models explain the experimentally observed aggregation behavior of peptides with different acyl chain lengths. The octameric micelles form a unique coiled-coil structure, with the acyl chains in a nearly frozen conformation inside the cylindrical assemblies. Two extreme models with helices either all parallel or in an alternating orientation suggest that the alternating orientation is energetically more favorable. The models suggest several new directions for further diversifying this new class of detergents for the structural studies of membrane proteins.     

Optically and thermally induced electron transfer pathways in hexakis[4-(N,N-diarylamino)phenyl]benzene derivatives

The optically and thermally induced electron transfer pathways of highly symmetrical (D(3)) hexaarylbenzene systems with six triarylamine redox sites have been investigated. Owing to slightly different local redox potentials, the radical trication could be selectively generated by electrochemical methods. This trication shows a strong intervalence charge-transfer band in the near infrared (NIR) that was measured by spectroelectrochemistry and analysed using multi-dimensional Mulliken-Hush theory. Quantum chemical AM1 CI calculations indicate that there is no optically induced concerted three-electron transfer that transforms the ground state into a state in which all three positively charged triarylamine moieties change place with their neutral neighbours. The potential energy surface of the ground state was constructed by using quadratic potentials. From this potential surface it is apparent that there is also no thermally allowed concerted three-electron transfer pathway. Instead, three consecutive one-electron transfer steps are necessary for this process.     

The preparation of a new "safety catch" ester linker for solid-phase synthesis

A new "safety catch" linker for esters has been synthesized on polystyrene resin. This 2-tert-butoxyphenol resin 10 may be acylated to give a relatively stable ester that will allow nucleophilic chemistry without reaction at the linking ester group. Removal of the tert-butyl group with acid unmasks a highly reactive 2-hydroxyphenyl ester that reacts readily with nucleophiles to cause release of the product from the resin. This sequence has been exemplified by acylating the resin with various bromo acids, carrying out nucleophilic displacements with thiols, phenols, or amines, activating the ester with trifluoroacetic acid and cleaving from the resin with amines to give the (nucleophile) substituted carboxamides in high yield and purity. Kinetic studies with a model ester revealed half-lives for reaction with morpholine of 119 h for the tert-butoxyphenyl ester and 1 min for the corresponding phenol.     

Amperometric method for determining the degree of complexation of polyelectrolytes with cationic surfactants

The complexation of sodium polystyrene sulfonate with monovalent cationic surfactants at a microsized liquid/liquid interface has been studied using electrochemistry. The method is based on measurement of surfactant ion transfer across the interface between two immiscible electrolyte solutions (ITIES). The complexation of various cationic surfactants (alkylpyridinium- and trimethylammonium-) with oligosized polystyrene sulfonate was measured. Binding isotherms were used to determine the degree of binding as a function of the surfactant chain length and type of head group. It was found that the hydrophobicity of the surfactant was the predominant factor. The effect of the polyelectrolyte chain length on the binding mechanism was studied using cetylpyridinium chloride as a complexing agent. It was found that binding affinity, as well as cooperativity of the binding process, decreases with decreasing polyelectrolyte chain length. Thermodynamics of surfactant binding was measured using titration microcalorimetry. The thermodynamic data obtained show that the enthalpy of surfactant binding is not dependent on polymer chain length, but an increase in chain length makes the binding process entropically more favorable.     

Selective separations by reactive ion exchange : Part III. Preconcentration and separation of oxo anions

A new procedure for simultaneous preconcentration, separation, and determination of permanganate, chromate and vanadate was developed for aqueous solutions containing p.p.b. levels of these anions (and molybdate). The presence of other (non-complexing) ions in p.p.m. concentrations did not interfere. The procedure consisted of reactive adsorption of all three anions as their reduced cations, on iron(II)-treated resin, followed by elution of vanadium(IV) with 0.01 M HNO₃/H₂O₂, manganese(II) with 0.35 M HCl, and chromium(III) with 4M HCl. Concentration factors of 40 were obtained with 1-l “samples” 10 p.p.b. concentrations were determined with standard deviations of 4–5 % by a.a.s. and conventional spectrophotometry.     

ETUDE EN SPECTROMETRIE DE MASSE DE PHOSPHONATES ENYNIQUES ET DE β-CETOPHOSPHONATES-γ, δ-ETHYLENIQUES

The mass spectrometric study of acetylenic dialkylphosphonates (I) and of γ, δ-éthylenic-β-ketophosphonates (II) is characterised by rearrangement processes of the phosphorylated group and cleavage of the phosphorus-carbon bond. The fragmentation patterns of compounds II were identical to those found for I by loss of water.