Sparse generalized Fourier transforms

Block-diagonalization of sparse equivariant discretization matrices is studied. Such matrices typically arise when partial differential equations that evolve in symmetric geometries are discretized via the finite element method or via finite differences. By considering sparse equivariant matrices as equivariant graphs, we identify a condition for when block-diagonalization via a sparse variant of a generalized Fourier transform (GFT) becomes particularly simple and fast. Characterizations for finite element triangulations of a symmetric domain are given, and formulas for assembling the block-diagonalized matrix directly are presented. It is emphasized that the GFT preserves symmetric (Hermitian) properties of an equivariant matrix. By simulating the heat equation at the surface of a sphere discretized by an icosahedral grid, it is demonstrated that the block-diagonalization is beneficial. The gain is significant for a direct method, and modest for an iterative method. A comparison with a block-diagonalization approach based upon the continuous formulation is made. It is found that the sparse GFT method is an appropriate way to discretize the resulting continuous subsystems, since the spectrum and the symmetry are preserved. AMS subject classification (2000)  43A30, 65T99, 20B25     

Adsorption of dicarboxylates on nano-sized gibbsite particles: effects of ligand structure on bonding mechanisms

The adsorption of the dicarboxylates o-phthalate, maleate, fumarate, malonate, and oxalate (representing ligands with the general composition ⁻O₂C---C_n---CO₂⁻; n=0, 1, or 2) on gibbsite were studied by means of quantitative batch adsorption experiments and attenuated total reflectance Fourier transform infrared (ATR-FTIR) spectroscopy. The interpretations of ATR-FTIR spectra were aided by comparison with IR spectra of solution species and by results from theoretical frequency calculations. The main objectives of the study were to identify the molecular level bonding mechanisms of the dicarboxylates to gibbsite, and to investigate how these were influenced by the composition and structure of the ligands. Carboxylates with n=2 formed predominantly outer sphere complexes, whereas the importance of inner sphere complexes progressively increased for n=1 and 0. The inner sphere structures were identified as mononuclear chelates with one oxygen from each carboxylate group bonded to Al(III) at the surface. This showed the importance of chelate ring structure for the formation of inner sphere surface complexes, with stabilities of the complexes increasing in the order seven-membered ring less than six-membered ring less than five-membered ring. For ligands with n=2 only small variations in surface speciation were observed as a function of steric factors; irrespective of the relative positions of the carboxylate groups and bulkiness of the ligands outer sphere bonding is the dominating adsorption mode. Adsorption experiments were also conducted with gibbsite particles exhibiting differences in shape and surface roughness. These experiments showed that inner sphere complexes were favored on the less well-developed and more irregular gibbsite particles.     

Atomic layer deposition in nanometer-level replication of cellulosic substances and preparation of photocatalytic TiO2/cellulose composites

TiO2 replicas of filter paper with nanometer-level accuracy were prepared by atomic layer deposition of thin conformal TiO2 coating, followed by a removal of the paper by air-anneal at 450 degrees C. Photocatalytic anatase TiO2/cellulose composites were also made by leaving the paper intact. The TiO2 films were deposited from Ti(OMe)4 and H2O at 150-250 degrees C. The photocatalytic activity of the TiO2/cellulose composite was verified by photocatalytic reduction of Ag(I) from an aqueous solution to Ag nanoparticles on the TiO2 surface. The TiO2/cellulose composites are mechanically more stable than the free-standing TiO2 replicas and are therefore potentially suitable as lightweight, high surface area photocatalysts.     

The reactions of tellurium tetrahalides with triphenylphosphine under ambient conditions

The reactions of tellurium tetrahalides and triphenylphosphine in tetrahydrofuran have been carried out under ambient conditions and afford [(Ph(3)PO)(2)H](2)[Te(2)X(10)] [X = Cl (1), Br (2)] and [(Ph(3)PO)(3)(OH(3)])(2)[TeI(6)] (4). The X-ray structures of 1 and 2 show that they are isostructural and contain discrete [Te(2)X(10)](2-) anions exhibiting octahedral coordination around both tellurium atoms with one shared edge and [Ph(3)POH...OPPh(3)](+) cations that show strong hydrogen bonds (the O...O distances are 2.399 and 2.404 A for 1 and 2, respectively). The compound 4 is built up with discrete octahedral hexaiodotellurate anions and [(Ph(3)PO)(3)(OH(3))](+) cations. The reaction of TeBr(4) and PPh(3) also results in the formation of formally zwitterionic Ph(3)PO(CH(2))(4)TeBr(4) (3). This reaction involves an unprecedented THF ring opening in which the oxygen atom becomes bonded to the phosphorus atom of triphenylphosphine and the carbon atom at the other end of the five-atomic chain becomes bonded to the tellurium atom of TeBr(4). The ring opening of the solvent THF is also taking place in the reaction involving tellurium tetraiodide, as indicated by the formation of C(4)H(8)TeI(2) (5). The reaction may initially lead to Ph(3)PI(2) that reacts with THF yielding Ph(3)PO and ICH(2)(CH(2))(2)CH(2)I. The latter species reacts with elemental tellurium producing 5. Depending on the conditions upon crystallization, two polymorphs of C(4)H(8)TeI(2) (5a and 5b) are observed. While the molecular structures of the two forms are virtually identical, their packing and intermolecular contacts are different. Two further minor products (6a and 6b) were isolated in the reaction of TeI(4) and PPh(3): Both are formally 1:1 adducts of 5 and TeI(4), but they differ considerably in their structures. 6a can be formulated as [C(4)H(8)TeI(+)](2)[Te(2)I(10)(2-)] and 6b as [C(4)H(8)TeI(+)](2)(TeI(3)(+))(2)(I(-))(4). The latter compound exhibits framework similar to that of the tetramers in gamma- and delta-TeI(4).     

Study of metal-loaded activated charcoals for the separation and determination of selenium species by energy dispersive X-ray fluorescence analysis

Conversion of adsorption properties of activated charcoal (C^∗) by metal modification towards selenite, selenate and seleno-dl-methionine (Se-Met) was studied. Several metals were included in modification studies and also a procedure was developed for the separation and independent determination of microgram quantities of these selenium species in aqueous solution. Selenium species were collected one by one from the same sample solution onto collector material by adsorbing them directly or after complex formation onto different metal-loaded activated charcoals (MgC^∗ and FeC^∗). The mass of selenium in these collectors was measured directly by energy dispersive X-ray fluorescence spectrometry (EDXRF). Detection limits for all three selenium species were better than 6 μg corresponding to concentration of 60 μg/l with 0.100 l of initial sample volume.     

Organo-noble-gas hydride compounds HKrCCH, HXeCCH, HXeCC, and HXeCCXeH: formation mechanisms and effect of 13C isotope substitution on the vibrational properties

We investigate the formation mechanism of HXeCCXeH in a Xe matrix. Our experimental results show that the HXeCCXeH molecules are formed in the secondary reactions involving HXeCC radicals. The experimental data on the formation of HXeCCXeH is fully explained based on the model involving the HXeCC+Xe+H-->HXeCCXeH reaction. This reaction is the first case when a noble-gas hydride molecule is formed from another noble-gas molecule. In addition, we investigate the (12)C/(13)C isotope effect on the vibrational properties of organo-noble-gas hydrides (HKrCCH, HXeCCH, HXeCC, and HXeCCXeH) in noble-gas matrixes. The present experimental results and ab initio calculations on carbon isotope shifts of the vibrational modes support the previous assignments of these molecules. Upon (12)C to (13)C isotope substitution, we observed a pronounced effect on the H-Kr stretching mode of HKrCCH (downshift of 1.0-3.6 cm(-1), depending on the matrix site) and a small anomalous shift (+0.1 cm(-1)) of the H-Xe stretching mode of HXeCCH and HXeCCXeH.     

IR-induced processes for allyl alcohol dimers involving an intermolecular OH...π Bond in low-temperature matrices

Sharp peaks were observed in the IR spectra of allyl alcohol dimers in the νOH and τOH regions in low-temperature Ar and N₂ matrices. In Ar a peak at 3587 cm^?1 was found to increase in intensity upon annealing and decrease upon IR irradiation; it is concluded that this peak is due to an OH...π interaction (of a cyclic structure) present in part of the dimers. This is the first time such an interaction and photoprocess have been found for an alcohol in matrices. Also a faster photo-process was found. Confomer changes in the monomers accompany the photoprocess in allyl alcohol dimers.     

Conformational analysis 29: Preparation and1H and13C NMR,FTIR, MS,and crystallographic conformational and configurational study of 3-Oxo-1,3-oxathiane and its monomethyl derivatives

3-Oxo-1,3-oxathiane (1) and its monomethyl derivatives were prepared by oxidation of the corresponding 1,3-oxathianes. The structural analysis was carried out by¹H and¹³C NMR, FTIR, and mass spectrometry. At 298 K compound1 was a 1 1 (at 173 K a 3 1) mixture of the SO(ax) and SO(eq) chair forms. The major oxidation products of methyl 1,3-oxathianes attained exclusively the SO(ax), Me(eq) chair forms except that of the 5-methyl derivative, which consisted of 7% of the SO(eq), Me(ax) chair conformation in CDCl₃ solution. The minor products of oxidation existed in anancomeric SO(eq), Me(eq) chair conformations. The oxidation of 2-methyl- 1,3-oxathiane, however, led to 3,3-dioxo derivative (6) in addition to thetrans [SO(eq)] monoxide. The crystal structures of6 andtrans-3-oxo-5-methyl-1,3-oxathiane were solved by X-ray diffractometry.