A classification of continuous wavelet transforms in dimension three

This paper presents a full catalogue, up to conjugacy and subgroups of finite index, of all matrix groups $H<\mathrm{GL}(3,\mathbb{R})$ that give rise to a continuous wavelet transform with associated irreducible quasi-regular representation. For each group in this class, coorbit theory allows to consistently define spaces of sparse signals, and to construct atomic decompositions converging simultaneously in a whole range of these spaces. As an application of the classification, we investigate the existence of compactly supported admissible vectors and atoms for the groups.     

Ma?analytische Bestimmung des Quecksilbers

Ohne Zusammenfassung     

Ueber die Bestimmung des Acetons

Ohne Zusammenfassung     

Characterization of Shift-Invariant Spaces on a Class of Nilpotent Lie Groups with Applications

Given a simply connected nilpotent Lie group having unitary irreducible representations that are square-integrable modulo the center, we use operator-valued periodization to give a range-function type characterization of shift-invariant spaces of function on the group. We then give characterizations of frame and Riesz families for shift-invariant spaces.     

Mercerization of primary wall cellulose and its implication for the conversion of cellulose I→cellulose II

The mercerization of homogenized primary wall cellulose extracted fromsugar beet pulp was investigated by transmission electron microscopy (TEM),X-ray diffraction together with ¹³C CP-MAS NMR, and FT-IR spectroscopy.For samples resulting from acid extraction, mercerization began at 9% NaOH, whereasfor samples purified by alkaline treatment, the mercerization started at 10%NaOH. The change in morphology when going from cellulose I to cellulose II wasspectacular, as all the microfibrillar cellulose morphology disappeared duringthe treatment. This change in morphology was very drastic as soon as the NaOHconcentrations were increased beyond 8 and 9% for the acid and alkalinepreparedsamples, respectively. On the other hand, the conversion was found to be moreprogressive in terms of increasing NaOH concentration when the transformationwas analyzed by X-ray diffraction or spectroscopy. Our observations of themercerization of isolated cellulose microfibrils are consistent with theconceptof cellulose microfibrils made of parallel chains in cellulose I and crystalsofcellulose II consisting of antiparallel chains.     

Dynamic chirality: keen selection in the face of stereochemical diversity in mechanically bonded compounds

The template-directed syntheses, employing bisparaphenylene-[34]crown-10 (BPP34C10), 1,5-dinaphthoparaphenylene-[36]crown-10 (1/5NPPP36C10), and 1,5-dinaphtho-[38]crown-10 (1/5DNP38C10) as templates, of three [2]catenanes, whereby one of the two bipyridinium units in cyclobis(paraquat-p-phenylene) is replaced by a bipicolinium unit, are described. The crude reaction mixtures comprising the [2]catenanes all contain slightly more of the homologous [3]catenanes, wherein a "dimeric" octacationic cyclophane has the crown ether macrocycles encircling the alternating bipyridinium units with the bipicolinium units completely unfettered. X-ray crystallography, performed on all three [2]catenanes and two of the three [3]catenanes reveals co-conformational and stereochemical preferences that are stark and pronounced. Both the [3]catenanes crystallize as mixtures of diastereoisomers on account of the axial chirality associated with the picolinium units in the solid state. Dynamic (1)H NMR spectroscopy is employed to probe in solution the relative energy barriers for rotations by the phenylene and pyridinium rings in the tetracationic cyclophane component of the [2]catenanes. Where there are co-conformational changes that are stereochemically "allowed", crown ether circumrotation and rocking processes are also investigated for the relative rates of their occurrence. The outcome is one whereby the three [2]catenanes containing BPP34C10, 1/5NPPP36C10, and 1/5DNP38C10 exist as one major enantiomeric pair of diastereoisomers amongst two, four, and eight diastereoisomeric pairs of enantiomers, respectively. The diastereoisomerism is a consequence of the presence of axial chirality together with helical and/or planar chirality in the same interlocked molecule. These [2]catenanes constitute a rich reserve of new stereochemical types that might be tapped for their switching and mechanical properties.     

Controllable donor-acceptor neutral [2]rotaxanes

In pursuit of a neutral bistable [2]rotaxane made up of two tetraarylmethane stoppers--both carrying one isopropyl and two tert-butyl groups located at the para positions on each of three of the four aryl rings--known to permit the slippage of the pi-electron-donating 1,5-dinaphtho[38]crown-10 (1/5DNP38C10) at the thermodynamic instigation of pi-electron-accepting recognition sites, in this case, pyromellitic diimide (PmI) and 1,4,5,8-naphthalenetetracarboxylate diimide (NpI) units separated from each other along the rod section of the rotaxane's dumbbell component, and from the para positions of the fourth aryl group of the two stoppers by pentamethylene chains, a modular approach was employed in the synthesis of the dumbbell-shaped compound NpPmD, as well as of its two degenerate counterparts, one (PmPmD) which contains two PmI units and the other (NpNpD) which contains two NpI units. The bistable [2]rotaxane NpPmR, as well as its two degenerate analogues PmPmR and NpNpR, were obtained from the corresponding dumbbell-shaped compounds NpPmD, PmPmD, and NpNpD and 1/5DNP38C10 by slippage. Dynamic 1H NMR spectroscopy in CD2Cl2 revealed that shuttling of the 1/5DNP38C10 ring occurs in NpNpR and PmPmR, with activation barriers of 277 K of 14.0 and 10.9 kcal mol(-1), respectively, reflecting a much more pronounced donor-acceptor stabilizing interaction involving the NpI units over the PmI ones. The photophysical and electrochemical properties of the three neutral [2]rotaxanes and their dumbbell-shaped precursors have also been investigated in CH2Cl2. Interactions between 1/5DNP38C10 and PmI and NpI units located within the rod section of the dumbbell components of the [2]rotaxane give rise to the appearance of charge-transfer bands, the energies of which correlate with the electron-accepting properties of the two diimide moieties. Comparison between the positions of the visible absorption bands in the three [2]rotaxanes shows that, in NpPmR, the major translational isomer is the one in which 1/5DNP38C10 encircles the NpI unit. Correlations of the reduction potentials for all the compounds studied confirm that, in this non-degenerate [2]rotaxane, one of the translational isomers predominates. Furthermore, after deactivation of the NpI unit by one-electron reduction, the 1/5DNP38C10 macrocycle moves to the PmI unit. Li+ ions have been found to strengthen the interaction between the electron-donating crown ether and the electron-accepting diimide units, particularly the PmI one. Titration experiments show that two Li+ ions are involved in the strengthening of the donor-acceptor interaction. Addition of Li+ ions to NpPmR induces the 1/5DNP38C10 macrocycle to move from the NpI to the PmI unit. The Li+-ion-promoted switching of NpPmR in a 4:1 mixture of CD2Cl2 and CD3COCD3 has also been shown by 1H NMR spectroscopy to involve the mechanical movement of the 1/5DNP38C10 macrocycle from the NpI to the PmI unit, a process that can be reversed by adding an excess of [12]crown-4 to sequester the Li+ ions.