The Class of Clifford-Fourier Transforms

Recently, there has been an increasing interest in the study of hypercomplex signals and their Fourier transforms. This paper aims to study such integral transforms from general principles, using 4 different yet equivalent definitions of the classical Fourier transform. This is applied to the so-called Clifford-Fourier transform (see Brackx et al., J. Fourier Anal. Appl. 11:669–681, 2005). The integral kernel of this transform is a particular solution of a system of PDEs in a Clifford algebra, but is, contrary to the classical Fourier transform, not the unique solution. Here we determine an entire class of solutions of this system of PDEs, under certain constraints. For each solution, series expressions in terms of Gegenbauer polynomials and Bessel functions are obtained. This allows to compute explicitly the eigenvalues of the associated integral transforms. In the even-dimensional case, this also yields the inverse transform for each of the solutions. Finally, several properties of the entire class of solutions are proven.     

Quest to enhance up-conversion efficiency: a comparison of anhydrous vs. hydrous synthesis of NaGdF4: Yb3+ and Tm3+ nanoparticles

A major challenge in the field of up-converting (UC) nanomaterials is to enhance their efficiencies. The –OH defects on the surface of the nanoparticles are thought to be the main cause of luminescence quenching, but there are no comparative studies in the literature showing the impact of anhydrous vs. hydrous synthesis on up-conversion efficiency. In this article, we present the synthesis of up-converting NaGdF₄: Yb⁺³, Tm⁺³ nanoparticles by two different methods: thermal decomposition of single source metal-organic anhydrous precursors [NaLn(TFA)₄(diglyme)] (Ln = Gd, Tm, Yb; TFA = trifluoroacetate) and room temperature co-precipitation using hydrated inorganic salts Ln(NO₃)₃·5H₂O (Ln = Gd, Tm, Yb), NaNO₃ and NH₄F in ethylene glycol. After a detailed study on the influence of solvents and the percentage of lanthanide dopant on the crystal phase of the up-converting nanoparticles (NPs) and their complete characterization, a comparative up-conversion study was carried out which revealed that the uniform nanospheres (av. size ～13 nm) obtained from the anhydrous SSP had significantly higher up-conversion efficiency than agglomerated nanorods (～197 nm in length and ～95 nm in width) produced from hydrated inorganic salts. An enhanced up-conversion quantum yield of 1.8% for the anhydrous sample validates the anhydrous precursor approach as a strategy to obtain small but highly emitting up-converting particles without requiring a silica or undoped matrix surface passivation layer.     

Wetting of a self-assembled 1,4-benzenedimethanethiol monolayer on gold and silver: An adhesion force spectroscopy study

Using atomic force microscopy we investigated how local capillary phenomena are affected by the deposition of a self-assembled 1,4-benzenedimethanethiol (BDMT) layer on epitaxially grown Au(1 1 1) and Ag(1 1 1) films. Force-distance curves monitored at varying relative humidity show clear differences in the adhesion forces on the different samples, which can be explained in terms of a change in the wetting behavior due to the presence of the molecules. Moreover, we found that not only the chemical structure of the molecules but also their orientation strongly influences the strength of the capillary forces. A detailed analysis of the measurements shows that condensation of water vapor on Au(1 1 1) films is drastically enhanced due to the vertically aligned BDMT molecules, while on Ag(1 1 1) water condensation is reduced due to a parallel molecule orientation.     

Ruthenium complexes bearing bidentate Schiff base ligands as efficient catalysts for organic and polymer syntheses

A concise overview is given on mononuclear and dinuclear, bidentate Schiff base ruthenium complexes with different additional ligands and on their applications in various chemical transformations such as Kharasch addition, enol-ester synthesis, alkyne dimerization, olefin metathesis and atom transfer radical polymerization. These new ruthenium complexes, conveniently prepared from commonly available ruthenium compounds, are very stable, exhibit a good tolerance towards organic functionalities, air and moisture and display high activity and chemoselectivity in chemical transformations. Relevant features of coordination chemistry connected with the reaction mechanism and chemoselectivity are also fully described. Since the nature of Schiff bases can be changed in a variety of ways, appealing routes for designing and preparing novel ruthenium complexes can be foreseen in the future.     

Transformées de Burkholder et sommabilité de Martingales à deux paramètres

Sans résumé     

From Brunn–Minkowski to sharp Sobolev inequalities

We present a simple direct proof of the classical Sobolev inequality in with best constant from the geometric Brunn–Minkowski–Lusternik inequality. Research supported in part by NSF Gr. No. 0405587 and B. Zegarlinski’s Pierre de Fermat Grant 2006 from the Région Midi-Pyrénées, France.     

Autoassembly of nanofibrous zeolite crystals via silicon carbide substrate self-transformation

ZSM-5 zeolite nanofibers with a size of 90 nm and lengths up to several micrometers were prepared via in-situ silicon carbide support self-transformation. The morphology and aggregation degree of these zeolite nanofibers could be modified by adjusting the pH conditions, the nature of the mineralizer (OH- or F-), or the synthesis duration. The novelty consists of the preparation of zeolite nanowires without the use of any organogelating agent, along with controlled macroscopic shapes (extrudates, foam monolith) for direct use as a structured reactor. Finally, these materials are catalytically active in the conversion of methanol to gasoline range hydrocarbons (MTG process) and hence exhibit the typical solid acidity of zeolitic materials.     

Dilithiation of Bis(benzene)molybdenum and subsequent isolation of a molybdenum-containing paracyclophane

The homoleptic sandwich complex bis(benzene)molybdenum, [Mo(eta6-C6H6)2], was successfully dilithiated by employing an excess of BuLi in the presence of N,N,N',N'-tetramethylethylenediamine (up to 6 equiv each) at slightly elevated temperatures furnishing the highly reactive, ring metalated species [Mo(eta6-C6H5Li)2].tmeda in high yields. Alternatively, this compound was synthesized upon prolonged sonication with 5 equiv of tBuLi/tmeda without heating. An X-ray crystal structure determination revealed a symmetrical, dimeric composition in the solid state, i.e., a formula of [Mo(eta6-C6H5Li)2]2.(thf)6, where the six-membered rings are connected by two pairs of bridging lithium atoms. The synthesis of an elusive ansa-bridged complex failed in the case of a [1]bora and a [1]sila bridge due to the thermal lability of the resulting compounds. Instead, reverse addition of the dilithio precursor to an excess of the appropriate element dihalide facilitated the isolation of several unstrained, 1,1'-disubstituted derivatives, namely, [Mo{eta6-C6H5(BN(SiMe3)2X)}2] (X = Cl, Br) and [Mo{eta6-C6H5(SiiPr2Cl)}2], respectively. However, the incorporation of a less congesting [2]sila bridge was accomplished. In addition to the formation of [Mo{(eta6-C6H5)2Si2Me4}], a molybdenum-containing paracylophane complex was isolated and characterized by means of crystal structure analysis. The ancillary formation of 1 equiv of bis(benzene)molybdenum strongly suggests that this species is generated by deprotonation of the ansa-bridged complex by the dilithiated precursor and subsequent reaction with a second equivalent of the disilane.     

Stability studies of high-stable water-in-oil model emulsions

Different compositions and emulsification protocols were used to prepare stable water-in-oil (w/o) emulsions. Water, mineral oil, and a mixture of Span 80 and Tween 80 surfactants were combined to form emulsions that can be used as reference for electrolyte-free systems. Here, we have proposed emulsions wherein different properties were evaluated. Electrical conductivity measurements indicated that conductivity increases linearly with increasing surfactant content. The emulsions’ flow curves and viscoelastic behaviors were delineated by rheological measurements. Stability studies by centrifugal testing have shown that smaller the surfactant content, lower the stability, for any used stirring speeds. Furthermore, higher the applied mixing rate to make the emulsion, higher the stability, regardless of the amount of surfactant. Electrical field stability analysis showed, for all systems, that critical electric field (CEF) values were dependent on either surfactant amount and emulsion elastic modulus.     

Chromophores from hexeneuronic acids (HexA): synthesis of model compounds and primary degradation intermediates

Hexeneuronic acid (HexA) is formed under pulping conditions from 4-O-methyl-glucuronic acid residues in xylans by methanol elimination. It is usually removed by an acidic washing treatment (A-stage) within the pulp bleaching sequence. Hexeneuronic acid has long been recognized as a source of color generation in pulps, but the chemical structure of the actual chromophoric compounds remained elusive. We report the synthesis of isotopically (¹³C) labeled HexA model units carrying a label at any of the six carbon atoms. Confirming pertinent literature accounts, it is shown that HexA forms three primary degradation intermediates, 2-furancarboxylic acid, 5-formyl-2-furancarboxylic acid, and formic acid, under mildly acidic conditions, and their formation mechanism is discussed. 2-Furancarboxylic acid is demonstrated to be deformylation product of 5-formyl-2-furancarboxylic acid. The three primary intermediates are colorless and do not represent chromophores themselves. Their mixture, upon thermal or acidic treatment, gives rise to the same chromophores that are also directly formed from HexA.