Hausdorff measures and dimension on

We consider the Hausdorff measures , , defined on with the topology induced by the metric

for all . We study its properties, their relation to the ``Lebesgue measure" defined on by R. Baker in 1991, and the associated Hausdorff dimension. Finally, we give some examples.

     

Some Properties of the Bel and Bel-Robinson Tensors

The properties of the Bel and Bel-Robinson tensors seem to indicate that they are closely related to the gravitational energy-momentum. We present some new properties of these tensors which might throw some light onto this relationship. First, for any spacetime we find a decomposition of the Bel tensor in terms of the Bel-Robinson tensor and two other tensors, which we call the pure matter super-energy tensor and the matter-gravity coupling super-energy tensor. We show that the pure matter super-energy tensor of any Einstein-Maxwell field is simply the square of the usual energy-momentum tensor. This, together with the fact that the Bel-Robinson tensor has dimensions of energy density square, leads us to the definition of square root for the Bel-Robinson tensor: a two-covariant symmetric traceless tensor with dimensions of energy density and such that its square gives the Bel-Robinson tensor. We prove that this square root exists if and only if the spacetime is of Petrov type O, N or D, and its general expression is explicitly presented. The properties of this new tensor are examined and some interesting explicit examples are analyzed. Of particular interest are an invariant function that appears in the spherically symmetric metrics and an expression for the energy carried out by pure plane gravitational waves. We also examine the decomposition of the whole Bel tensor for Vaidya's radiating metric and Kerr-Newman's solution. Finally, we generalize the definition of square root to a factorization of the Bel-Robinson tensor and get the general solution for all Petrov types.     

A molecular mechanics study of conformational trends in simple alcohols and ethers. Part I: Geometric trends

We have modified the MM2(80) molecular mechanics force field for alcohols and ethers to reproduce geometric trends predicted by ab initio calculations with the 4-21G basis set. The results obtained with the modified force field are consistent with experimental evidence.     

Aliphatic organolithiums by fluorine-lithium exchange: n-octyllithium

The reaction of 1-fluorooctane (1) with an excess of lithium powder (4-10 equiv.) and DTBB (2-4 equiv.) in THP at 0°C for 5 min gives a solution of the corresponding 1-octyllithium (2), which reacts then with different electrophiles at 0°C (D₂O, MeSiCl, Bu^tCHO, Et₂CO), or −78°C [ClCO₂Me, (PhCH₂S)₂] or −40°C (CO₂) to room temperature to give, after hydrolysis, the expected products (3). The same process applied to 2-fluorooctane gives mainly octane as reaction product, independently on the electrophile used, resulting from a proton abstraction by 2-lithiooctane formed from the reaction medium before addition of the electrophilic reagent.     

Photonucleophilic aromatic substitution of 6-fluoroquinolones in basic media: triplet quenching by hydroxide anion

Photoreaction of 1-ethyl-6-fluoro-7-(1-piperazinyl)-1,4-dihydro-4-oxoquinoline-3-carboxylic acid (norfloxacin, NFX) and other 6-fluoroquinolones in aqueous solution gives rise to the corresponding 6-hydroxy derivatives. Although two mechanisms have been proposed for this photonucleophilic aromatic substitution, direct evidence for any of them is still missing. Obtaining such evidence requires work in basic media, where intramolecular electron transfer from the piperazine ring to the quinolone system is the almost exclusive singlet deactivation pathway. To overcome this problem, the 4'-N-acetyl derivative of norfloxacin (ANFX) has been employed in the present paper due to the lower availability of the N lone pair. The photochemical and photophysical properties of ANFX have been studied in aqueous solutions at pH between 7.4 and 13. As expected, fluorescence of ANFX is not significantly quenched in basic media. Furthermore, the excited triplet state (lambda(max) = 620 nm) reacts with hydroxide anions with a rate constant of (0.3 +/- 0.1) x 10(6) M(-)(1) s(-)(1). This supports a direct attack by hydroxide anions to the excited triplet state with subsequent release of fluoride as the operating mechanism. The fact that the reaction is inhibited by the presence of naproxen (a water-soluble naphthalene derivative) as triplet quencher clearly confirms the mechanistic assignment.     

Surface selectivities in wet chemically modified PVC films. Influence of reaction conditions

Wet chemical modification reactions of PVC films in solvent/nonsolvent mixtures were performed. Three thiol compounds of different size and reactivity were used varying the solvent quality, temperature, and reaction times. The evolution of the concentration gradients within the films was studied using confocal Raman spectroscopy and attenuated total reflectance Fourier transform infrared spectroscopy. Surface selectivities were calculated and discussed in terms of the different relative reaction rate constants.     

An unprecedented hybrid scorpionate/cyclopentadienyl ligand

An efficient method for the preparation of the first hybrid scorpionate/cyclopentadienyllithium compound as a new class of tridentate ligand is described. This compound is an excellent reagent for the introduction of this ligand into transition metal complexes.     

Automatic screening method for the rapid and simple discrimination between synthetic and natural colorants in foods

A simple screening method was developed to discriminate between synthetic and natural colorants present in foods in order to reduce the use of expensive instruments such as a liquid chromatograph with diode array detection. A rapid flow system was proposed in which samples containing natural and synthetic colorants in an acetic acid medium were passed through a wool/cotton column, where only synthetic colorants were retained and were then eluted with dilute ammonia. Yellow, red and green-blue-brown additives can be monitored at 400, 530 and 610 nm, respectively, with a conventional spectrophotometer. Complete discrimination (no false positives) between natural and synthetic colorants can be obtained for molar concentrations of natural colorants (in the absence of synthetic ones) up to 2000 (yellow), 2000 (red) and 10 000 (brown) times that of the detection limit (DL) of synthetic additives. The reliability of the method was established at five concentrations (between 0.5 and 3 DL) of the synthetic colorants tartrazine (yellow), erythrosin B (red) and brilliant black BN (brown). For a cut-off concentration of 2 DL, the percentage of false negatives ranges from 8 to 12%. Finally, the method was applied to screening several fruit drinks and candies for the determination of synthetic colorants with a sampling frequency of 10 h⁻¹.     

Synthesis and diels-alder reactions of (S)-3-chloro and (S)-3-ethyl-2-p-tolylsulfinyl-1,4-benzoquinones

The synthesis of the title compounds in enantiomerically pure form is described. Their cycloadditions with cyclopentadiene and 2,3-dimethyl-1,3-butadiene in the presence of ZnBr₂ take place on the unsubstituted dienophilic double bond C₅-C₆ giving access to optically active 4a,5,8,8a-tetrahydronaphthoquinones with a diastereoisomeric excess ranging from 40 to 72%.