Lines of curvature near umbilical points on surfaces immersed in R4

LetM be a compact orientable surface,I:M R⁴ an immersion of classC ^r and a normal unitary smooth vector field. We prove that in the space of pairs (I, ) the family of pairs all whose umbilical points are Darbouxian is generic.     

Chaos in classical cosmology (II)

We present a new dynamical calculation about the Friedman-Robertson-Walker universe considered as an autonomous Hamiltonian. The time evolution of this Hamiltonian presents numerical instabilities so we apply a symplectic integration via infinitesimal canonical transformations of the phase space time evolution that preserves the Poincaré invariant. In this way, we have also obtained a sensitive improvement in the accuracy of the Hamiltonian constraint, as well as in the computing time. We confirm our previous results; in a spatially closed universe, the route to chaos is reached by sucessive breakage of the resonant tori due to the action of 11 resonances.     

Spectrophotometric determination of zinc in copper-base alloys with TAN

TAN reacts with zinc(II) forming a red complex with composition 1:2 Zn(II)-TAN and absorption maximum at 582 nm. Zinc can be determined with this reagent in the presence of Triton X-100, in the pH range 6.20-8.00 with a molar absorptivity of 4.5×10⁴ l/mol/cm Beer's Law was obeyed up to least 1.55 g/ml. Copper interference was eliminated with a mixture of thiosulfate and ascorbic acid and nickel separated by precipitation with dimethylglyoxime. The proposed method was used for zinc determination in several copper-base alloys and the results of analysis in comparison with certified values indicated that the procedure was accurate and precise. A derivative procedure is also proposed, allowing zinc determination with high sensitivity (5-400 ng/ml).     

Kinetics and mechanism of the anation of (αβ)-hydroxo(tetraethylenepentamine)cobalt(III) by thiosulfate, the base hydrolysis of the (sulfur-bonded) thiosulfato(tetraethylenepentamine)-cobalt(III) and photochemistry of oxygen and sulfur-bonded thiosulfato complexes

Summary The reaction of ()-(tetren)CoOH²⁺ with S₂O ₃ ^2- in the 7.25–8.28 pH range at 20–40 °C yielded S- (yellow) and O- (purple) bonded thiosulfato(tetren)cobalt(III) complexes, the former in larger quantities. The rate determining step is preceded by diffusion-controlled ion-pair [(tetren)CoOH²⁺,S₂O ₃ ^2- ] formation. Replacement of coordinated OH^- by S₂O ₃ ^2- is interpreted in terms of an internal conjugate base mechanism: (tetren)CoOH²⁺ (tetren-H)CoOH ₂ ²⁺ , the reactive amido conjugate base being generated by intramolecular proton transfer from the coordinated NH site.In acid medium the S-bonded (tetren)Co(S₂O₃)⁺ is highly stable to redox decomposition, in contrast to its pentaammine analogue. The complex however, undergoes base hydrolysis yielding the corresponding hydroxo complex. The rate and activation parameters for the base hydrolysis have been reported. Photolysis of O- and S-bonded isomers of [(tetren)CoS₂O₃]⁺ in acidic medium at 254 and 313 nm, respectively, yielded aquation products accompanied by some decomposition of S₂O ₃ ^2- .     

On the cube problem of Las Vergnas

We prove a conjecture of Las Vergnas in dimensions d7: The matroid of the d-dimensional cube C ^d has a unique reorientation class. This extends a result of Las Vergnas, Roudneff and Salaün in dimension 4. Moreover, we determine the automorphism group G ^d of the matroid of the d-cube C ^d for arbitrary dimension d, and we discuss its relation to the Coxeter group of C ^d . We introduce matroid facets of the matroid of the d-cube in order to evaluate the order of G ^d . These matroid facets turn out to be arbitrary pairs of parallel subfacets of the cube. We show that the Euclidean automorphism group W ^d is a proper subgroup of the group G ^d of all matroid symmetries of the d-cube by describing genuine matroid symmetries for each Euclidean facet. A main theorem asserts that any one of these matroid symmetries together with the Euclidean Coxeter symmetries generate the full automorphism group G ^d . For the proof of Las Vergnas' conjecture we use essentially these symmetry results together with the fact that the reorientation class of an oriented matroid is determined by the labeled lower rank contractions of the oriented matroid. We also describe the Folkman-Lawrence representation of the vertex figure of the d-cube and a contraction of it. Finally, we apply our method of proof to show a result of Las Vergnas, Roudneff, and Salaün that the matroid of the 24-cell has a unique reorientation class, too.     

On the Claisen Rearrangement of Allyl Ethyl Ketene Acetals Generated in situ via Benzeneselenenic Acid Eliminiation

Mixed acetals 7 of benzeneseleninylacetaldehyde, prepared by a simple 2-step procedure from mono- and bicyclic allylic alcohols 5 , undergo benzeneselenenic acid elimination to transient ketene acetals 8 which afford γ, δ-unsaturated esters 9 via the ester Claisen rearrangement (Scheme 2). Under the same conditions selenoxide 7h derived from benzyl alcohol 5h is converted back to benzyl alcohol with the concomitant formation of ethylphenylselenoacetate 12 .