On the geometries of the rational unfoldings of X k

With a view to applications to self-regulating dynamical processes in biology, we determine the geometric structure of what we call isotangent curves, i.e. curves parametrized by the slopes of their points. They come up naturally as bifurcation curves of rational unfoldings of X ^k, and we classify them according to degree and number of cusps. They, as well as their isotangent involute curves, turn up in simulations of these processes.     

Pair formation in age-structured populations

The modelling problems associated with pair formation in age-structured populations are discussed in this paper. The author also formulates a class of models for pair formation in human populations and derives some results on persistent distributions.     

Some inequalities for partial orders

We establish some inequalities connecting natural parameters of a partial order P. For example, if every interval [a,b] contains at most maximal chains, if some antichain has cardinality v, and if there are ₁ chains whose union is cofinal and coinitial in P, then the chain decomposition number for P is ₁v (Theorem 2.2), and the inequality is sharp in a certain sense (Section 3).This paper was written while the authors were visitors at the Laboratoire d'algèbre ordinale, Département de Mathématiques, Université Claude Bernard, Lyon 1, France.Research supported by NSERC grant # A5198.     

Why is space three-dimensional and how may groups be seen?

First we propose a model of visual perception essentially based on the Keldysh-Chernavsky-Sossinsky three-channel theorem, from which three-dimensionality of space follows. Second, we associate with a system of subgroups H ₁, ..., H_s of a given group G a geometric object, called a group crystal, in order to visualize G. How this notion works is illustrated via the Burnside problem.     

A novel synthetic route to substituted pyranoanthocyanins with unique colour properties

Anthocyanins, isolated from natural sources by countercurrent chromatography, were reacted with cinnamic acids bearing at least one electron-donating substituent at the para-position. The resulting pyranoanthocyanins obtained by this simple one-step reaction were much less susceptible to pH shifts and retained their original colour over a wide pH-range. Through reaction with p-dimethylamino cinnamic acid, synthetic malvidin- and cyanidin-based anthocyanins with a unique violet hue were prepared.     

The use of fluorescence in determining formation constants of complexes : Part II. Complexes which fluoresce

Four different methods for the calculation of the formation constants of fluorescing complexes are discussed. The methods were used to determine constants for the complexes of aluminum(III), gallium(III) and indium(III) with 8-quinolinol-5-sulfonate anion.     

Molecular structure and pseudopolymorphism of squamtin A from Annona squamosa

The cyclopeptide, squamtin A (1, formula: C₃₉H₆₀O₁₁N₈S), was found to crystallized in two pseudopolymorphisms, i.e. 1·(H₂O)_3.5 and 1·(H₂O)_3.9. The composition of the amino acids and their linkage sequences are the same. The main differences between the two kinds of crystals lie in the positions and occupancies of the water molecules, the positions of the sulfur atoms and the conformation of the side chains. The absolute configuration of 1 is established by X-ray analysis in combination with the Marfey's analysis of its hydrolysates.     

Determination of the valence distribution of vanadium in polyolefin catalysts prepared from vanadium trichloride

An electrochemical method has been developed for determining the relative amounts of V(II) and V(III) in polyolefin catalysts prepared using aluminum alkyls and vanadium trichloride. The catalyst, usually obtained as a slurry in n-heptane, is titrated potentiometrically with ceric sulfate using a platinum indicator electrode. Under proper experimental conditions, 2 breaks are obtained, corresponding to the reactions V(II) →V(III) + e and V(III) →V(IV) + e. Interferences from other components of the catalyst are negligible. The method is useful for the analysis of both laboratory catalysts and grab samples taken from pilot plant reactors.     

A facile three-step synthesis of (±)-crispine A via an acyliminium ion cyclisation

A high yielding cyclisation of the readily available N-(4,4-diethoxybutyl)-2-(3,4-dimethoxyphenyl)acetamide to 8,9-bis(methyloxy)-2,3,6,10b-tetrahydropyrrolo[2,1-a]isoquinolin-5(1H)-one is described. The latter can be reduced with either AlH₃ or BH₃ to (±)-crispine A in an overall yield of 55%.     

Vibrational and high pressure conformational studies of 1,1,2-trichloroethane

The infrared spectra of 1,1,2-trichloroethane were recorded from 4000 to 50 cm^?1 in the vapour and liquid states. Additional spectra above 200 cm^?1 of the low temperature crystal and of two crystalline solids, prepared by compressing the sample in a diamond anvil cell, were obtained. Also, infrared spectra of the liquid and of the compound dissolved in CS₂ were recorded at increased pressures. Raman spectral data of the liquid (including polarization measurements) and of the low temperature crystal were obtained.The fundamental frequencies for each of the two conformers, C₁ and C_s, were assigned and the results checked by normal coordinate analysis. The same diagonal and off-diagonal force constants were employed for the two conformers and the force fields were derived by means of a least squares refinement, including data for chloroethane, 1,1-dichloroethane and various deuterated species. A standard deviation of 3 % was obtained in the final fit.From the changes in relative intensities for infrared bands belonging to the C₁ and C_s conformers with increasing pressure, the volume differences (ΔV¯between the conformers were determined in CS₂ solution and in the pure liquid.