The <Emphasis Type="Italic">n</Emphasis>-Body Problem in Spaces of Constant Curvature. Part I: Relative Equilibria

We extend the Newtonian n-body problem of celestial mechanics to spaces of curvature κ=constant and provide a unified framework for studying the motion. In the 2-dimensional case, we prove the existence of several classes of relative equilibria, including the Lagrangian and Eulerian solutions for any κ≠0 and the hyperbolic rotations for κ<0. These results lead to a new way of understanding the geometry of the physical space. In the end we prove Saari’s conjecture when the bodies are on a geodesic that rotates elliptically or hyperbolically.     

Formamide and the origin of life

The complexity of life boils down to the definition: “self-sustained chemical system capable of undergoing Darwinian evolution” (Joyce, 1994) [1]. The term “self-sustained” implies a set of chemical reactions capable of harnessing energy from the environment, using it to carry out programmed anabolic and catabolic functions. We briefly present our opinion on the general validity of this definition.Running anabolic and catabolic functions entails complex chemical information whose stability, reproducibility and evolution constitute the core of what is dubbed genetics.Life as-we-know-it is made of the intimate interaction of metabolism and genetics, both built around the chemistry of the most common elements of the Universe (hydrogen, oxygen, nitrogen, carbon). Other elements like phosphorus and sulphur play important but ancillary and potentially replaceable roles.The reproducible interaction of metabolic and genetic cycles results in the hypercycles of organization and de-organization of chemical information that we consider living entities. In order to approach the problem of the origin of life it is therefore reasonable to start from the assumption that both metabolism and genetics had a common origin, shared a common chemical frame, were embedded in physical–chemical conditions favourable for the onset of both.The most abundant three-atoms organic compound in interstellar environment is hydrogen cyanide HCN, the most abundant three-atoms inorganic compound is water H₂O. The combination of the two results in the formation of formamide H₂NCOH. We have explored the chemistry of formamide in conditions compatible with the synthesis and the stability of compounds of potential pre-genetic and pre-metabolic interest. We discuss evidence showing (i) that all the compounds necessary for the build-up of nucleic acids are easily obtained abiotically, (ii) that essentially all the steps leading to the spontaneous generation of RNA are abiotically possible, (iii) that the key compounds of extant metabolic cycles are obtained in the same chemical frame, often in the same test tube.How close are these observations to a plausible scenario for the origin of life?     

The effect of twin‐tailed sidearms on sodium cation transport in synthetic hydraphile cation channels

The preparation of novel tris(macrocycle)s that transport Na⁺ through phospholipid bilayers is reported. All of the reported structures have the following structural elements: sidechain‐macrocycle‐spacer‐macro‐cycle‐spacer‐macrocycle‐sidechain. The overall extension of the channel system is defined by the lengths of the spacer chains. The flexible sidearms are thought to be aligned with the lipid axis and opposite to the spacer chains. To the extent the spacer chains contribute to organization of the structure and isolation of the cation‐containing pore from the surrounding lipid chains, more hydrophobic chains are expected to enhance cation transport. A comparison of C₁₂H₂₅<N18N>(CH₂)₁₂<N18N>(CH₂)₁₂<N18N> C₁₂H₂₅, 1, with (C₁₂H₂₅)₂N(CH₂)₂<N18N>(CH₂)₁₂<N18N>(CH₂)₁₂<N18N>(CH₂)2N(C₁₂H₂₅)₂, 11, shows that the Na⁺ transport is, indeed, enhanced.     

Numerical approximation of non-homogeneous, non-convex vector variational problems

Summary. In this paper we study a numerical scheme for non-convex vector variational problems allowing for microstructure, based on the approximation of gradient Young measures. We present a convergence result and some numerical experiments. Received March 26, 2000 / Revised version received November 13, 2000 / Published online March 20, 2001     

On discrete frames associated with semidirect products

For groups which are the semidirect product of some vector group with a unimodular group we prove that the existence of a discrete frame obtained from an at-most countable set of vectors through the action of a given unitary representation implies that the representation in use has to be square-integrable.     

β‐Pyridinium di­chloro­iodide

The β modification of pyridinium di­chloro­iodide, C₅H₆N⁺·Cl₂I^?, was obtained as yellow crystals by the reaction of (C₅NH₅)AuCl₃, C₅H₆N⁺·Cl^? and I₂ in a vacuum‐sealed ampoule. The di­chloro­iodide ion is nearly symmetric and linear with I—Cl bond lengths of 2.544 (3) and 2.550 (3) Å and a Cl—I—Cl angle of 179.68 (12)°.     

Aspects of xylitol formation in sugarcane bagasse hydrolysate by Candida guillie rmondii in the presence of tetracycline

The biocon version of xylose intoxylitol using pH values of 4.0, 5.5 and 7.0 and tetracycline concentrations of 20 and 40 mg/L was carried out to verify the influence of these parameters on Candida guilliermondii metabolism for xylitol production. Experiments were performed with sugarcane bagasse hemicellulosi chydrolysate (48.5 g/L of xylose) in 125-mL Erlenmeyer flasks, at 30°C, 200 rpm, during 88 h. The results demostrated that the bioconversion of xylose into xylitol was significantly influenced by the pH. On the other hand, in media containing 20 or 40 mg/L of tetracycline, this bioconversion was not significantly affected. The best results of xylitol production were obtained in hemicellulosic hydrolysate without tetracycline, at pH 7.0 In these conditions, the maxim um specific growth rate was 0.014/h and the yield factor of xylitol and volumetric productivity were 0.85g/g and 0.70g/L/h respectively. Xylitol and cell growth occureed simultaneously.     

Molecular modelling of ring‐opening polymerisation of lactamimide‐containing macrocyclic esters

The molecular modelling of anionic (MeO^– initiated) and cationic (H⁺ initiated) polymerisation of lactamimide‐containing macrocyclic esters has been carried out at PM3 and HF/3‐21G levels of theory. It was found that it is strain release that drives the polymerisation of these macrocycles. Most likely that entropy gain on the ring‐opening can not compensate entropy loss on polymer formation due to the relative rigidity of cyclic molecules. Methanolate initiated anionic polymerisation will not lead to high molecular weight product due to the side reactions involving the attack of growing anionic species at carbonyls of lactamimide moiety, while the cationic polymerisation should proceed smoothly provided that polymerisation temperature is less than ceiling temperature.