On the geometrical structure of Euler-Lagrange equations

Summary We show that the Euler-Lagrange operator and the Poincaré-Cartan form arise in a very simple and natural way from the contact structure of the second order jet space, in a purely differential context, without any reference to a variational problem. By the way we obtain an intrinsic expression of the Euler-Lagrange operator.     

Microcanonical and canonical distributions and finite exchangeable random processes

A pure probabilistic approach to Gibbs' distributions is given, starting from the notion of finite exchangeable random process. The differences between bosons, fermions, and classical particles are ascribed to different values of correlation. The relationship between exchangeability and constraints on the energy distribution is investigated.     

The lattice automorphisms of simple algebraic groups over\bar F_2

We prove that if u is a unipotent element of a connected reductive algebraic group G over , there exists an involution σ in G such that σuσ=u⁻¹. We use this result to determine the group of lattice automorphisms of G, when G is simple.     

Removal of organic pollutants from water via molecular inclusion within a cavitand

The molecular inclusion chemistry of cavitands provides a useful way for the removal of organic pollutants from water. A wide range of lipophilic organic compounds, present at trace level in water, are efficiently extracted by cavitand1, which can be easily reactivated and reused.     

Easy hydrolysis of white phosphorus coordinated to ruthenium

The P4 molecule bound to ruthenium as an eta1-ligand in [CpRu(PPh3)2(eta1-P4)]Y (Y = PF6, CF3SO3) undergoes an easy reaction with water in exceedingly mild conditions to yield PH3, which remains coordinated to the [CpRu(PPh3)2] fragment, and oxygenated derivatives.     

On Constrained Nonlinear Hermite Subdivision

We determine shape-preserving regions and we describe a general setting to generate shape-preserving families for the 2-points Hermite subdivision scheme introduced by Merrien (Numer. Algorithms 2:187–200, [1992]). This general construction includes the shape-preserving families presented in Merrien and Sablonníere (Constr. Approx. 19:279–298, [2003]) and Pelosi and Sablonníere (C ¹ GP Hermite Interpolants Generated by a Subdivision Scheme, Prépublication IRMAR 06–23, Rennes, [2006]). New special families are presented as particular examples. Nonstationary and nonuniform versions of such schemes, which produce smoother limits, are discussed.      

Crystal chemistry,magnetic, and electrical properties of the tetragonal plutonium oxide telluride Pu2O2Te

We describe the preparation and some physical properties of Pu₂O₂Te. The plutonium oxide telluride is isostructural with the corresponding rare-earth oxide tellurides which crystallize in the tetragonal system of La₂O₂Te-type. Magnetic susceptibility data from 4 K to room temperature are reported together with resistivity measurements. Pu₂O₂Te is found to be an antiferromagnet below 56 K and a semiconductor with an intrinsic energy gap of 0.65 eV. The magnetic behavior is interpreted in terms of superexchange coupling interactions via nonmetal p orbitals, i.e., in terms of 5f-p overlaps. This conclusion is supported by crystal chemistry considerations by comparison of cell volumes of Pu₂O₂Te and Nd₂O₂Te. In Pu₂O₂Te, the Pu crystal radius is found to be much lower than that of Nd in Nd₂O₂Te, suggesting some 5f electron “delocalization” leading to a crystal radius shrinkage. As for the hexagonal Pu₂O₂X compounds, with X = O, S, Se, the measured gap may be considered as the energy separation between the chalcogen np band and the 6d-7s conduction band, the occupied 5f states lying just below the np band with some 5f-np overlap.     

Lateral motion of SiGe islands driven by surface-mediated alloying

SiGe islands move laterally on a Si(001) substrate during in situ postgrowth annealing. This surprising behavior is revealed by an analysis of the substrate surface morphology after island removal using wet chemical etching. We explain the island motion by asymmetric surface-mediated alloying. Material leaves one side of the island by surface diffusion, and mixes with additional Si from the surrounding surface as it redeposits on the other side. Thus the island moves laterally while becoming larger and more dilute.     

Brillouin statistics vindicated

Brillouin statistics have been recently derived from a general theory of indistinguishable classical particles. We derive Brillouin's formulas by a random process ruling the growth of the physical system. This growth process is concerned with distinguishable and observable events, which have nothing to do with (in)distinguishability of particles.     

Self‐Assembly of Decoupled Borazines on Metal Surfaces: The Role of the Peripheral Groups

Two borazine derivatives have been synthesised to investigate their self‐assembly behaviour on Au(111) and Cu(111) surfaces by scanning tunnelling microscopy (STM) and theoretical simulations. Both borazines form extended 2D networks upon adsorption on both substrates at room temperature. Whereas the more compact triphenyl borazine 1 arranges into close‐packed ordered molecular islands with an extremely low density of defects on both substrates, the tris(phenyl‐4‐phenylethynyl) derivative 2 assembles into porous molecular networks due to its longer lateral substituents. For both species, the steric hindrance between the phenyl and mesityl substituents results in an effective decoupling of the central borazine core from the surface. For borazine 1 , this is enough to weaken the molecule–substrate interaction, so that the assemblies are only driven by attractive van der Waals intermolecular forces. For the longer and more flexible borazine 2 , a stronger molecule–substrate interaction becomes possible through its peripheral substituents on the more reactive copper surface.