Kinetic approach to chemical reactions and inelastic transitions in a rarefied gas

A kinetic approach is presented for the analysis of a gas mixture with two kinds of nonconservative interactions. In a bimolecular chemical reaction, mass transfer and energy of chemical link arise, and in inelastic mechanical encounters, molecules get excited or de‐excited due to their quantized structure. Molecules undergo transitions between energy levels also by absorption and emission of photons of the self‐consistent radiation field. From the kinetic Boltzmann‐type equations, the problem of equilibria and of their stability is addressed. A detailed balance principle is proved and a Lyapunov functional is constructed; mass action law and Planck's law of radiation are recovered. This revised version was published online in July 2006 with corrections to the Cover Date.     

Compositions of n satisfying some coprimality conditions

An ?-composition of n is a sequence of length ? of positive integers summing up to n. In this paper, we investigate the number of ?-compositions of n satisfying two natural coprimality conditions. Namely, we first give an exact asymptotic formula for the number of ?-compositions having the first summand coprime to the others. Then, we estimate the number of ?-compositions whose summands are all pairwise coprime.     

Cayley graphs on abelian groups

Let A be an abelian group and let ι be the automorphism of A defined by: ι: a ? a^?1. A Cayley graph Γ = Cay(A,S) is said to have an automorphism group as small as possible if Aut(Γ)=A?<ι>. In this paper, we show that almost all Cayley graphs on abelian groups have automorphism group as small as possible, proving a conjecture of Babai and Godsil.     

Synthesis of peptides containing 1,2,3,4-tetrahydroquinoline-2-carboxylic acid. Part 3. Cyclization of dipeptides and amides with acetic anhydride

Treatment of protected dipeptides containing 1,2,3,4-tetrahydroquinoline-2-carboxylic acid with acetic anhydride affords only 1H,3H,5H-oxazolo[3,4-a]quinolin-3-one derivatives. The formation of a-acylaminomethylketones, arising from the competitive Dakin-West reaction, was generally observed when the cyclization procedure was extended to some amides of the cyclic imino acid. The preferential stabilization of one of two probable mesoionic intermediates seems to determine the preferred pathway.     

Mathematical Structures of Space-Time

At first we introduce the space-time manifold and we compare some aspects of Riemannian and Lorentzian geometry such as the distance function and the relations between topology and curvature. We then define spinor structures in general relativity, and the conditions for their existence are discussed. The causality conditions are studied through an analysis of strong causality, stable causality and global hyperbolicity. In looking at the asymptotic structure of space-time, we focus on the asymptotic symmetry group of Bondi, Metzner and Sachs, and the b-boundary construction of Schmidt. The Hamiltonian structure of space-time is also analyzed, with emphasis on Ashtekar's spinorial variables. Finally, the question of a rigorous theory of singularities in space-times with torsion is addressed, describing in detail recent work by the author. We define geodesics as curves whose tangent vector moves by parallel transport. This is different from what other authors do, because their definition of geodesics only involves the Christofel symbols, though studying theories with torsion. We then prove how to extend Hawking's singularity theorem without causality assumptions to the space-time of the ECSK theory. This is achieved studying the generalized Raychauduri equation in the ECSK theory, the conditions for the existence of conjugate points and properties of maximal timelike geodesics. Our result can also be interpreted as a no-singularity theorem if the torsion tensor does not obey some additional conditions. Namely, it seems that the occurrence of singularities in closed cosmological models based on the ECSK theory is less generic than in general relativity. Our work is to be compared with important previous papers. There are some relevant differences, because we rely on a different definition of geodesics, we keep the field equations of the ECSK theory in their original form rather than casting them in a form similar to general relativity with a modified energy-momentum tensor, and we emphasize the role played by the full extrinsic curvature tensor and by the variation formulae.     

Density profiles of charged test particles diffusing in a slab of finite thickness

Summary In the paper a detailed study of the diffusion of charged test particles in the interior of a slab of finite thickness is carried out upon a suitable functional formulation of the stationary Boltzmann equation. Either the boundary effects and the action of an external conservative force are accounted for. Numerical results for the shape function and the total density of the test particles considered are at last reported for an appropriate specialization of the relevant physical parameters.

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Sommario Nel presente lavoro viene sviluppato-tramite una opportuna formulazione funzionale dell'equazione di Boltzmann nel caso stazionario — uno studio dettagliato della diffusione di particelle cavia cariche all'interno di una lastra piana di spessore finito. Sia gli effetti del contorno che l'azione esercitata da una forza conservativa esterna sono portati in conto. In corrispondenza di una appropriata specifica dei parametri fisici di interesse vengono infine illustrati i risultati numerici ottenuti per la funzione di forma e per la densità totale delle particelle cavia considerate.

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Work performed during the preparation of the thesis for the degree of Dottore in Nuclear Engineering.     

Boundary terms for massless fermionic fields

Local supersymmetry leads to boundary conditions for fermionic fields in one-loop quantum cosmology involving the Euclidean normal _e n _A/A to the boundary and a pair of independent spinor fields ^A and . This paper studies the corresponding classical properties, i.e., the classical boundary-value problem and boundary terms in the variational problem. If is set to zero on a 3-sphere bounding flat Euclidean 4-space, the modes of the massless spin–1/2 field multiplying harmonics having positive eigenvalues for the intrinsic 3-dimensional Dirac operator onS ³ should vanish onS ³. Remarkably, this coincides with the property of the classical boundary-value problem when spectral boundary conditions are imposed onS ³ in the massless case. Moreover, the boundary term in the action functional is proportional to the integral on the boundary of ^A _e n _AA ^A .     

Similarity solutions in space dependent extended kinetic theory

Summary An exact analytical solution to a simple problem of extended kinetic theory for a spatially inhomogeneous binary gas mixture is determined in this paper by resorting to Lie group theoretical methods. It is shown that there are cases in which the nonlinear evolution equations for the densities allow transformation groups of infinite order, where the corresponding similarity solution actually represents the general solution to the problem. Results are then briefly commented on.

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Zusammenfassung In dieser Abhandlung wird eine exakte analytische Lösung für ein einfaches Problem aus der erweiterten kinetischen Theorie räumlich inhomogener binärer Gasmischungen angegeben. Es wird gezeigt, daß die nichtlinearen Evolutionsgleichungen für die Dichten in bestimmten Fällen Lie-Gruppen von unendlicher Ordnung zulassen. Die zugehörigen Similarity-Lösungen stellen dann allgemeine Lösungen des zugrundeliegenden Problems dar. Die Ergebnisse werden kurz erläutert.

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On leave from: Institute of Mathematical Physics, University of Ulm, Oberer Eselsberg, 7900 Ulm, F.R.G.