Vector potential

Abstract: A theory of the long-wavelength low-energy electronic structure of graphite-derived nanotubules is presented. The propagating $\ensuremath{\pi}$ electrons are described by wrapping a massless two dimensional Dirac Hamiltonian onto a curved surface. The effects of the tubule size, shape, and symmetry are included through an effective vector potential which we derive for this model. The rich gap structure for all straight single wall cylindrical tubes is obtained analytically in this theory, and the effects of inhomogeneous shape deformations on nominally metallic armchair tubes are analyzed.

Abstract: On presente dans cet article un certain nombre de resultats concernant le potentiel vecteur associe a une fonction a divergence nulle dans un ouvert borne de dimension trois. En particulier, plusieurs types de conditions aux limites sont proposes, pour lesquels on discute l'existence, l'unicite et la regularite du potentiel vecteur. On analyse la convergence d'une discretisation par elements finis de ces potentiels et on indique une application concernant l'approximation de fluides visqueux incompressibles.

Abstract: Let D be an arbitrary set of Cc vector fields on the Cc manifold M. It is shown that the orbits of D are C' submanifolds of M, and that, moreover, they are the maximal integral submanifolds of a certain C9? distribution PD. (In general, the dimension of PD(m) will not be the same for all m EM.) The second main result gives necessary and sufficient conditions for a distribution to be integrable. These two results imply as easy corollaries the theorem of Chow about the points attainable by broken integral curves of a family of vector fields, and all the known results about integrability of distributions (i.e. the classical theorem of Frobenius for the case of constant dimension and the more recent work of Hermann, Nagano, Lobry and Matsuda). Hermann and Lobry studied orbits in connection with their work on the accessibility problem in control theory. Their method was to apply Chow's theorem to the maximal integral submanifolds of the smallest distribution A such that every vector field X in the Lie algebra generated by D belongs to A (i.e. X(m) e A(m) for every m EM). Their work therefore requires the additional assumption that A be integrable. Here the opposite approach is taken. The orbits are studied directly, and the integrability of A is not assumed in proving the first main result. It turns out that A is integrable if and only if A = PD' and this fact makes it possible to derive a characterization of integrability and Chow's theorem. Therefore, the approach presented here generalizes and unifies the work of the authors quoted above.