MPEJ Volume 2, No.1, 43pp Received: June 14, 1995, Revised: January 15, 1996, Accepted: January 26, 1996 Federico Bonetto, Vieri Mastropietro Filled band Fermi systems ABSTRACT: Extending the result in (B.M.) on one dimensional interacting fermions in a periodic potential we study the infrared behaviour of the two points Schwinger function in the filled band case. If the fermions are spinless such behaviour is completely determined and it depends on the ratio between the amplitude of the gap and the strength of the interaction. If the ratio is large the Schwinger function behaviour is similar to the one in the free non interacting case while if it is small the Schwinger function is deeply modified and it depends on two anomaly indices, in term of which the occupation number discontinuity and the spectral gap are expressed. A heuristic second order analysis of the spinning case is also performed. ------------------------------ MPEJ Volume 2, No.2, 16pp Received: August 29, 1995, Revised: March 19, 1996, Accepted: March 20, 1996 Roy R. Douglas A Canonical Construction Yielding a Global View of Twistor Theory ABSTRACT: The construction investigated in this paper begins with an ordered, finite set of closed subgroups of some compact Lie group; from this data, the construction produces a topological space. Using a combination of fibration and cofibration techniques, it is possible to describe both the global and the local topological structure for this space. The construction yields novel, canonical decompositions of some compact manifolds (including certain spheres), as well as other interesting spaces with more exotic local topological structure. With this approach, the correspondences of twistor theory can be seen in their global geometric context, as a 1-parameter family of such correspondences, which canonically fit together to form $S^{14}$, a (constant radius) 14-dimensional sphere in a 15-dimensional Euclidean space. ------------------------------ MPEJ Volume 2, No.3, 9pp Received: April 22, 1996, Accepted: May 23, 1996 G. Benfatto Renormalization group approach to zero temperature Bose condensation ABSTRACT: We study the problem of Bose condensation at zero temperature and weak coupling for a three dimensional system of bosons, interacting with a repulsive short range potential, in the Bogoliubov approximation. We prove that the properties of the model can be explained in terms of an anomalous asymptotically free renormalization group flow and we show that the two-point correlation function has the typical superfluid behaviour at long wavelengths, as generally expected. The proof is, for the moment, only at the level of perturbation theory in the running coupling constants. We also obtain an expression for the sound speed, whose leading term (when the coupling goes to zero) coincides with the sound speed in the exactly soluble Bogoliubov model. ------------------------------ MPEJ Volume 2, No.4, 33pp Received: June 19, 1996, Accepted: June 21, 1996 L. H. Eliasson Absolutely Convergent Series Expansions for Quasi Periodic Motions ABSTRACT: CONTENTS. I. INTRODUCTION. * The Hamiltonian problem II. THE CLASSICAL APPROACH TO THE HAMILTONIAN PROBLEM - ABSOLUTELY DIVERGENT SERIES. * The formal solution and its series expansion * "Killing the constants" and the Lindstedt series * Index sets * Description of the coefficients * Convergence and divergence of the formal solution III. SIEGEL'S METHOD. * Siegel's first lemma * Siegel's second lemma * Siegel's third lemma IV. GENERALIZATION OF SIEGEL'S FIRST LEMMA. * Resonances on linear index sets * An equivalence relation on $ad(\gamma)$ * Generalization of Siegel's first lemma - proposition 4 * The basic compensations * Proof of proposition 4 V. GENERALIZATION OF SIEGEL'S THIRD LEMMA. * Resonances on index sets * Generalization of Siegel's third lemma - proposition 5 VI. ABSOLUTELY CONVERGENT SERIES. * Interpretation of the series * "Killing the constants" * Theorem * Other examples ------------------------------ MPEJ Volume 2, No.5, 8pp Received: Jul 17, 1996, Revised: Oct 6, 1966, Accepted: Oct 13, 1996 Andrzej Sitarz, Piotr Zgliczynski On a Matched Pair of Lie Groups for the $\kappa$-Poincar\'e in 2-Dimensions ABSTRACT: We present the construction of the $\kappa$-Poincar\'e in two dimensions from matched pairs of Lie groups: $SO(1,1)$ and a $\kappa$-deformed group of translations.