Classical Singularities In Chaotic Atom-Surface Scattering

In this paper we show that the diffraction condition for the scattering of atoms from surfaces leads to the appearance of a distinct type of classical singularity. Moreover, it is also shown that the onset of classical trapping or classical chaos is closely related to the bifurcation set of the diffraction-order function around the surface points presenting the rainbow effect. As an illustration of this dynamic, application to the scattering of He atoms by the stepped Cu(115) surface is presented using both a hard corrugated one-dimensional wall and a soft corrugated Morse potential

Analysis of a three-component model phase diagram by Catastrophe Theory: Potentials with two Order Parameters

In this work we classify the singularities obtained from the Gibbs potential of a lattice gas model with three components, two order parameters and five control parameters applying the general theorems provided by Catastrophe Theory. In particular, we clearly establish the existence of Landau potentials in two variables or, in other words, corank 2 canonical forms that are associated to the hyperbolic umbilic, D_{+4}, its dual the elliptic umbilic, D_{-4}, and the parabolic umbilic, D_5, catastrophes. The transversality of the potential with two order parameters is explicitely shown for each case. Thus we complete the Catastrophe Theory analysis of the three-component lattice model, initiated in a previous paper.Comment: 17 pages, 3 EPS figures, Latex file, continuation of Phys. Rev. B57, 13527 (1998) (cond-mat/9707015), submitted to Phys. Rev.

Analysis of a three-component model phase diagram by Catastrophe Theory

We analyze the thermodynamical potential of a lattice gas model with three components and five parameters using the methods of Catastrophe Theory. We find the highest singularity, which has codimension five, and establish its transversality. Hence the corresponding seven-degree Landau potential, the canonical form Wigwam or $A_6$, constitutes the adequate starting point to study the overall phase diagram of this model.Comment: 16 pages, Latex file, submitted to Phys. Rev.

Connections on Infinite Dimensional Manifolds with Corners

In this paper we study connections for surjective C p \Gammasubmersions on manifolds with corners, invariant by C p \Gammaactions of Lie groups which are compatible with the equivalence relation defined by the submersion. In this context the principal connections and linear connections are studied as particular cases of these connections. Previously we adapt the vector bundle theory to be used in the paper, to the field of infinite dimensional manifolds with corners. 1991. Mathematics Subject Classification. Primary: 53C05,55R05, 55R10,57N20,57R22,57S25. Secundary: 58E46 Key words and phrases: Connection, Principal connection, Linear connection. 1 Introduction In [5], P. Liberman defined connections for surjective submersions ß : M ! B; as excisions of the exact sequence of vector bundles 0 ! V M ! TM ! ß (TM) ! 0: An analogous definition of connections on smooth vector bundles was given by J. Vilms in [11], and a systematic study of this type of connections on fibre bundles..

Embedding Of Hilbert Manifolds With Smooth Boundary Into Semispaces Of Hilbert Spaces

. In this paper we prove the existence of a closed neat embedding of a Hausdorff paracompact Hilbert manifold with smooth boundary into H \Theta [0; +1), where H is a Hilbert space, such that the normal space in each point of a certain neighbourhood of the boundary is contained in H \Theta f0g. Then, we give a neccesary and sufficient condition that a Hausdorff paracompact topological space could admit a differentiable structure of class 1 with smooth boundary. 0. Introduction A generalization of Whitney's embedding theorem was given by J. Mc Alpin on 1965 [1] and [8]: "Every separable C r --manifold without boundary modeled on a separable Hilbert space can be C r --embedded as a closed submanifold of a separable Hilbert space". On 1970 J. Eells and K.D. Elworthy [4] proved the following immersion theorem: "Let E be a C 1 --smooth Banach space of infinite dimension, with a Shauder base. Suppose that X is a separable metrizable C 1 --manifold without boundary modeled on E. If..

Embedding of Hilbert manifolds with smooth boundary into semispaces of Hilbert spaces

summary:In this paper we prove the existence of a closed neat embedding of a Hausdorff paracompact Hilbert manifold with smooth boundary into $H \times [0, + \infty)$, where $H$ is a Hilbert space, such that the normal space in each point of a certain neighbourhood of the boundary is contained in $H \times \{ 0 \}$. Then, we give a neccesary and sufficient condition that a Hausdorff paracompact topological space could admit a differentiable structure of class $\infty$ with smooth boundary

Differential topology

...there are reasons enough to warrant a coherent treatment of the main body of differential topology in the realm of Banach manifolds, which is at the same time correct and complete. This book fills the gap: whenever possible the manifolds treated are Banach manifolds with corners. Corners add to the complications and the authors have carefully fathomed the validity of all main results at corners. Even in finite dimensions some results at corners are more complete and better thought out here than elsewhere in the literature. The proofs are correct and with all details. I see this book as a reliable monograph of a well-defined subject; the possibility to fall back to it adds to the feeling of security when climbing in the more dangerous realms of infinite dimensional differential geometry. Peter W. Micho

Density of the transversality on manifolds with corners

In this paper we establish a new concept of transversality on manifolds with corners, which is more restrictive than, and not equivalent to, the classical one. Then, we prove for this transversality an analogue of the Thom’s transversality theorem