MPEJ Volume 3, No.1, 36pp
Received: September 24, 1996, Accepted: March 18, 1997

V.A. Malyshev, F.M. Spieksma
Dynamics in Binary Neural Networks with a Finite Number of Patterns.
Part 1. General Picture of the Asynchronous Zero Temperature Dynamics.

ABSTRACT:  We rigorously define and study the limiting dynamics for
Pastur-Figotin-Hopfield models of neural networks with N nodes and
p patterns in the (thermodynamic) limit N to infinity, p constant.
We study local and global properties of this limiting dynamics.

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MPEJ Volume 3, No.2, 22pp
Received: March 4, 1997, Revised: May 9, 1997, Accepted: May 19, 1997

Walter Craig
Microlocal moments and regularity of solutions of Schroedinger's equation

ABSTRACT: There is a connection between the smoothness of solutions
of Schr\"odinger's equation and the moments of the initial data.
This relationship is microlocal in character, and extends on
asymptotically flat Riemannian manifolds to a connection between
the global scattering behavior of the geodesic flow, the moments 
of initial data properly microlocalized along bicharacteristics,
and the microlocal regularity of the solution. A proof of these 
results involves an interesting class of symbols of pseudodifferential
operators. This article gives an outline of the above results and the 
microlocal analysis of these symbols. It also contains a study of 
the evolution operator for the Schr\"odinger equation on weighted 
Sobolev spaces, and presents a series of results for the 
non-selfadjoint case. This article is an extension of seminar talks 
on the linear Schr\"odinger equation given at the 
Ecole Polytechnique on 9 April 1996 (s\'eminaire `\'equations 
aux d\'eriv\'ees partielles') and at the Universit\"at Bonn 
on 2 May 1996 (`Oberseminar zur Analysis').

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MPEJ Volume 3, No.3, 19pp
Received: May 17, 1997, Revised Jun 17, 1997, Accepted: Jun 27, 1997

Pierre Collet, Jean-Pierre Eckmann
Oscillations of Observables in 1-Dimensional Lattice Systems

ABSTRACT: Using, and extending, striking inequalities by V.V. Ivanov
on the down-crossings of monotone functions and ergodic sums, we give
universal bounds on the probability of finding oscillations of observables
in 1-dimensional lattice gases in infinite volume. In particular, we study
the finite volume average of the  occupation number as one runs through an
increasing sequence of boxes of size $2n$ centered at the origin. We show
that the probability to see $k$ oscillations of this average between two
values $\beta $ and $0<\alpha <\beta $ is bounded by $C R^k$, with $R<1$,
where the constants $C$ and $R$ do {\em not} depend on any detail of the
model, nor on the state one observes, but only on the ratio $\alpha/\beta$.

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MPEJ Volume 3, No.4, 40pp
Received: July 28, 1997, Revised: Sep 10, 1997, Accepted: Sep 30, 1997

Amadeu Delshams, Tere M. Seara
Splitting of separatrices in Hamiltonian systems
with one and a half degrees of freedom

ABSTRACT:  The splitting of separatrices for Hamiltonians
with $1{1\over 2}$ degrees of freedom
$$h(x,t/\varepsilon)=h^0(x)+\mu\varepsilon^p h^1(x,t/\varepsilon)$$
is measured. We assume that $h^0(x)=h^0(x_1,x_2)=x_2^2/2+V(x_1)$
has a separatrix $x^0(t)$, $h^1(x,\theta)$ is $2\pi$-periodic in $\theta$,
$\mu$ and $\varepsilon>0$ are independent small parameters, and $p\ge 0$.
Under suitable conditions of meromorphicity for $x_2^0(u)$ and the
perturbation $h^1(x^0(u),\theta)$, the order $\ell$ of the perturbation
on the separatrix is introduced, and it is proved that, for $p\ge\ell$,
the splitting is exponentially small in $\varepsilon$, and is given
in first order by the Melnikov function.

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MPEJ Volume 3, No.5, 25pp
Received: Mar 18, 1997, Revised: Jul 28, 1997, Accepted: Oct 1, 1997

Antonio Giorgilli, Ugo Locatelli
On classical series expansions for quasi-periodic motions

ABSTRACT:  We reconsider the problem of convergence of classical 
expansions in a parameter $\epsilon$ for quasiperiodic motions on
invariant tori in nearly integrable Hamiltonian systems. Using a
reformulation of the algorithm proposed by Kolmogorov, we show that if
the frequencies satisfy the nonresonance condition proposed by Bruno,
then one can construct a normal form such that the coefficient of
$\epsilon^s$ is a sum of $O(C^s)$ terms each of which is bounded by
$O(C^s)$. This allows us to produce a direct proof of the classical
$\epsilon$ expansions.  We also discuss some relations between our
expansions and the Lindstedt's ones.