MPEJ Volume 6, No.1, 18 pp.
Received June 21 1999, Revised Jan 9 2000, Accepted Jan 9 2000

M. Salmhofer, Chr. Wieczerkowski
Construction of the renormalized $GN_{2-\epsilon}$ trajectory


ABSTRACT:
We construct the renormalized Gross-Neveu trajectory in 
$2-\epsilon$ dimensions. Our construction uses a contraction mapping
for an extended renormalization group. The extension is a running 
coupling constant with linear step $\beta$ function. The contraction
mapping relies on norm estimates for a fermionic momentum space
renormalization group.
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MPEJ Volume 6, No.2, 18 pp.
Received Nov 19 1999, Revised Feb 10  2000, Accepted Feb 15 2000

E. Valdinoci
Families of whiskered tori for a-priori stable/unstable
Hamiltonian systems and construction of unstable orbits


ABSTRACT:
We give a detailed statement of a KAM theorem about the conservation 
of partially hyperbolic tori on a fixed energy level for an analytic 
Hamiltonian $H(I,\f,p,q)=h(I,pq;\m)+\m f(I,\f,p,q;\m)$,
where $\f$ is a $({d}-1)-$dimensional angle, $I$ is in a domain of
$\RR^{{d}-1}$, $p$ and $q$ are real in a neighborhood $0$, and $\m$ 
is a small parameter.

We show that invariant whiskered tori covering a large measure exist for  
sufficiently small perturbations. The associated stable and unstable manifolds
also cover a large measure.  Moreover, we show that there is a geometric 
organization to these tori.  Roughly, the  whiskered tori we construct 
are organized in smooth families, indexed by a Cantor parameter.
The whole set of tori as well as their stable and unstable manifolds
is smoothly interpolated.

In particular, we emphasize the following items: 
sharp estimates on the relative measure of
the surviving tori on the energy level,
analyticity properties, including dependence upon
parameters, geometric structures.

We  apply these results to both ``a-priori unstable'' and ``a-priori stable''
systems.  We also show how to use the information obtained in the KAM Theorem 
we prove to construct unstable orbits. 
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MPEJ Volume 6, No.3, 67 pp.
Received: Oct 28  1999, Revised: Mar 21 2000, Accepted: Mar 22 2000

A. Schenkel, J. Wehr, P. Wittwer:
Computer-assisted proofs for fixed point problems in Sobolev spaces


ABSTRACT:
In this paper we extend the technique of computer-assisted proofs
to fixed point problems in Sobolev spaces. Up to now, the method was
limited to spaces of analytic functions. The possibility to work with
Sobolev spaces is an important progress and opens up many new domains
of applications. Our discussion is centered around a concrete problem
that arises in the theory of critical phenomena and describes the phase
transition in a hierarchical system of random resistors.  For this problem
we have implemented in particular the convolution product based on the
fast Fourier transform (FFT) algorithm with rigorous error estimates.
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MPEJ Volume 6, No.4, 14 pp.
Received June 17 2000, Revised Jul 17 2000, Accepted Aug 14 2000

K. Borchsenius
Degenerate space-time paths and the non-locality of
quantum mechanics in a Clifford substructure of space-time


ABSTRACT:
The quantized canonical space-time coordinates of a relativistic point
particle are expressed in terms of the elements of a complex Clifford
algebra which combines the complex properties of $SL(2.C)$ and quantum
mechanics. When the quantum measurement principle is adapted to the
generating space of the Clifford algebra we find that the transition
probabilities for twofold degenerate paths in space-time equal the
transition amplitudes for the underlying paths in Clifford space. This
property is used to show that the apparent non-locality of quantum
mechanics in a double slit experiment and in an EPR type of
measurement is resolved when analyzed in terms of the full paths in
the underlying Clifford space. We comment on the relationship of this
model to the time symmetric formulation of quantum mechanics and to
the Wheeler-Feynman model.

.