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#### Dynamics and entropy in the Zhang model of Self-Organized Criticality

(Working paper; Arbeidsnotat, 2005-09-12)

We give a detailed study of dynamical properties of the Zhang model,
including evaluation of topological entropy and estimates for the Lyapunov
exponents and the dimension of the attractor. In the thermodynamic
limit the entropy goes to zero and the Lyapunov spectrum collapses.1

#### Entropy via multiplicity

(Working paper; Arbeidsnotat, 2005-09-30)

The topological entropy of piecewise affine maps is studied. It is shown
that singularities may contribute to the entropy only if there is angular expansion
and we bound the entropy via the expansion rates of the map. As
a corollary we deduce that non-expanding conformal piecewise affine maps
have zero topological entropy. We estimate the entropy of piecewise affine
skew-products. Examples of ...

#### A piece-wise affine contracting map with positive entropy

(Working paper; Arbeidsnotat, 2005-04-10)

We construct the simplest chaotic system with a two-point attractor.

#### SDiff(2) and uniqueness of the Plebanski equation

(Journal article; Tidsskriftartikkel; Peer reviewed, 2012)

The group of area preserving diffeomorphisms showed importance in the problems of self-dual gravity and integrability theory. We discuss how representations of this infinite-dimensional Lie group can arise in mathematical physics from pure local considerations. Then using Lie algebra extensions and cohomology we derive the second Plebański equation and its geometry. We do not use Kähler or other ...

#### Compatibility, multi-brackets and integrability of systems of PDEs

(Journal article; Tidsskriftartikkel; Peer reviewed, 2008-02-20)

We establish an efficient compatibility criterion for a system of generalized
complete intersection type in terms of certain multi-brackets of
differential operators. These multi-brackets generalize the higher Jacobi-
Mayer brackets, important in the study of evolutionary equations and the
integrability problem. We also calculate Spencer δ-cohomology of generalized
complete intersections and ...

#### Differential invariants of the motion group actions

(Working paper; Arbeidsnotat, 2007-12-20)

Differential invariants of a (pseudo)group action can vary when restricted
to invariant submanifolds (differential equations). The algebra
is still governed by the Lie-Tresse theorem, but may change a lot. We
describe in details the case of the motion group O(n) ⋉ R<sup>n</sup> acting on the
full (unconstraint) jet-space as well as on some invariant equations.

#### Invariants of pseudogroup actions: Homological methods and Finiteness theorem

(Working paper; Arbeidsnotat, 2005-12-07)

We study the equivalence problem of submanifolds with respect to a
transitive pseudogroup action. The corresponding differential invariants
are determined via formal theory and lead to the notions of l-variants and
l-covariants, even in the case of non-integrable pseudogroup. Their calculation
is based on the cohomological machinery: We introduce a complex
for covariants, define their cohomology ...

#### Spencer δ-cohomology, restrictions, characteristics and involutive symbolic PDEs

(Working paper; Arbeidsnotat, 2005-03-07)

We generalize the notion of involutivity to systems of differential equations
of different orders and show that the classical results relating involutivity,
restrictions, characteristics and characteristicity, known for first
order systems, extend to the general context. This involves, in particular,
a new definition of strong characteristicity. The proof exploits a spectral
sequence relating ...

#### Dimension of the solutions space of PDEs

(Conference object; Konferansebidrag, 2006-10-26)

We discuss the dimensional characterization of the solutions
space of a formally integrable system of partial differential equations and
provide certain formulas for calculations of these dimensional quantities.

#### Tangent and normal bundles in almost complex geometry

(Journal article; Tidsskriftartikkel; Peer reviewed, 2005-06-10)

We define and study pseudoholomorphic vector bundles structures,
particular cases of which are tangent and normal bundle almost complex
structures. As an application we deduce normal forms of 1-jets of almost
complex structures along a pseudoholomorphic submanifold. In dimension
four we relate these normal forms to the problem of pseudoholomorphic
foliation of a neighborhood of a curve and the ...