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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.
Evidence of low-dimensional chaos in magnetized plasma turbulence
(Working paper; Arbeidsnotat, 2008-03-10)
We analyze probe data obtained from a toroidal magnetized plasma configuration
suitable for studies of low-frequency gradient-driven instabilities. These instabilities give
rise to field-aligned convection rolls analogous to Rayleigh-Benard cells in neutral fluids,
and may theoretically develop similar routes to chaos. When using mean-field dimension
analysis, we observe low dimensionality, but ...
Stretched exponential relaxation and ac universality in disordered dielectrics
(Working paper; Arbeidsnotat, 2007-05-30)
This paper is concerned with the connection between the properties of dielectric relaxation and ac
(alternating-current) conduction in disordered dielectrics. The discussion is divided between the classical
linear-response theory and a self-consistent dynamical modeling. The key issues are, stretched
exponential character of dielectric relaxation, power-law power spectral density, and anomalous ...
Nature reserves as a bioeconomic management tool. A simplified modeling approach
(Working paper; Arbeidsnotat, 2006-02)
This paper demonstrates analytically how a nature reserve may protect the total population, realize
maximum sustainable yield (MSY), maximum economic yield (MEY) and consumer surplus (CS)
and how this depends on biological growth, migration, reserve size and economic parameters. The
pre-reserve population is assumed to follow the logistic growth law and two post-reserve growth
models are discussed. ...
Involutivity of field equations
(Working paper; Arbeidsnotat, 2009-02-10)
We prove involutivity of Einstein and Einstein-Maxwell equations by calculating
the Spencer cohomology of these systems. Relation with Cartan method is traced in details.
Basic implications through Cartan-Kähler theory are derived.
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.
The categorical theory of relations and quantization
(Working paper; Arbeidsnotat, 2001-10-30)
In this paper we develops a categorical theory of relations and use this
formulation to define the notion of quantization for relations. Categories
of relations are defined in the context of symmetric monoidal categories.
They are shown to be symmetric monoidal categories in their own right
and are found to be isomorphic to certain categories of A−A bicomodules.
Properties of relations are ...
Modeling temporal fluctuations in avalanching system
(Working paper; Arbeidsnotat, 2008-07-22)
We demonstrate how to model the toppling activity in avalanching systems by stochastic differential
equations (SDEs). The theory is developed as a generalization of the classical mean field
approach to sandpile dynamics by formulating it as a generalization of Itoh’s SDE. This equation
contains a fractional Gaussian noise term representing the branching of an avalanche into small
active clusters, ...