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About Dynamical Systems Appearing in the Microscopic Traffic Modeling
Motivated by microscopic traffic modeling, we analyze dynamical systems which
have a piecewise linear concave dynamics not necessarily monotonic. We
introduce a deterministic Petri net extension where edges may have negative
weights. The dynamics of these Petri nets are well-defined and may be described
by a generalized matrix with a submatrix in the standard algebra with possibly
negative entries, and another submatrix in the minplus algebra. When the
dynamics is additively homogeneous, a generalized additive eigenvalue may be
introduced, and the ergodic theory may be used to define a growth rate under
additional technical assumptions. In the traffic example of two roads with one
junction, we compute explicitly the eigenvalue and we show, by numerical
simulations, that these two quantities (the additive eigenvalue and the growth
rate) are not equal, but are close to each other. With this result, we are able
to extend the well-studied notion of fundamental traffic diagram (the average
flow as a function of the car density on a road) to the case of two roads with
one junction and give a very simple analytic approximation of this diagram
where four phases appear with clear traffic interpretations. Simulations show
that the fundamental diagram shape obtained is also valid for systems with many
junctions. To simulate these systems, we have to compute their dynamics, which
are not quite simple. For building them in a modular way, we introduce
generalized parallel, series and feedback compositions of piecewise linear
concave dynamics.Comment: PDF 38 page
Implicit solutions with consistent additive and multiplicative components
Use of multiple-point-constraint
Frameworks, Symmetry and Rigidity
Symmetry equations are obtained for the rigidity matrix of a bar-joint
framework in R^d. These form the basis for a short proof of the Fowler-Guest
symmetry group generalisation of the Calladine-Maxwell counting rules. Similar
symmetry equations are obtained for the Jacobian of diverse framework systems,
including constrained point-line systems that appear in CAD, body-pin
frameworks, hybrid systems of distance constrained objects and infinite
bar-joint frameworks. This leads to generalised forms of the Fowler-Guest
character formula together with counting rules in terms of counts of
symmetry-fixed elements. Necessary conditions for isostaticity are obtained for
asymmetric frameworks, both when symmetries are present in subframeworks and
when symmetries occur in partition-derived frameworks.Comment: 5 Figures. Replaces Dec. 2008 version. To appear in IJCG
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