11,120 research outputs found
A Combinatorial classification of postcritically fixed Newton maps
We give a combinatorial classification for the class of postcritically fixed
Newton maps of polynomials as dynamical systems. This lays the foundation for
classification results of more general classes of Newton maps.
A fundamental ingredient is the proof that for every Newton map
(postcritically finite or not) every connected component of the basin of an
attracting fixed point can be connected to through a finite chain of
such components.Comment: 37 pages, 5 figures, published in Ergodic Theory and Dynamical
Systems (2018). This is the final author file before publication. Text
overlap with earlier arxiv file observed by arxiv system relates to an
earlier version that was erroneously uploaded separately. arXiv admin note:
text overlap with arXiv:math/070117
Topological Reverberations in Flat Space-times
We study the role played by multiply-connectedness in the time evolution of
the energy E(t) of a radiating system that lies in static flat space-time
manifolds M_4 whose t=const spacelike sections M_3 are compact in at least one
spatial direction. The radiation reaction equation of the radiating source is
derived for the case where M_3 has any non-trivial flat topology, and an exact
solution is obtained. We also show that when the spacelike sections are
multiply-connected flat 3-manifolds the energy E(t) exhibits a reverberation
pattern with discontinuities in the derivative of E(t) and a set of relative
minima and maxima, followed by a growth of E(t). It emerges from this result
that the compactness in at least one spatial direction of Minkowski space-time
is sufficient to induce this type of topological reverberation, making clear
that our radiating system is topologically fragile. An explicit solution of the
radiation reaction equation for the case where M_3 = R^2 x S^1 is discussed,
and graphs which reveal how the energy varies with the time are presented and
analyzed.Comment: 16 pages, 4 figures, REVTEX; Added five references and inserted
clarifying details. Version to appear in Int. J. Mod. Phys. A (2000
Conditions for synchronizability in arrays of coupled linear systems
Synchronization control in arrays of identical output-coupled continuous-time
linear systems is studied. Sufficiency of new conditions for the existence of a
synchronizing feedback law are analyzed. It is shown that for neutrally stable
systems that are detectable form their outputs, a linear feedback law exists
under which any number of coupled systems synchronize provided that the
(directed, weighted) graph describing the interconnection is fixed and
connected. An algorithm generating one such feedback law is presented. It is
also shown that for critically unstable systems detectability is not
sufficient, whereas full-state coupling is, for the existence of a linear
feedback law that is synchronizing for all connected coupling configurations
Some NP-complete edge packing and partitioning problems in planar graphs
Graph packing and partitioning problems have been studied in many contexts,
including from the algorithmic complexity perspective. Consider the packing
problem of determining whether a graph contains a spanning tree and a cycle
that do not share edges. Bern\'ath and Kir\'aly proved that this decision
problem is NP-complete and asked if the same result holds when restricting to
planar graphs. Similarly, they showed that the packing problem with a spanning
tree and a path between two distinguished vertices is NP-complete. They also
established the NP-completeness of the partitioning problem of determining
whether the edge set of a graph can be partitioned into a spanning tree and a
(not-necessarily spanning) tree. We prove that all three problems remain
NP-complete even when restricted to planar graphs.Comment: 6 pages, 2 figure
Distribution of sizes of erased loops for loop-erased random walks
We study the distribution of sizes of erased loops for loop-erased random
walks on regular and fractal lattices. We show that for arbitrary graphs the
probability of generating a loop of perimeter is expressible in
terms of the probability of forming a loop of perimeter when a
bond is added to a random spanning tree on the same graph by the simple
relation . On -dimensional hypercubical lattices,
varies as for large , where for , where
z is the fractal dimension of the loop-erased walks on the graph. On
recursively constructed fractals with this relation is modified
to , where is the hausdorff and
is the spectral dimension of the fractal.Comment: 4 pages, RevTex, 3 figure
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