846 research outputs found
1-Safe Petri nets and special cube complexes: equivalence and applications
Nielsen, Plotkin, and Winskel (1981) proved that every 1-safe Petri net
unfolds into an event structure . By a result of Thiagarajan
(1996 and 2002), these unfoldings are exactly the trace regular event
structures. Thiagarajan (1996 and 2002) conjectured that regular event
structures correspond exactly to trace regular event structures. In a recent
paper (Chalopin and Chepoi, 2017, 2018), we disproved this conjecture, based on
the striking bijection between domains of event structures, median graphs, and
CAT(0) cube complexes. On the other hand, in Chalopin and Chepoi (2018) we
proved that Thiagarajan's conjecture is true for regular event structures whose
domains are principal filters of universal covers of (virtually) finite special
cube complexes.
In the current paper, we prove the converse: to any finite 1-safe Petri net
one can associate a finite special cube complex such that the
domain of the event structure (obtained as the unfolding of
) is a principal filter of the universal cover of .
This establishes a bijection between 1-safe Petri nets and finite special cube
complexes and provides a combinatorial characterization of trace regular event
structures.
Using this bijection and techniques from graph theory and geometry (MSO
theory of graphs, bounded treewidth, and bounded hyperbolicity) we disprove yet
another conjecture by Thiagarajan (from the paper with S. Yang from 2014) that
the monadic second order logic of a 1-safe Petri net is decidable if and only
if its unfolding is grid-free.
Our counterexample is the trace regular event structure
which arises from a virtually special square complex . The domain of
is grid-free (because it is hyperbolic), but the MSO
theory of the event structure is undecidable
Branched Coverings, Triangulations, and 3-Manifolds
A canonical branched covering over each sufficiently good simplicial complex
is constructed. Its structure depends on the combinatorial type of the complex.
In this way, each closed orientable 3-manifold arises as a branched covering
over the 3-sphere from some triangulation of S^3. This result is related to a
theorem of Hilden and Montesinos. The branched coverings introduced admit a
rich theory in which the group of projectivities plays a central role.Comment: v2: several changes to the text body; minor correction
Hyperbolic Unfoldings of Minimal Hypersurfaces
We study the intrinsic geometry of area minimizing (and also of almost
minimizing) hypersurfaces from a new point of view by relating this subject to
quasiconformal geometry. For any such hypersurface we define and construct a
so-called S-structure which reveals some unexpected geometric and analytic
properties of the hypersurface and its singularity set. In this paper, this is
used to prove the existence of hyperbolic unfoldings: canonical conformal
deformations of the regular part of these hypersurfaces into complete Gromov
hyperbolic spaces of bounded geometry with Gromov boundary homeomorphic to the
singular set
The moduli space of germs of generic families of analytic diffeomorphisms unfolding a parabolic fixed point
In this paper we describe the moduli space of germs of generic families of
analytic diffeomorphisms which unfold a parabolic fixed point of codimension 1.
In [MRR] (and also [R]), it was shown that the Ecalle-Voronin modulus can be
unfolded to give a complete modulus for such germs. The modulus is defined on a
ramified sector in the canonical perturbation parameter \eps. As in the case
of the Ecalle-Voronin modulus, the modulus is defined up to a linear scaling
depending only on \eps.
Here, we characterize the moduli space for such unfoldings by finding the
compatibility conditions on the modulus which are necessary and sufficient for
realization as the modulus of an unfolding.
The compatibility condition is obtained by considering the region of
sectorial overlap in \eps-space. This lies in the Glutsyuk sector where the
two fixed points are hyperbolic and connected by the orbits of the
diffeomorphism. In this region we have two representatives of the modulus which
describe the same dynamics. We identify the necessary compatibility condition
between these two representatives by comparing them both with their common
Glutsyuk modulus.
The compatibility condition implies the existence of a linear scaling for
which the modulus is 1/2-summable in \eps, whose direction of non-summability
coincides with the direction of real multipliers at the fixed points.
Conversely, we show that the compatibility condition (which implies the
summability property) is sufficient to realize the modulus as coming from an
analytic unfolding, thus giving a complete description of the space of moduli.Comment: 48 page
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