2,656 research outputs found
Lower bounds for regular genus and gem-complexity of PL 4-manifolds
Within crystallization theory, two interesting PL invariants for
-manifolds have been introduced and studied, namely {\it gem-complexity} and
{\it regular genus}. In the present paper we prove that, for any closed
connected PL -manifold , its gem-complexity and its
regular genus satisfy:
where These lower bounds enable to strictly improve
previously known estimations for regular genus and gem-complexity of product
4-manifolds. Moreover, the class of {\it semi-simple crystallizations} is
introduced, so that the represented PL 4-manifolds attain the above lower
bounds. The additivity of both gem-complexity and regular genus with respect to
connected sum is also proved for such a class of PL 4-manifolds, which
comprehends all ones of "standard type", involved in existing crystallization
catalogues, and their connected sums.Comment: 17 pages, 3 figures. To appear in Forum Mathematicu
The Phase Diagram of 1-in-3 Satisfiability Problem
We study the typical case properties of the 1-in-3 satisfiability problem,
the boolean satisfaction problem where a clause is satisfied by exactly one
literal, in an enlarged random ensemble parametrized by average connectivity
and probability of negation of a variable in a clause. Random 1-in-3
Satisfiability and Exact 3-Cover are special cases of this ensemble. We
interpolate between these cases from a region where satisfiability can be
typically decided for all connectivities in polynomial time to a region where
deciding satisfiability is hard, in some interval of connectivities. We derive
several rigorous results in the first region, and develop the
one-step--replica-symmetry-breaking cavity analysis in the second one. We
discuss the prediction for the transition between the almost surely satisfiable
and the almost surely unsatisfiable phase, and other structural properties of
the phase diagram, in light of cavity method results.Comment: 30 pages, 12 figure
Simple crystallizations of 4-manifolds
Minimal crystallizations of simply connected PL 4-manifolds are very natural
objects. Many of their topological features are reflected in their
combinatorial structure which, in addition, is preserved under the connected
sum operation. We present a minimal crystallization of the standard PL K3
surface. In combination with known results this yields minimal crystallizations
of all simply connected PL 4-manifolds of "standard" type, that is, all
connected sums of , , and the K3 surface. In
particular, we obtain minimal crystallizations of a pair of homeomorphic but
non-PL-homeomorphic 4-manifolds. In addition, we give an elementary proof that
the minimal 8-vertex crystallization of is unique and its
associated pseudotriangulation is related to the 9-vertex combinatorial
triangulation of by the minimum of four edge contractions.Comment: 23 pages, 7 figures. Minor update, replacement of Figure 7. To appear
in Advances in Geometr
On Norm-Based Estimations for Domains of Attraction in Nonlinear Time-Delay Systems
For nonlinear time-delay systems, domains of attraction are rarely studied
despite their importance for technological applications. The present paper
provides methodological hints for the determination of an upper bound on the
radius of attraction by numerical means. Thereby, the respective Banach space
for initial functions has to be selected and primary initial functions have to
be chosen. The latter are used in time-forward simulations to determine a first
upper bound on the radius of attraction. Thereafter, this upper bound is
refined by secondary initial functions, which result a posteriori from the
preceding simulations. Additionally, a bifurcation analysis should be
undertaken. This analysis results in a possible improvement of the previous
estimation. An example of a time-delayed swing equation demonstrates the
various aspects.Comment: 33 pages, 8 figures, "This is a pre-print of an article published in
'Nonlinear Dynamics'. The final authenticated version is available online at
https://doi.org/10.1007/s11071-020-05620-8
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