787 research outputs found
The Complexity of Simultaneous Geometric Graph Embedding
Given a collection of planar graphs on the same set of
vertices, the simultaneous geometric embedding (with mapping) problem, or
simply -SGE, is to find a set of points in the plane and a bijection
such that the induced straight-line drawings of
under are all plane.
This problem is polynomial-time equivalent to weak rectilinear realizability
of abstract topological graphs, which Kyn\v{c}l (doi:10.1007/s00454-010-9320-x)
proved to be complete for , the existential theory of the
reals. Hence the problem -SGE is polynomial-time equivalent to several other
problems in computational geometry, such as recognizing intersection graphs of
line segments or finding the rectilinear crossing number of a graph.
We give an elementary reduction from the pseudoline stretchability problem to
-SGE, with the property that both numbers and are linear in the
number of pseudolines. This implies not only the -hardness
result, but also a lower bound on the minimum size of a
grid on which any such simultaneous embedding can be drawn. This bound is
tight. Hence there exists such collections of graphs that can be simultaneously
embedded, but every simultaneous drawing requires an exponential number of bits
per coordinates. The best value that can be extracted from Kyn\v{c}l's proof is
only
Amoebas of algebraic varieties and tropical geometry
This survey consists of two parts. Part 1 is devoted to amoebas. These are
images of algebraic subvarieties in the complex torus under the logarithmic
moment map. The amoebas have essentially piecewise-linear shape if viewed at
large. Furthermore, they degenerate to certain piecewise-linear objects called
tropical varieties whose behavior is governed by algebraic geometry over the
so-called tropical semifield. Geometric aspects of tropical algebraic geometry
are the content of Part 2. We pay special attention to tropical curves. Both
parts also include relevant applications of the theories. Part 1 of this survey
is a revised and updated version of an earlier prepreint of 2001.Comment: 40 pages, 15 figures, a survey for the volume "Different faces in
Geometry
Observing trajectories with weak measurements in quantum systems in the semiclassical regime
We propose a scheme allowing to observe the evolution of a quantum system in
the semiclassical regime along the paths generated by the propagator. The
scheme relies on performing consecutive weak measurements of the position. We
show how weak trajectories" can be extracted from the pointers of a series of
measurement devices having weakly interacted with the system. The properties of
these "weak trajectories" are investigated and illustrated in the case of a
time-dependent model system.Comment: v2: Several minor corrections were made. Added Appendix (that will
appear as Suppl. Material). To be published in Phys Rev Let
Generalized explicit descent and its application to curves of genus 3
We introduce a common generalization of essentially all known methods for
explicit computation of Selmer groups, which are used to bound the ranks of
abelian varieties over global fields. We also simplify and extend the proofs
relating what is computed to the cohomologically-defined Selmer groups. Selmer
group computations have been practical for many Jacobians of curves over Q of
genus up to 2 since the 1990s, but our approach is the first to be practical
for general curves of genus 3. We show that our approach succeeds on some
genus-3 examples defined by polynomials with small coefficients.Comment: 58 pages; added a few references, and updated a few other
Khovanov homology is an unknot-detector
We prove that a knot is the unknot if and only if its reduced Khovanov
cohomology has rank 1. The proof has two steps. We show first that there is a
spectral sequence beginning with the reduced Khovanov cohomology and abutting
to a knot homology defined using singular instantons. We then show that the
latter homology is isomorphic to the instanton Floer homology of the sutured
knot complement: an invariant that is already known to detect the unknot.Comment: 124 pages, 13 figure
Excluding a Weakly 4-connected Minor
A 3-connected graph is called weakly 4-connected if min holds for all 3-separations of . A 3-connected graph is called quasi 4-connected if min . We first discuss how to decompose a 3-connected graph into quasi 4-connected components. We will establish a chain theorem which will allow us to easily generate the set of all quasi 4-connected graphs. Finally, we will apply these results to characterizing all graphs which do not contain the Pyramid as a minor, where the Pyramid is the weakly 4-connected graph obtained by performing a transformation to the octahedron. This result can be used to show an interesting characterization of quasi 4-connected, outer-projective graphs
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