74 research outputs found
Braids, posets and orthoschemes
In this article we study the curvature properties of the order complex of a
graded poset under a metric that we call the ``orthoscheme metric''. In
addition to other results, we characterize which rank 4 posets have CAT(0)
orthoscheme complexes and by applying this theorem to standard posets and
complexes associated with four-generator Artin groups, we are able to show that
the 5-string braid group is the fundamental group of a compact nonpositively
curved space.Comment: 33 pages, 16 figure
Beyond Worst-Case Analysis for Joins with Minesweeper
We describe a new algorithm, Minesweeper, that is able to satisfy stronger
runtime guarantees than previous join algorithms (colloquially, `beyond
worst-case guarantees') for data in indexed search trees. Our first
contribution is developing a framework to measure this stronger notion of
complexity, which we call {\it certificate complexity}, that extends notions of
Barbay et al. and Demaine et al.; a certificate is a set of propositional
formulae that certifies that the output is correct. This notion captures a
natural class of join algorithms. In addition, the certificate allows us to
define a strictly stronger notion of runtime complexity than traditional
worst-case guarantees. Our second contribution is to develop a dichotomy
theorem for the certificate-based notion of complexity. Roughly, we show that
Minesweeper evaluates -acyclic queries in time linear in the certificate
plus the output size, while for any -cyclic query there is some instance
that takes superlinear time in the certificate (and for which the output is no
larger than the certificate size). We also extend our certificate-complexity
analysis to queries with bounded treewidth and the triangle query.Comment: [This is the full version of our PODS'2014 paper.
Multifraction reduction III: The case of interval monoids
We investigate gcd-monoids, which are cancellative monoids in which any two
elements admit a left and a right gcd, and the associated reduction of
multifractions (arXiv:1606.08991 and 1606.08995), a general approach to the
word problem for the enveloping group. Here we consider the particular case of
interval monoids associated with finite posets. In this way, we construct
gcd-monoids, in which reduction of multifractions has prescribed properties not
yet known to be compatible: semi-convergence of reduction without convergence,
semi-convergence up to some level but not beyond, non-embeddability into the
enveloping group (a strong negation of semi-convergence).Comment: 23 pages ; v2 : cross-references updated ; v3 : one example added,
typos corrected; final version due to appear in Journal of Combinatorial
Algebr
Geometric Properties of Closed Three Manifolds and Hyperbolic Links
The Geometrization Theorem for 3-manifolds states that every closed orientable 3-manifold can be cut along spheres and tori into pieces which have a geometric structure modeled on one of the eight, 3-dimensional geometries. In joint work with Dennis Sullivan, we combine the different geometries on the toroidal ends of 3-manifolds to describe a uniform geometric structure for all oriented closed prime 3-manifolds. Hyperbolic structures on links in the thickened torus and their geometric properties have been of great interest recently. We discuss geometric properties of augmented and fully augmented links in the thickened torus. We show how sequences of fully augmented links in the 3-sphere which diagrammatically converge to a biperiodic fully augmented link have interesting asymptotic volume growth
Testing vertex connectivity of bowtie 1-plane graphs
A separating set of a connected graph is a set of vertices such that is disconnected. is a minimum separating set of if there is no separating set of with fewer vertices than . The size of a minimum separating set of is called the vertex connectivity of . A separating set of that is a cycle is called a separating cycle of .
Let be a planar graph with a given planar embedding. Let be a supergraph of obtained by inserting a face vertex in each face of and connecting the face vertex to all vertices on the boundary of the face. It is well known that a set is a minimum separating set of a planar graph if and only if the vertices of can be connected together using face vertices to get a cycle of length that is separating in .
We extend this correspondence between separating sets and separating cycles from planar graphs to the class of bowtie 1-plane graphs. These are graphs that are embedded on the plane such that each edge is crossed at most once by another edge, and the endpoints of each such crossing induce either , or . Using this result, we give an algorithm to compute the vertex connectivity of a bowtie 1-plane graph in linear time
Revealing the Landscape of Globally Color-Dual Multi-loop Integrands
We report on progress in understanding how to construct color-dual multi-loop
amplitudes. First we identify a cubic theory, semi-abelian Yang-Mills, that
unifies many of the color-dual theories studied in the literature, and provides
a prescriptive approach for constructing -dimensional color-dual numerators
through one-loop directly from Feynman rules. By a simple weight counting
argument, this approach does not further generalize to two-loops. As a first
step in understanding the two-loop challenge, we use a -dimensional
color-dual bootstrap to successfully construct globally color-dual local
two-loop four-point nonlinear sigma model (NLSM) numerators. The double-copy of
these NLSM numerators with themselves, pure Yang-Mills, and
super-Yang-Mills correctly reproduce the known unitarity constructed integrands
of special Galileons, Born-Infeld theory, and Dirac-Born-Infeld-Volkov-Akulov
theory, respectively. Applying our bootstrap to two-loop four-point pure
Yang-Mills, we exhaustively search the space of local numerators and find that
it fails to satisfy global color-kinematics duality, completing a search
previously initiated in the literature. We pinpoint the failure to the bowtie
unitarity cut, and discuss a path forward towards non-local construction of
color-dual integrands at generic loop order.Comment: 42 pages, 4 figures, ancillary fil
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