810 research outputs found
Geometric contextuality from the Maclachlan-Martin Kleinian groups
There are contextual sets of multiple qubits whose commutation is
parametrized thanks to the coset geometry of a subgroup of
the two-generator free group . One defines
geometric contextuality from the discrepancy between the commutativity of
cosets on and that of quantum observables.It is shown in this
paper that Kleinian subgroups that are
non-compact, arithmetic, and generated by two elliptic isometries and
(the Martin-Maclachlan classification), are appropriate contextuality filters.
Standard contextual geometries such as some thin generalized polygons (starting
with Mermin's grid) belong to this frame. The Bianchi groups
, defined over the imaginary quadratic field
play a special role
The nonexistence of regular near octagons with parameters (s, t, t(2), t(3)) = (2,24,0,8)
Let S be a regular near octagon with s + 1 = 3 points per line, let t + 1 denote the constant number of lines through a given point of S and for every two points x and y at distance i is an element of {2, 3} from each other, let t(i) + 1 denote the constant number of lines through y containing a (necessarily unique) point at distance i - 1 from x. It is known, using algebraic combinatorial techniques, that (t(2), t(3), t) must be equal to either (0, 0, 1), (0, 0, 4), (0, 3, 4), (0, 8, 24), (1, 2, 3), (2, 6, 14) or (4, 20, 84). For all but one of these cases, there is a unique example of a regular near octagon known. In this paper, we deal with the existence question for the remaining case. We prove that no regular near octagons with parameters (s, t, t(2), t(3)) = (2, 24, 0, 8) can exist
Polygonal valuations
AbstractWe develop a valuation theory for generalized polygons similar to the existing theory for dense near polygons. This valuation theory has applications for the study and classification of generalized polygons that have full subpolygons as subgeometries
Characterizations of the Suzuki tower near polygons
In recent work, we constructed a new near octagon from certain
involutions of the finite simple group and showed a correspondence
between the Suzuki tower of finite simple groups, , and the tower of near polygons, . Here we characterize
each of these near polygons (except for the first one) as the unique near
polygon of the given order and diameter containing an isometrically embedded
copy of the previous near polygon of the tower. In particular, our
characterization of the Hall-Janko near octagon is similar to an
earlier characterization due to Cohen and Tits who proved that it is the unique
regular near octagon with parameters , but instead of regularity
we assume existence of an isometrically embedded dual split Cayley hexagon,
. We also give a complete classification of near hexagons of
order and use it to prove the uniqueness result for .Comment: 20 pages; some revisions based on referee reports; added more
references; added remarks 1.4 and 1.5; corrected typos; improved the overall
expositio
Star Integrals, Convolutions and Simplices
We explore single and multi-loop conformal integrals, such as the ones
appearing in dual conformal theories in flat space. Using Mellin amplitudes, a
large class of higher loop integrals can be written as simple
integro-differential operators on star integrals: one-loop -gon integrals in
dimensions. These are known to be given by volumes of hyperbolic simplices.
We explicitly compute the five-dimensional pentagon integral in full generality
using Schl\"afli's formula. Then, as a first step to understanding higher
loops, we use spline technology to construct explicitly the hexagon and
octagon integrals in two-dimensional kinematics. The fully massive hexagon
and octagon integrals are then related to the double box and triple box
integrals respectively. We comment on the classes of functions needed to
express these integrals in general kinematics, involving elliptic functions and
beyond.Comment: 23 page
Trees and Matchings
In this article, Temperley's bijection between spanning trees of the square
grid on the one hand, and perfect matchings (also known as dimer coverings) of
the square grid on the other, is extended to the setting of general planar
directed (and undirected) graphs, where edges carry nonnegative weights that
induce a weighting on the set of spanning trees. We show that the weighted,
directed spanning trees (often called arborescences) of any planar graph G can
be put into a one-to-one weight-preserving correspondence with the perfect
matchings of a related planar graph H.
One special case of this result is a bijection between perfect matchings of
the hexagonal honeycomb lattice and directed spanning trees of a triangular
lattice. Another special case gives a correspondence between perfect matchings
of the ``square-octagon'' lattice and directed weighted spanning trees on a
directed weighted version of the cartesian lattice.
In conjunction with results of Kenyon, our main theorem allows us to compute
the measures of all cylinder events for random spanning trees on any (directed,
weighted) planar graph. Conversely, in cases where the perfect matching model
arises from a tree model, Wilson's algorithm allows us to quickly generate
random samples of perfect matchings.Comment: 32 pages, 19 figures (minor revisions from version 1
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