19,560 research outputs found
Antimatroids and Balanced Pairs
We generalize the 1/3-2/3 conjecture from partially ordered sets to
antimatroids: we conjecture that any antimatroid has a pair of elements x,y
such that x has probability between 1/3 and 2/3 of appearing earlier than y in
a uniformly random basic word of the antimatroid. We prove the conjecture for
antimatroids of convex dimension two (the antimatroid-theoretic analogue of
partial orders of width two), for antimatroids of height two, for antimatroids
with an independent element, and for the perfect elimination antimatroids and
node search antimatroids of several classes of graphs. A computer search shows
that the conjecture is true for all antimatroids with at most six elements.Comment: 16 pages, 5 figure
On the existence of asymptotically good linear codes in minor-closed classes
Let be a sequence of codes such that each
is a linear -code over some fixed finite field
, where is the length of the codewords, is the
dimension, and is the minimum distance. We say that is
asymptotically good if, for some and for all , , , and . Sequences of
asymptotically good codes exist. We prove that if is a class of
GF-linear codes (where is prime and ), closed under
puncturing and shortening, and if contains an asymptotically good
sequence, then must contain all GF-linear codes. Our proof
relies on a powerful new result from matroid structure theory
Eulerian digraphs and toric Calabi-Yau varieties
We investigate the structure of a simple class of affine toric Calabi-Yau
varieties that are defined from quiver representations based on finite eulerian
directed graphs (digraphs). The vanishing first Chern class of these varieties
just follows from the characterisation of eulerian digraphs as being connected
with all vertices balanced. Some structure theory is used to show how any
eulerian digraph can be generated by iterating combinations of just a few
canonical graph-theoretic moves. We describe the effect of each of these moves
on the lattice polytopes which encode the toric Calabi-Yau varieties and
illustrate the construction in several examples. We comment on physical
applications of the construction in the context of moduli spaces for
superconformal gauged linear sigma models.Comment: 27 pages, 8 figure
Holographic entropy relations
We develop a framework for the derivation of new information theoretic
quantities which are natural from a holographic perspective. We demonstrate the
utility of our techniques by deriving the tripartite information (the quantity
associated to monogamy of mutual information) using a set of abstract arguments
involving bulk extremal surfaces. Our arguments rely on formal manipulations of
surfaces and not on local surgery or explicit computation of entropies through
the holographic entanglement entropy prescriptions. As an application, we show
how to derive a family of similar information quantities for an arbitrary
number of parties. The present work establishes the foundation of a broader
program that aims at the understanding of the entanglement structures of
geometric states for an arbitrary number of parties. We stress that our method
is completely democratic with respect to bulk geometries and is equally valid
in static and dynamical situations. While rooted in holography, we expect that
our construction will provide a useful characterization of multipartite
correlations in quantum field theories.Comment: v1: 58 pages, 1 pdf figur
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