610 research outputs found
Causal Set Dynamics: A Toy Model
We construct a quantum measure on the power set of non-cyclic oriented graphs
of N points, drawing inspiration from 1-dimensional directed percolation.
Quantum interference patterns lead to properties which do not appear to have
any analogue in classical percolation. Most notably, instead of the single
phase transition of classical percolation, the quantum model displays two
distinct crossover points. Between these two points, spacetime questions such
as "does the network percolate" have no definite or probabilistic answer.Comment: 28 pages incl. 5 figure
Coxeter Complexes and Graph-Associahedra
Given a graph G, we construct a simple, convex polytope whose face poset is
based on the connected subgraphs of G. This provides a natural generalization
of the Stasheff associahedron and the Bott-Taubes cyclohedron. Moreover, we
show that for any simplicial Coxeter system, the minimal blow-ups of its
associated Coxeter complex has a tiling by graph-associahedra. The geometric
and combinatorial properties of the complex as well as of the polyhedra are
given. These spaces are natural generalizations of the Deligne-Knudsen-Mumford
compactification of the real moduli space of curves.Comment: 18 pages, 9 figures; revised content and reference
Countable locally 2-arc-transitive bipartite graphs
We present an order-theoretic approach to the study of countably infinite
locally 2-arc-transitive bipartite graphs. Our approach is motivated by
techniques developed by Warren and others during the study of cycle-free
partial orders. We give several new families of previously unknown countably
infinite locally-2-arc-transitive graphs, each family containing continuum many
members. These examples are obtained by gluing together copies of incidence
graphs of semilinear spaces, satisfying a certain symmetry property, in a
tree-like way. In one case we show how the classification problem for that
family relates to the problem of determining a certain family of highly
arc-transitive digraphs. Numerous illustrative examples are given.Comment: 29 page
Discrete Morse theory for computing cellular sheaf cohomology
Sheaves and sheaf cohomology are powerful tools in computational topology,
greatly generalizing persistent homology. We develop an algorithm for
simplifying the computation of cellular sheaf cohomology via (discrete)
Morse-theoretic techniques. As a consequence, we derive efficient techniques
for distributed computation of (ordinary) cohomology of a cell complex.Comment: 19 pages, 1 Figure. Added Section 5.
Algebraic description of spacetime foam
A mathematical formalism for treating spacetime topology as a quantum
observable is provided. We describe spacetime foam entirely in algebraic terms.
To implement the correspondence principle we express the classical spacetime
manifold of general relativity and the commutative coordinates of its events by
means of appropriate limit constructions.Comment: 34 pages, LaTeX2e, the section concerning classical spacetimes in the
limit essentially correcte
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