18,607 research outputs found
Elusive extremal graphs
We study the uniqueness of optimal solutions to extremal graph theory
problems. Lovasz conjectured that every finite feasible set of subgraph density
constraints can be extended further by a finite set of density constraints so
that the resulting set is satisfied by an asymptotically unique graph. This
statement is often referred to as saying that `every extremal graph theory
problem has a finitely forcible optimum'. We present a counterexample to the
conjecture. Our techniques also extend to a more general setting involving
other types of constraints
Meyer sets, topological eigenvalues, and Cantor fiber bundles
We introduce two new characterizations of Meyer sets. A repetitive Delone set
in with finite local complexity is topologically conjugate to a Meyer
set if and only if it has linearly independent topological eigenvalues,
which is if and only if it is topologically conjugate to a bundle over a
-torus with totally disconnected compact fiber and expansive canonical
action. "Conjugate to" is a non-trivial condition, as we show that there exist
sets that are topologically conjugate to Meyer sets but are not themselves
Meyer. We also exhibit a diffractive set that is not Meyer, answering in the
negative a question posed by Lagarias, and exhibit a Meyer set for which the
measurable and topological eigenvalues are different.Comment: minor errors corrected, references added. To appear in the Journal of
the LM
Intrinsic Universality in Self-Assembly
We show that the Tile Assembly Model exhibits a strong notion of universality
where the goal is to give a single tile assembly system that simulates the
behavior of any other tile assembly system. We give a tile assembly system that
is capable of simulating a very wide class of tile systems, including itself.
Specifically, we give a tile set that simulates the assembly of any tile
assembly system in a class of systems that we call \emph{locally consistent}:
each tile binds with exactly the strength needed to stay attached, and that
there are no glue mismatches between tiles in any produced assembly.
Our construction is reminiscent of the studies of \emph{intrinsic
universality} of cellular automata by Ollinger and others, in the sense that
our simulation of a tile system by a tile system represents each tile
in an assembly produced by by a block of tiles in , where
is a constant depending on but not on the size of the assembly
produces (which may in fact be infinite). Also, our construction improves on
earlier simulations of tile assembly systems by other tile assembly systems (in
particular, those of Soloveichik and Winfree, and of Demaine et al.) in that we
simulate the actual process of self-assembly, not just the end result, as in
Soloveichik and Winfree's construction, and we do not discriminate against
infinite structures. Both previous results simulate only temperature 1 systems,
whereas our construction simulates tile assembly systems operating at
temperature 2
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