955 research outputs found
Tiling Problems on Baumslag-Solitar groups
We exhibit a weakly aperiodic tile set for Baumslag-Solitar groups, and prove
that the domino problem is undecidable on these groups. A consequence of our
construction is the existence of an arecursive tile set on Baumslag-Solitar
groups.Comment: In Proceedings MCU 2013, arXiv:1309.104
Fixed Parameter Undecidability for Wang Tilesets
Deciding if a given set of Wang tiles admits a tiling of the plane is
decidable if the number of Wang tiles (or the number of colors) is bounded, for
a trivial reason, as there are only finitely many such tilesets. We prove
however that the tiling problem remains undecidable if the difference between
the number of tiles and the number of colors is bounded by 43.
One of the main new tool is the concept of Wang bars, which are equivalently
inflated Wang tiles or thin polyominoes.Comment: In Proceedings AUTOMATA&JAC 2012, arXiv:1208.249
Aperiodic Subshifts of Finite Type on Groups
In this note we prove the following results:
If a finitely presented group admits a strongly aperiodic SFT,
then has decidable word problem. More generally, for f.g. groups that are
not recursively presented, there exists a computable obstruction for them to
admit strongly aperiodic SFTs.
On the positive side, we build strongly aperiodic SFTs on some new
classes of groups. We show in particular that some particular monster groups
admits strongly aperiodic SFTs for trivial reasons. Then, for a large class of
group , we show how to build strongly aperiodic SFTs over . In particular, this is true for the free group with 2 generators,
Thompson's groups and , and any f.g. group of
rational matrices which is bounded.Comment: New version. Adding results about monster group
Microstructural enrichment functions based on stochastic Wang tilings
This paper presents an approach to constructing microstructural enrichment
functions to local fields in non-periodic heterogeneous materials with
applications in Partition of Unity and Hybrid Finite Element schemes. It is
based on a concept of aperiodic tilings by the Wang tiles, designed to produce
microstructures morphologically similar to original media and enrichment
functions that satisfy the underlying governing equations. An appealing feature
of this approach is that the enrichment functions are defined only on a small
set of square tiles and extended to larger domains by an inexpensive stochastic
tiling algorithm in a non-periodic manner. Feasibility of the proposed
methodology is demonstrated on constructions of stress enrichment functions for
two-dimensional mono-disperse particulate media.Comment: 27 pages, 12 figures; v2: completely re-written after the first
revie
Quasiperiodicity and non-computability in tilings
We study tilings of the plane that combine strong properties of different
nature: combinatorial and algorithmic. We prove existence of a tile set that
accepts only quasiperiodic and non-recursive tilings. Our construction is based
on the fixed point construction; we improve this general technique and make it
enforce the property of local regularity of tilings needed for
quasiperiodicity. We prove also a stronger result: any effectively closed set
can be recursively transformed into a tile set so that the Turing degrees of
the resulted tilings consists exactly of the upper cone based on the Turing
degrees of the later.Comment: v3: the version accepted to MFCS 201
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