Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userCombinatorial substitutions and sofic tilings
152
0
0.0
(
0
)
Added by
Thomas Fernique
Publication date
2010
fields
Informatics Engineering
and research's language is
English
Authors
Thomas Fernique
Ask ChatGPT about the research
No Arabic abstract
A combinatorial substitution is a map over tilings which allows to define sets of tilings with a strong hierarchical structure. In this paper, we show that such sets of tilings are sofic, that is, can be enforced by finitely many local constraints. This extends some similar previous results (Mozes90, Goodman-Strauss98) in a much shorter presentation.
rate research
Read More
An aperiodic tile set was first constructed by R.Berger while proving the undecidability of the domino problem. It turned out that aperiodic tile sets appear in many topics ranging from logic (the Entscheidungsproblem) to physics (quasicrystals) We present a new construction of an aperiodic tile set that is based on Kleenes fixed-point construction instead of geometric arguments. This construction is similar to J. von Neumann self-reproducing automata; similar ideas were also used by P. Gacs in the context of error-correcting computations. The flexibility of this construction allows us to construct a robust aperiodic tile set that does not have periodic (or close to periodic) tilings even if we allow some (sparse enough) tiling errors. This property was not known for any of the existing aperiodic tile sets.
We introduce a counterpart to the notion of vertex disjoint tilings by copy of a fixed graph F to the setting of graphons. The case F=K_2 gives the notion of matchings in graphons. We give a transference statement that allows us to switch between the finite and limit notion, and derive several favorable properties, including the LP-duality counterpart to the classical relation between the fractional vertex covers and fractional matchings/tilings, and discuss connections with property testing. As an application of our theory, we determine the asymptotically almost sure F-tiling number of inhomogeneous random graphs mathbb{G}(n,W). As another application, in an accompanying paper [Hladky, Hu, Piguet: Komloss tiling theorem via graphon covers, preprint] we give a proof of a strengthening of a theorem of Komlos [Komlos: Tiling Turan Theorems, Combinatorica, 2000].
We describe limits of line bundles on nodal curves in terms of toric arrangements associated to Voronoi tilings of Euclidean spaces. These tilings encode information on the relationship between the possibly infinitely many limits, and ultimately give rise to a new definition of limit linear series. This paper and its second and third companion parts are the first in a series aimed to explore this new approach. In the present article, we set up the combinatorial framework and show how graphs with integer lengths associated to the edges provide tilings of Euclidean spaces by certain polytopes associated to the graph itself and to certain of its subgraphs. We further provide a description of the combinatorial structure of these polytopes and the way they are glued together in the tiling. In the second part of the series, we describe the arrangements of toric varieties associated to these tilings. These results will be of use in the third part to achieve our goal of describing all stable limits of a family of line bundles along a degenerating family of curves.
Combinatorial algorithms such as those that arise in graph analysis, modeling of discrete systems, bioinformatics, and chemistry, are often hard to parallelize. The Combinatorial BLAS library implements key computational primitives for rapid development of combinatorial algorithms in distributed-memory systems. During the decade since its first introduction, the Combinatorial BLAS library has evolved and expanded significantly. This paper details many of the key technical features of Combinatorial BLAS version 2.0, such as communication avoidance, hierarchical parallelism via in-node multithreading, accelerator support via GPU kernels, generalized semiring support, implementations of key data structures and functions, and scalable distributed I/O operations for human-readable files. Our paper also presents several rules of thumb for choosing the right data structures and functions in Combinatorial BLAS 2.0, under various common application scenarios.
A semi-regular tiling of the hyperbolic plane is a tessellation by regular geodesic polygons with the property that each vertex has the same vertex-type, which is a cyclic tuple of integers that determine the number of sides of the polygons surrounding the vertex. We determine combinatorial criteria for the existence, and uniqueness, of a semi-regular tiling with a given vertex-type, and pose some open questions.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
