Related papers: On the structure of Ammann A2 tilings
Surfaces in i-Al68Pd23Mn9 as observed with STM and LEED experiments show atomic terraces in a Fibonacci spacing. We analyze them in a bulk tiling model due to Elser which incorporates many experimental data. The model has dodecahedral…
We study the structure of families of theories in the language of arithmetic extended to allow these families to refer to one another and to themselves. If a theory contains schemata expressing its own truth and expressing a specific Turing…
The Abelian Sandpile Model, seen as a deterministic lattice automaton, on two-dimensional periodic graphs generates complex regular patterns displaying (fractal) self-similarity. In particular, on a variety of lattices and initial…
This essay aims to propose construction theory, a new domain of theoretical research on machine construction, and use it to shed light on a fundamental relationship between living and computational systems. Specifically, we argue that…
We prove a version of uniqueness theorem for Cuntz-Pimsner algebras of discrete product systems over semigroups of Ore type. To this end, we introduce Doplicher-Roberts picture of Cuntz-Pimsner algebras, and the semigroup dual to a product…
Let $M$ be a complete, connected Riemannian surface and suppose that $\mathcal{S} \subset M$ is a discrete subset. What can we learn about $M$ from the knowledge of all distances in the surface between pairs of points of $\mathcal{S}$? We…
We present a technique to lift some tilings of the discrete hyperbolic plane -- tilings defined by a 1D substitution -- into a zero entropy subshift of finite type (SFT) on non-abelian amenable Baumslag-Solitar groups $BS(1,n)$ for…
We find explicit optimal vertex, edge and face coulourings for the chair tiling, the Ammann--Beenker tiling, the rational pinwheel tiling and the pinwheel tiling.
Can the entire plane be paved with a single tile that forces aperiodicity? This is known as the ein Stein problem (in German, ein Stein means one tile). This paper presents a monotile that delivers aperiodic tiling by design. It is based on…
We develop a general framework of Euclidean patterns and pattern spaces of translational finite local complexity (FLC), analogues of translational tiling spaces. The notion of a self affine substitution of tilings is extended to both…
We establish two versions of a central theorem, the Family Colimit Theorem, for the coarse coherence property of metric spaces. This is a coarse geometric property and so is well-defined for finitely generated groups with word metrics. It…
We study asymptotic properties of the modular representation theory of symmetric groups and investigate modular analogs of stabilization phenomena in characteristic zero. The main results are equivalences of categories between certain…
The consistency problem for a class of algebraic structures asks for an algorithm to decide for any given conjunction of equations whether it admits a non-trivial satisfying assignment within some member of the class. By Adyan (1955) and…
To initiate a systematic study on the applications of perfectoid methods to Noetherian rings, we introduce the notions of perfectoid towers and their tilts. We mainly show that the tilting operation preserves several homological invariants…
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…
A finite set of integers $A$ tiles the integers by translations if $\mathbb{Z}$ can be covered by pairwise disjoint translated copies of $A$. Restricting attention to one tiling period, we have $A\oplus B=\mathbb{Z}_M$ for some…
In this paper we propose two guiding principles that suggest a number of conjectures (some now proved) about various forms of rigidity for moduli spaces arising in algebraic geometry. Such conjectures have group-theoretic, topological and…
Recently Taylor and Socolar introduced an aperiodic mono-tile. The associated tiling can be viewed as a substitution tiling. We use the substitution rule for this tiling and apply the algorithm of \cite{AL} to check overlap coincidence. It…
We prove a new kind of homological stability theorem for automorphism groups of finitely-generated projective modules over Dedekind domains, which takes into account all possible stabilisation maps between these, rather than only…
This paper establishes combinatorial characterisations of forced-symmetric and forced-periodic rigidity (under a fixed lattice) of bar-joint frameworks in non-Euclidean normed planes. In $\ell_q$-planes for $q\in(1,\infty)\backslash\{2\}$,…