We show that for any class of Boolean algebras with an associative operator, if it contains the complex algebra of $(\mathcal {P}(\mathbb {N}), \cup )$, its equational theory is undecidable. Equivalently, any associative normal modal logic valid over the frame $(\mathcal {P}(\mathbb {N}), \cup )$ is undecidable. This settles a long-open question on the decidability of hyperboolean modal logic [16], and addresses several related problems.

In this article, we investigate the possibility of generating all the configurations of a subshift in a local way. We propose two definitions of local generation, explore their properties and develop techniques to determine whether a subshift satisfies these definitions. We illustrate the results with several examples.

For a given graph $H$, we say that a graph $G$ has a perfect $H$-subdivision tiling if $G$ contains a collection of vertex-disjoint subdivisions of $H$ covering all vertices of $G.$ Let $\delta _{\mathrm {sub}}(n, H)$ be the smallest integer $k$ such that any $n$-vertex graph $G$ with minimum degree at least $k$ has a perfect $H$-subdivision tiling. For every graph $H$, we asymptotically determined the value of $\delta _{\mathrm {sub}}(n, H)$. More precisely, for every graph $H$ with at least one edge, there is an integer $\mathrm {hcf}_{\xi }(H)$ and a constant $1 \lt \xi ^*(H)\leq 2$ that can be explicitly determined by structural properties of $H$ such that $\delta _{\mathrm {sub}}(n, H) = \left (1 - \frac {1}{\xi ^*(H)} + o(1) \right )n$ holds for all $n$ and $H$ unless $\mathrm {hcf}_{\xi }(H) = 2$ and $n$ is odd. When $\mathrm {hcf}_{\xi }(H) = 2$ and $n$ is odd, then we show that $\delta _{\mathrm {sub}}(n, H) = \left (\frac {1}{2} + o(1) \right )n$.

We use a special tiling for the hyperbolic d-space $\mathbb {H}^d$ for $d=2,3,4$ to construct an (almost) explicit isomorphism between the Lipschitz-free space $\mathcal {F}(\mathbb {H}^d)$ and $\mathcal {F}(P)\oplus \mathcal {F}(\mathcal {N})$, where P is a polytope in $\mathbb {R}^d$ and $\mathcal {N}$ a net in $\mathbb {H}^d$ coming from the tiling. This implies that the spaces $\mathcal {F}(\mathbb {H}^d)$ and $\mathcal {F}(\mathbb {R}^d)\oplus \mathcal {F}(\mathcal {M})$ are isomorphic for every net $\mathcal {M}$ in $\mathbb {H}^d$. In particular, we obtain that, for $d=2,3,4$, $\mathcal {F}(\mathbb {H}^d)$ has a Schauder basis. Moreover, using a similar method, we also give an explicit isomorphism between $\mathrm {Lip}(\mathbb {H}^d)$ and $\mathrm {Lip}(\mathbb {R}^d)$.

We demonstrate that every difference set in a finite Abelian group is equivalent to a certain ‘regular’ covering of the lattice $ A_n = \{ \boldsymbol {x} \in \mathbb {Z} ^{n+1} : \sum _{i} x_i = 0 \} $ with balls of radius $ 2 $ under the $ \ell _1 $ metric (or, equivalently, a covering of the integer lattice $ \mathbb {Z} ^n $ with balls of radius $ 1 $ under a slightly different metric). For planar difference sets, the covering is also a packing, and therefore a tiling, of $ A_n $. This observation leads to a geometric reformulation of the prime power conjecture and of other statements involving Abelian difference sets.

We study the repetition of patches in self-affine tilings in ${\mathbb {R}}^d$. In particular, we study the existence and non-existence of arithmetic progressions. We first show that an arithmetic condition of the expansion map for a self-affine tiling implies the non-existence of certain one-dimensional arithmetic progressions. Next, we show that the existence of full-rank infinite arithmetic progressions, pure discrete dynamical spectrum, and limit-periodicity are all equivalent for a certain class of self-affine tilings. We finish by giving a complete picture for the existence or non-existence of full-rank infinite arithmetic progressions in the self-similar tilings in ${\mathbb {R}}^d$.

It was recently shown in Gaidashev and Yampolsky [Golden mean Siegel disk universality and renormalization. Preprint, 2016, arXiv:1604.00717] that appropriately defined renormalizations of a sufficiently dissipative golden-mean semi-Siegel Hénon map converge super-exponentially fast to a one-dimensional renormalization fixed point. In this paper, we show that the asymptotic two-dimensional form of these renormalizations is universal and is parameterized by the average Jacobian. This is similar to the limit behavior of period-doubling renormalizations in the Hénon family considered in de Carvalho et al [Renormalization in the Hénon family, I: universality but non-rigidity. J. Stat. Phys.121 (5/6) (2006), 611–669]. As an application of our result, we prove that the boundary of the golden-mean Siegel disk of a dissipative Hénon map is non-smoothly rigid.

A one-dimensional tiling is a bi-infinite string on a finite alphabet, and its tiling semigroup is an inverse semigroup whose elements are marked finite substrings of the tiling. We characterize the structure of these semigroups in the periodic case, in which the tiling is obtained by repetition of a fixed primitive word.

An n-hedral tiling of ℝd is a tiling with each tile congruent to one of the n distinct sets. In this paper, we use the iterated function systems (IFS) to generate n-hedral tilings of ℝd. Each tile in the tiling is similar to the attractor of the IFS. These tiles are fractals and their boundaries have the Hausdorff dimension less than d. Our results generalize a result of Bandt.

This paper describes the development of a system that is capable of tiling mosaics using a robot according to requirements of the individual customer. The art of mosaic, in one form or another, has been practiced manually for thousands of years. Modern developments in materials and production techniques could be found as evidence that mosaic is very much alive in the new millennium. It is very costly and also difficult to find skilful people to tile mosaic. Therefore a computer-assisted robotic system has been constructed and applied. To activate the system a particular algorithm has been developed and successfully applied on an available SCARA robot.