Related papers: Selfdual Substitutions in Dimension One
It is shown that there are primitive substitution tilings with dense tile orientations invariant under n-fold rotation for n=2,3,4,5,6,8. The proof for dense tile orientations uses a general result about irrationality of angles in certain…
There are several classical characterisations of the valuative dimension of a commutative ring. Constructive versions of this dimension have been given and proven to be equivalent to the classical notion within classical mathematics, and…
We define the notion of duality categories as generalization of duality groups. Two examples are treated. The first is the Serre duality in the categories of strict polynomial functors. The second concerns finite complexes. We show in…
I argue that, on a judicious reading of two existing criteria--one syntactic and the other semantic--dual theories can be taken to be empirically equivalent. The judicious reading is straightforward, but leads to the surprising conclusion…
In this paper we propose a research programme for getting structural characterisations for 2-dimensional languages generated by self-assembling tiles. This is part of a larger programme on getting a formal foundation of parallel,…
It is often claimed [PST1] that the (Hodge type) duality operation is defined only in even dimensional spacetimes and that self-duality is further restricted to twice-odd dimensional spacetime theories. The purpose of this paper is to…
We study the problem of perfect tiling in the plane and exploring the possibility of tiling a rectangle using integral distinct squares. Assume a set of distinguishable squares (or equivalently a set of distinct natural numbers) is given,…
The subject of this paper is a simulation to that in [1] but here we consider substitutions corresponding to transpositions instead of replacements.
We associate in a canonical way a substitution to any abstract numeration system built on a regular language. In relationship with the growth order of the letters, we define the notion of two independent substitutions. Our main result is…
We obtain structural results on translational tilings of periodic functions in $\mathbb{Z}^d$ by finite tiles. In particular, we show that any level one tiling of a periodic set in $\mathbb{Z}^2$ must be weakly periodic (the disjoint union…
A recent elegant result of Akrobotu et al. states that a triangulation of any convex polyomino is word-representable if and only if it is 3-colorable. In this paper, we generalize a particular case of this result by showing that the result…
Several old and recent classes of picture grammars, that variously extend context-free string grammars in two dimensions, are based on rules that rewrite arrays of pixels. Such grammars can be unified and extended using a tiling based…
The question of whether all words in two real positive definite letters have only positive eigenvalues is addressed and settled (negatively). This question was raised some time ago in connection with a long-standing problem in theoretical…
Let $S$ be a set of arbitrary objects, and let $s\mapsto s'$ be a permutation of $S$ such that $s"=(s')'=s$ and $s'\neq s$. Let $S^d=\{v_1...v_d\colon v_i\in S\}$. Two words $v,w\in S^d$ are dichotomous if $v_i=w'_i$ for some $i\in [d]$,…
I summarize and discuss some recent results on formulating actions of six-dimensional superconformal field theories using the language of higher gauge theory. The latter guarantees mathematical consistency of our constructions and we review…
In this note, we establish a duality result under the residue paring between certain two-dimensional adelic spaces, which are associated to a closed point on an arithmetic surface.
Given a finite collection of two-dimensional tile types, the field of study concerned with covering the plane with tiles of these types exclusively has a long history, having enjoyed great prominence in the last six to seven decades. Much…
We generalize the notion of (geometric) substitution rule to obtain overlapping substitutions. Our motivating example is the substitution presented in Ziherl, Dotera and Bekku \cite{DBZ}, which features a substitution matrix with…
Siegel's action is generalized to the D=2(p+1) (p even) dimensional space-time. The investigation of self-duality of chiral p-forms is extended to the momentum frame, using Siegel's action of chiral bosons in two space-time dimensions and…
Substitutions play a crucial role in a wide range of contexts, from analyzing the dynamics of social opinions and conducting mathematical computations to engaging in game-theoretical analysis. For many situations, considering one-step…