Related papers: Canonical projection tilings defined by patterns
The category of finite dimensional (type 1) representations of a quantum affine algebra $U_q(\hat{{\mathfrak g}})$ is not semisimple. However, as any abelian category with finite-length objects, it admits a unique decomposition into a…
We give short and simple proofs of what seem to be folklore results: * the maximum cardinality of the intersection of a lattice cube with an affine subspace; * the minimum number of affine subspaces needed to cover a lattice cube.
An arc space of an affine cone over a projective toric variety is known to be non-reduced in general. It was demonstrated recently that the reduced scheme structure is worth studying due to various connections with representation theory and…
An algebraic scheme is suggested in which discretized spacetime turns out to be a quantum observable. As an example, a toy model producing spacetimes of four points with different topologies is presented. The possibility of incorporating…
This article considers some affine algebraic varieties attached to finite trees and closely related to cluster algebras. Their definition involves a canonical coloring of vertices of trees into three colors. These varieties are proved to be…
In the present paper, as we did previously in [7], we investigate the relations between the geometric properties of tilings and the algebraic properties of associated relational structures. Our study is motivated by the existence of…
This paper explores in some detail a recent proposal (the Rieffel induction/refined algebraic quantization scheme) for the quantization of constrained gauge systems. Below, the focus is on systems with a single constraint and, in this…
A quandle will be called quasi-affine, if it embeds into an affine quandle. Our main result is a characterization of quasi-affine quandles, by group-theoretic properties of their displacement group, by a universal algebraic condition coming…
We prove a conjecture of Kleinbock which gives a clear-cut classification of all extremal affine subspaces of $\mathbb{R}^n$. We also give an essentially complete classification of all Khintchine type affine subspaces, except for some…
A basic assumption of tiling theory is that adjacent tiles can meet in only a finite number of ways, up to rigid motions. However, there are many interesting tiling spaces that do not have this property. They have "fault lines", along which…
A set is said to tile the integers if and only if the integers can be written as a disjoint union of translates of that set. We consider the problem of finding necessary and sufficient conditions for a finite set to tile the integers. For…
We consider all compatible topologies of an arbitrary finite-dimensional vector space over a non-trivial valuation field whose metric completion is a locally compact space. We construct the canonical lattice isomorphism between the lattice…
Affine type systems are substructural type systems where copying of information is restricted, but discarding of information is permissible at all types. Such type systems are well-suited for describing quantum programming languages,…
In this paper we study certain homogeneous spaces, which we call generalised affine Stiefel algebraic varieties. The main aim is to characterise the canonical divisor of generalised affine Stiefel algebraic varieties in terms of group…
Cut and project sets are obtained by projecting an irrational slice through a lattice to a lower dimensional subspace. Under standard conditions, the resulting pattern has no translational periods even though it retains some regularity of…
We associate to each automorphism of the plane, a geometric construction with some properties, it is the {\it{canonical resolution}}. We study the geometry of the canonical resolution, we deduce from it an upper bound for a geometric…
Recent work by Forsg{\aa}rd indicates that not every convex lattice polygon arises as the characteristic polygon of an affine dimer or, equivalently, an admissible oriented line arrangement on the torus in general position. We begin the…
We address an apparent conflict between the traditional canonical quantization framework of quantum theory and the spatially restricted quantum dynamics, when the translation invariance of the otherwise free quantum system is broken by…
The similarity between classical and quantum physics is large enough to make an investigation of quantization methods a worthwhile endeavour. As history has shown, Dirac's canonical quantization method works reasonably well in the case of…
Positive configurations of points in the affine building were introduced in \cite{Le} as the basic object needed to define higher laminations. We start by giving a self-contained, elementary definition of positive configurations of points…