Related papers: Tiling Lattices with Sublattices, I

We describe how the usual supercharges of extended supersymmetry may be {\it twisted} to produce a BRST-like supercharge $Q$. The usual supersymmetry algebra is then replaced by a twisted algebra and the action of the twisted theory is…

Lattice simulation of supersymmetric gauge theories is not straightforward. In some cases the lack of manifest supersymmetry just necessitates cumbersome fine-tuning, but in the worse cases the chiral and/or Majorana nature of fermions…

We show that all balanced d-lattices must be complemented, answering a question of Chajda and Eigenthaler. (A bounded lattice is balanced if any two congruences agree on their 1-classes iff they agree on their 0-classes.) Our main tool is…

This paper is concerned with realizing Lattes maps as subdivision maps of finite subdivision rules. The main result is that the Lattes maps in all but finitely many analytic conjugacy classes can be realized as subdivision maps of finite…

We study the equational theory of the Weihrauch lattice with multiplication, meaning the collection of equations between terms built from variables, the lattice operations $\sqcup$, $\sqcap$, the product $\times$, and the finite…

Motivated by Iyama's higher representation theory, we introduce $n$-translation quivers and $n$-translation algebras. The classical $\mathbb Z Q$ construction of the translation quiver is generalized to construct an $(n+1)$-translation…

For a positive integer $s$, a lattice $L$ is said to be $s$-integrable if $\sqrt{s}\cdot L$ is isometric to a sublattice of $\mathbb{Z}^n$ for some integer $n$. Conway and Sloane found two minimal non $2$-integrable lattices of rank $12$…

A dyadic tile of order n is any rectangle obtained from the unit square by n successive bisections by horizontal or vertical cuts. Let each dyadic tile of order n be available with probability p, independently of the others. We prove that…

Given a graph $G$ and collection of subgraphs $T$ (called tiles), we consider covering $G$ with copies of tiles in $T$ so that each vertex $v\in G$ is covered with a predetermined multiplicity. The multinomial tiling model is a natural…

We construct a unilateral lattice tiling of $\mathbb{R}^n$ into hypercubes of two differnet side lengths $p$ or $q$. This generalizes the Pythagorean tiling in $\mathbb{R}^2$. We also show that this tiling is unique up to symmetries, which…

In this work, we introduce a definition of the Discrete Fourier Transform (DFT) on Euclidean lattices in $\R^n$, that generalizes the $n$-th fold DFT of the integer lattice $\Z^n$ to arbitrary lattices. This definition is not applicable for…

We give an explicit formulae for obtaining the translation symmetries in the cartesian product $X^N$, where $N$ is some positive integer and $X$ is some finite set. Moreover, we obtain some fundamental results from elementary number theory.

In this study we follow a new framework for the theory that offers us, other than traditional, a new angle to observe and investigate some relations between finite sets, F-lattice L and their elements. The theory is based on the Fuzzy…

In this article we introduce theory and algorithms for learning discrete representations that take on a lattice that is embedded in an Euclidean space. Lattice representations possess an interesting combination of properties: a) they can be…

The classical Chung-Feller theorem [2] tells us that the number of Dyck paths of length $n$ with $m$ flaws is the $n$-th Catalan number and independent on $m$. In this paper, we consider the refinements of Dyck paths with flaws by four…

Looking at some monoids and (semi)rings (natural numbers, integers and p-adic integers), and more generally, residually finite algebras (in a strong sense), we prove the equivalence of two ways for a function on such an algebra to behave…

In a capacitated directed graph, it is known that the set of all min-cuts forms a distributive lattice [1], [2]. Here, we describe this lattice as a regular predicate whose forbidden elements can be advanced in constant parallel time after…

We consider the ring $\mathbb Z_n$ (integers modulo $n$) with the partial order `$\leq$' given by `$a \leq b$ if either $a=b$ or $a\equiv ab~(mod~n)$'. In this paper, we obtain necessary and sufficient conditions for the poset ($\mathbb…

We prove that supports of a wide class of temperate distributions with uniformly discrete support and spectrum on Euclidean spaces are finite unions of translations of full-rank lattices. This result is a generalization of the corresponding…

Consider lattice paths in Z^2 taking unit steps north (N) and east (E). Fix positive integers r,s and put an equivalence relation on points of Z^2 by letting v,w be equivalent if v - w = m (r,s) for some m in Z. Call a lattice path valid if…