Consistency of multidimensional combinatorial substitutions

Consistency of Multidimensional Combinatorial Substitutions

Multidimensional combinatorial substitutions are rules that replace symbols by finite patterns of symbols in Zd. We focus on the case where the patterns are not necessarily rectangular, which requires a specific description of the way they are glued together in the image by a substitution. Two problems can arise when defining a substitution in such a way: it can fail to be consistent, and the p...

Substitutions on Multidimensional Sequen

Combinatorial Substitutions and Sofic Tilings

A combinatorial substitution is a map over tilings which allows to define sets of tilings with a strong hierarchical structure. In this paper, we show that such sets of tilings are sofic, that is, can be enforced by finitely many local constraints. This extends some similar previous results (Mozes’90, GoodmanStrauss’98) in a much shorter presentation.

Consistency of Multidimensional Convex Regression

Convex regression is concerned with computing the best fit of a convex function to a data set of n observations in which the independent variable is (possibly) multi–dimensional. Such regression problems arise in operations research, economics, and other disciplines in which imposing a convexity constraint on the regression function is natural. This paper studies a least squares estimator that ...

Multidimensional Euclidean algorithms, numeration and substitutions

The aim of this survey is to discuss multidimensional continued fraction and Euclidean algorithms from the viewpoint of numeration systems, substitutions, and the symbolic dynamical systems they generate. We will mainly focus on two types of multidimensional algorithms, namely, unimodular Markovian ones which include the most classical ones like e.g. Jacobi-Perron algorithm, and algorithms issu...