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Piecewise Affine Functions, Sturmian Sequences and Wang Tiles

Kari J.

Fundamenta Informaticae

2016

Vol. 145, nr 3

257--277

The tiling problem is the decision problem to determine if the infinite plane can be tiled by copies of finitely many given Wang tiles. The problem is known since the 1960's to be undecidable. The undecidability is closely related to the existence of aperiodic Wang tile sets. There is a known method to construct small aperiodic tile sets that simulate iterations of one-dimensional piecewise linear functions using encodings of real numbers as Sturmian sequences. In this paper we provide details of a similar simulation of two-dimensional piecewise affine functions byWang tiles. Mortality of such functions is undecidable, which directly yields another proof of the undecidability of the tiling problem. We apply the same technique on the hyperbolic plane to provide a strongly aperiodic hyperbolic Wang tile set and to prove that the hyperbolic tiling problem is undecidable. These results are known in the literature but using different methods.

Remarks about discrete Young measures and their Monte Carlo simulation

Grzybowski A., Puchała P.

Journal of Applied Mathematics and Computational Mechanics

2015

Vol. 14, nr 2

13--20

This article is devoted to the problem of simulation of random variables distributed according to Young measures associated with piecewise affine functions determined on bounded intervals. We start with simple functions which can take on a finite number of different values with inverse images being the intervals or their unions. We present some formal results connected with related discrete Young measures and propose an algorithm for generating random variables having such distributions. Next, based on these results we introduce an algorithm designed for approximation of Young measures in various, more general situations. We also present an example where a Young measure associated with a piecewise affine function is approximated with the help of computer simulations. In this benchmarking problem the theoretical results are compared with the ones obtained in the Monte Carlo experiment.