93 research outputs found
Quasiperiodicity and non-computability in tilings
We study tilings of the plane that combine strong properties of different
nature: combinatorial and algorithmic. We prove existence of a tile set that
accepts only quasiperiodic and non-recursive tilings. Our construction is based
on the fixed point construction; we improve this general technique and make it
enforce the property of local regularity of tilings needed for
quasiperiodicity. We prove also a stronger result: any effectively closed set
can be recursively transformed into a tile set so that the Turing degrees of
the resulted tilings consists exactly of the upper cone based on the Turing
degrees of the later.Comment: v3: the version accepted to MFCS 201
A new upper bound for the cross number of finite Abelian groups
In this paper, building among others on earlier works by U. Krause and C.
Zahlten (dealing with the case of cyclic groups), we obtain a new upper bound
for the little cross number valid in the general case of arbitrary finite
Abelian groups. Given a finite Abelian group, this upper bound appears to
depend only on the rank and on the number of distinct prime divisors of the
exponent. The main theorem of this paper allows us, among other consequences,
to prove that a classical conjecture concerning the cross and little cross
numbers of finite Abelian groups holds asymptotically in at least two different
directions.Comment: 21 pages, to appear in Israel Journal of Mathematic
Problem Solving Using Process Algebra Considered Insightful
Process algebras with data, such as LOTOS, PSF, FDR, and mCRL2, are very suitable to model and analyse combinatorial problems. Contrary to more traditional mathematics, many of these problems can very directly be formulated in process algebra. Using a wide range of techniques, such as behavioural reductions, model checking, and visualisation, the problems can subsequently be easily solved. With the advent of probabilistic process algebras this also extends to problems where probabilities play a role. In this paper we model and analyse a number of very well-known – yet tricky – problems and show the elegance of behavioural analysis
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