On the entropy and letter frequencies of ternary square-free words

We enumerate all ternary length-1 square-free words, which are words avoiding squares of words up to length 1, for 1<=24. We analyse the singular behaviour of the corresponding generating functions. This leads to new upper entropy bounds for ternary square-free words. We then consider ternary square-free words with fixed letter densities, thereby proving exponential growth for certain ensembles with various letter densities. We derive consequences for the free energy and entropy of ternary square-free words

A family of formulas with reversal of high avoidability index

We present an infinite family of formulas with reversal whose avoidability index is bounded between 4 and 5, and we show that several members of the family have avoidability index 5. This family is particularly interesting due to its size and the simple structure of its members. For each k ∈ {4,5}, there are several previously known avoidable formulas (without reversal) of avoidability index k, but they are small in number and they all have rather complex structure.http://dx.doi.org/10.1142/S021819671750024

Growth Rates of Complexity of Power-Free Languages

We present a new fast algorithm for calculating the growth rate of complexity for regular languages. Using this algorithm we develop a space and time efficient method to approximate growth rates of complexity of arbitrary power-free languages over finite alphabets. Through extensive computer-assisted studies we sufficiently improve all known upper bounds for growth rates of such languages, obtain a lot of new bounds and discover some general regularities. © 2010 Elsevier B.V

Growth Rates of Complexity of Power-Free Languages

We present a new fast algorithm for calculating the growth rate of complexity for regular languages. Using this algorithm we develop a space and time efficient method to approximate growth rates of complexity of arbitrary power-free languages over finite alphabets. Through extensive computer-assisted studies we sufficiently improve all known upper bounds for growth rates of such languages, obtain a lot of new bounds and discover some general regularities. © 2010 Elsevier B.V

Combinatorial aspects of root lattices and words

This thesis is concerned with two topics that are of interest for the theory of aperiodic order. In the first part, the similar sublattices and coincidence site lattices of the root lattice A4 are analysed by means of a particular quaternion algebra. Dirichlet series generating functions are derived, which count the number of similar sublattices, respectively coincidence site lattices, of each index. In the second part, several strategies to derive upper and lower bounds for the entropy of certain sets of powerfree words are presented. In particular, Kolpakov's arguments for the derivation of lower bounds for the entropy of powerfree words are generalised. For several explicit sets we derive very good upper and lower bounds for their entropy. Notably, Kolpakov's lower bounds for the entropy of ternary squarefree, binary cubefree and ternary minimally repetitive words are confirmed exactly

Errata and Addenda to Mathematical Constants

We humbly and briefly offer corrections and supplements to Mathematical Constants (2003) and Mathematical Constants II (2019), both published by Cambridge University Press. Comments are always welcome.Comment: 162 page