| dc.description.abstract | This habilitation thesis combines two fields of mathematics – the main one is Combinatorics on
Words and combinatorial results are applied in Random Number Generation. It is written in
the form of a collection of papers – all of them besides the last one have been already published
in scientific journals, the last one has been submitted. The collection is accompanied with a text
presenting in a nutshell and at the same time in a way comprehensive also for non-specialists
the content of the papers. The following six papers have been chosen
1. Sturmian Jungle (or Garden?) on Multiliteral Alphabets [8];
2. Infinite Words with Finite Defect [9];
3. On the Brlek–Reutenauer Conjecture [10];
4. Proof of the Brlek–Reutenauer Conjecture [11];
5. Infinite Words with Well Distributed Occurrences [5];
6. Pseudorandom Number Generators Based on Infinite Words [6]
in order to draw a picture that is self-consistent and illustrates very well the steps done by
the author in the last five years in her research. The first paper is purely combinatorial, it is
devoted to the study of a famous class of infinite words – Sturmian words – and in particular
to their generalization to multiliteral alphabets. Combinatorics on Words is treated here from
many perspectives and the topic of richness in palindromes is opened already in this study. The
following three papers study palindromes in infinite words, in particular, they deal with richness
and defect of infinite words. Recently, this subject has been very popular in Combinatorics on
Words and several research groups have contributed to its deeper exploration. One of our main
contribution is the proof of a conjecture stated by Brlek and Reutenauer. In the papers 2., 3.,
and 4. our thorny, but successful path to its complete proof is described. The last two papers
contain applications of Combinatorics on Words in Random Number Generation. We study
there a new property of infinite words – baptized well distributed occurrences – that was born
in connection with random number generation. It is a sufficient condition for absence of the
lattice structure when pseudorandom number generators are mixed according to an infinite word.
We succeeded to prove aperiodicity and absence of the lattice structure for arising generators,
however, their statistical testing shows – to our pleasure – that such generators are almost as fast
as the original generators and their other statistical properties resemble much more to a random
sequence in comparison to the original generators. This means that such generators are suitable
for use in practice.
Results of all above six papers were presented by the habilitation candidate in the form of
lectures at international conferences: Combinatorial generalizations of Sturmian words from the
first paper were introduced at NORCOM 2010 (the 10th Nordic Combinatorial Conference) in Reykjav´ık, Island, in May 2010, and at Joint Mathematical Conference CSASC 2010, session
Discrete Dynamical Systems, in Prague, in January 2010. The results on richness, defect, and the
Brlek–Reutenauer conjecture (items 2., 3., 4. of the above list) were presented at Cesko-slovensk´a ˇ
MELA (Meeting on Languages) in Telˇc as an invited lecture in September 2012, at RuFiDim
(Russian Finnish Symposium on Discrete Mathematics) in Turku, Finland, in September 2012,
and at Workshop on Challenges in Combinatorics on Words in Toronto, Canada, in April 2013.
The talks on pseudorandom number generators based on infinite words with well distributed
occurrences were given at WORDS 2013 in Turku, Finland, in September 2013, and at Cesko- ˇ
slovensk´a MELA in Telˇc in September 2013. Moreover, a lecture on application of Combinatorics
on Words in Random Number Generation was held in April 2014 within a research stay at the
University of Oulu, Finland. Last but not least, in 2013 the candidate was laureate of the award
L’Or´eal UNESCO For Women in Science for the project Application of Combinatorics on Words
in Cryptology that was closely related to the results summarized in the last two papers. | cze |
