The most unbalanced words 0q−p1p and majorization

A finite word w &isin; {0, 1}&lowast; is balanced if for every equal-length factors u and v of every cyclic shift of w we have ||u|1 &minus; |v|1| &le; 1. This new class of finite words was defined in [O. Jenkinson and L. Q. Zamboni, Characterisations of balanced words via orderings, Theoret. Comput. Sci. 310(1&ndash;3) (2004) 247&ndash;271]. In [O. Jenkinson, Balanced words and majorization, Discrete Math. Algorithms Appl. 1(4) (2009) 463&ndash;484], there was proved several results considering finite balanced words and majorization. One of the main results was that the base-2 orbit of the balanced word is the least element in the set of orbits with respect to partial sum. It was also proved that the product of the elements in the base-2 orbit of a word is maximized precisely when the word is balanced. It turns out that the words 0q&minus;p1p have similar extremal properties, opposite to the balanced words, which makes it meaningful to call these words the most unbalanced words. This paper contains the counterparts of the results mentioned above. We will prove that the orbit of the word u = 0q&minus;p1p is the greatest element in the set of orbits with respect to partial sum and that it has the smallest product. We will also prove that u is the greatest element in the set of orbits with respect to partial product.</p

Optimizing Properties of Balanced Words

In the past few decades there has been a good deal of papers which are concerned with optimization problems in different areas of mathematics (along 0-1 words, finite or infinite) and which yield - sometimes quite unexpectedly - balanced words as optimal. In this note we list some key results along these lines known to date.Comment: In Proceedings WORDS 2011, arXiv:1108.341

Rich Words and Balanced Words

This thesis is mostly focused on palindromes. Palindromes have been studied extensively, in recent years, in the field of combinatorics on words.Our main focus is on rich words, also known as full words. These are words which have maximum number of distinct palindromes as factors.We shed some more light on these words and investigate certain restricted problems. Finite rich words are known to be extendable to infinite rich words. We study more closely how many different ways, and in which situations, rich words can be extended so that they remain rich.The defect of a ord is defined to be the number of palindromes the word is lacking.We will generalize the definition of defect with respect to extending the word to be infinite.The number of rich words, on an alphabet of size $n$, is given an upper and a lower bound. Hof, Knill and Simon presented (Commun. Math. Phys. 174, 1995) a well-known question whether all palindromic subshifts which are enerated by primitive substitutions arise from substitutions which are in class P. Over the years, this question has transformed a bit and is nowadays called the class P conjecture. The main point of the conjecture is to attempt to explain how an infinite word can contain infinitely many palindromes.We will prove a partial result of the conjecture. Rich square-free words are known to be finite (Pelantov\'a and Sarosta, Discrete Math. 313, 2013). We will give another proof for that result. Since they are finite, there exists a longest such word on an $n$-ary alphabet.We give an upper and a lower bound for the length of that word. We study also balanced words. Oliver Jenkinson proved (Discrete Math., Alg. and Appl. 1(4), 2009) that if we take the partial sum of the lexicographically ordered orbit of a binary word, then the balanced word gives the least partial sum. The balanced word also gives the largest product. We will show that, at the other extreme, there are the words of the form $0^{q-p}1^p$ ($p$ and $q$ are integers with $1\leq p<q$), which we call the most unbalanced words. They give the greatest partial sum and the smallest product.Tässä väitöskirjassa käsitellään pääasiassa palindromeja. Palindromeja on tutkittu viime vuosina runsaasti sanojen kombinatoriikassa.Suurin kiinnostuksen kohde tässä tutkielmassa on rikkaissa sanoissa. Nämä ovat sanoja joissa on maksimaalinen määrä erilaisia palindromeja tekijöinä.Näitä sanoja tutkitaan monesta eri näkökulmasta. Äärellisiä rikkaita sanoja voidaan tunnetusti jatkaa äärettömiksi rikkaiksi sanoiksi.Työssä tutkitaan tarkemmin sitä, miten monella tavalla ja missä eri tilanteissa rikkaita sanoja voidaan jatkaa siten, että ne pysyvät rikkaina.Sanan vajauksella tarkoitetaan puuttuvien palindromien lukumäärää.Vajauksen käsite yleistetään tapaukseen, jossa sanaa on jatkettava äärettömäksi sanaksi.Rikkaiden sanojen lukumäärälle annetaan myös ylä- ja alaraja. Hof, Knill ja Simon esittivät kysymyksen (Commun. Math. Phys. 174, 1995), saadaanko kaikki äärettömät sanat joissa on ääretön määrä palindromeja tekijöinä ja jotka ovat primitiivisen morfismin generoimia, morfismeista jotka kuuluvat luokkaan P. Nykyään tätä ongelmaa kutsutaan luokan P konjektuuriksi ja sen tarkoitus on saada selitys sille,millä tavalla äärettömässä sanassa voi olla tekijöinä äärettömän monta palindromia. Osittainen tulos tästä konjektuurista todistetaan. Rikkaiden neliövapaiden sanojen tiedetään olevan äärellisiä (Pelantov\'a ja Starosta, Discrete Math. 313, 2013). Tälle tulokselle annetaan uudenlainen todistus.Koska kyseiset sanat ovat äärellisiä, voidaan selvittää mikä niistä on pisin.Ylä- ja alaraja annetaan tällaisen pisimmän sanan pituudelle. Työssä tutkitaan myös tasapainotettuja sanoja.Tasapainotetut sanat antavat pienimmän osittaissumman binäärisille sanoille (Jenkinson, Discrete Math., Alg. and Appl. 1(4), 2009).Lisäksi ne antavat suurimman tulon.Muotoa $0^{q-p}1^p$ ($p$ ja $q$ ovat kokonaislukuja joille $1\leq p<q$) olevien sanojen todistetaan vastaavasti antavan suurimman osittaissumman ja pienimmän tulon.Ne muodostavat täten toisen ääripään tasapainotetuille sanoille, ja asettavat kaikki muut sanat näiden väliin.Siirretty Doriast

Majorisation ordering of measures invariant under transformations of the interval

PhDMajorisation is a partial ordering that can be applied to the set of probability measures on the unit interval I = [0, 1). Its defining property is that one measure μ majorises another measure , written μ , if R I fdμ R I fd for every convex real-valued function f : I ! R. This means that studying the majorisation of MT , the set of measures invariant under a transformation T : I ! I, can give us insight into finding the maximising and minimising T-invariant measures for convex and concave f. In this thesis I look at the majorisation ordering of MT for four categories of transformations T: concave unimodal maps, the doubling map T : x 7! 2x (mod 1), the family of shifted doubling maps T : x 7! 2x + (mod 1), and the family of orientation-reversing weakly-expanding maps

Symbol ratio minimax sequences in the lexicographic order

Consider the space of sequences of letters ordered lexicographically. We study the set of all maximal sequences for which the asymptotic proportions of the letters are prescribed, where a sequence is said to be maximal if it is at least as great as all of its tails. The infimum of is called the -infimax sequence, or the -minimax sequence if the infimum is a minimum. We give an algorithm which yields all infimax sequences, and show that the infimax is not a minimax if and only if it is the -infimax for every in a simplex of dimension 1 or greater. These results have applications to the theory of rotation sets of beta-shifts and torus homeomorphisms

'Symbol ratio minimax sequences in the lexicographic order'

Consider the space of sequences of k letters ordered lexicographically. We study the set M(α) of all maximal sequences for which the asymptotic proportions α of the letters are prescribed, where a sequence is said to be maximal if it is at least as great as all of its tails. The infimum of M(α) is called the α-infimax sequence, or the α-minimax sequence if the infimum is a minimum. We give an algorithm which yields all infimax sequences, and show that the infimax is not a minimax if and only if it is the α-infimax for every α in a simplex of dimension 1 or greater. These results have applications to the theory of rotation sets of beta-shifts and torus homeomorphisms

Regular configurations and TBR graphs

PhD 2009 QMThis thesis consists of two parts: The first one is concerned with the theory and applications of regular configurations; the second one is devoted to TBR graphs. In the first part, a new approach is proposed to study regular configurations, an extremal arrangement of necklaces formed by a given number of red beads and black beads. We first show that this concept is closely related to several other concepts studied in the literature, such as balanced words, maximally even sets, and the ground states in the Kawasaki-Ising model. Then we apply regular configurations to solve the (vertex) cycle packing problem for shift digraphs, a family of Cayley digraphs. TBR is one of widely used tree rearrangement operationes, and plays an important role in heuristic algorithms for phylogenetic tree reconstruction. In the second part of this thesis we study various properties of TBR graphs, a family of graphs associated with the TBR operation. To investigate the degree distribution of the TBR graphs, we also study -index, a concept introduced to measure the shape of trees. As an interesting by-product, we obtain a structural characterization of good trees, a well-known family of trees that generalizes the complete binary trees

ERGODIC OPTIMIZATION IN DYNAMICAL SYSTEMS

Ergodic optimization is the study of problems relating to maximizing orbits and invariant measures, and maximum ergodic averages. An orbit of a dynamical system is called -maximizing if the time average of the real-valued function along the orbit is larger than along all other orbits, and an invariant probability measure is called -maximizing if it gives a larger space average than any other invariant probability measure. In this paper, we consider the main strands of ergodic optimization, beginning with an influential model problem, and the interpretation of ergodic optimization as the zero temperature limit of thermodynamic formalism. We describe typical properties of maximizing measures for various spaces of functions, the key tool of adding a coboundary so as to reveal properties of these measures, as well as certain classes of functions where the maximizing measure is known to be Sturmian