On winning shifts of marked uniform substitutions

The second author introduced with I. T\"orm\"a a two-player word-building game [Playing with Subshifts, Fund. Inform. 132 (2014), 131--152]. The game has a predetermined (possibly finite) choice sequence $\alpha_1$, $\alpha_2$, $\ldots$ of integers such that on round $n$ the player $A$ chooses a subset $S_n$ of size $\alpha_n$ of some fixed finite alphabet and the player $B$ picks a letter from the set $S_n$. The outcome is determined by whether the word obtained by concatenating the letters $B$ picked lies in a prescribed target set $X$ (a win for player $A$) or not (a win for player $B$). Typically, we consider $X$ to be a subshift. The winning shift $W(X)$ of a subshift $X$ is defined as the set of choice sequences for which $A$ has a winning strategy when the target set is the language of $X$. The winning shift $W(X)$ mirrors some properties of $X$. For instance, $W(X)$ and $X$ have the same entropy. Virtually nothing is known about the structure of the winning shifts of subshifts common in combinatorics on words. In this paper, we study the winning shifts of subshifts generated by marked uniform substitutions, and show that these winning shifts, viewed as subshifts, also have a substitutive structure. Particularly, we give an explicit description of the winning shift for the generalized Thue-Morse substitutions. It is known that $W(X)$ and $X$ have the same factor complexity. As an example application, we exploit this connection to give a simple derivation of the first difference and factor complexity functions of subshifts generated by marked substitutions. We describe these functions in particular detail for the generalized Thue-Morse substitutions.Comment: Extended version of a paper presented at RuFiDiM I

More on the dynamics of the symbolic square root map

In our earlier paper [A square root map on Sturmian words, Electron. J. Combin. 24.1 (2017)], we introduced a symbolic square root map. Every optimal squareful infinite word $s$ contains exactly six minimal squares and can be written as a product of these squares: $s = X_1^2 X_2^2 \cdots$. The square root $\sqrt{s}$ of $s$ is the infinite word $X_1 X_2 \cdots$ obtained by deleting half of each square. We proved that the square root map preserves the languages of Sturmian words (which are optimal squareful words). The dynamics of the square root map on a Sturmian subshift are well understood. In our earlier work, we introduced another type of subshift of optimal squareful words which together with the square root map form a dynamical system. In this paper, we study these dynamical systems in more detail and compare their properties to the Sturmian case. The main results are characterizations of periodic points and the limit set. The results show that while there is some similarity it is possible for the square root map to exhibit quite different behavior compared to the Sturmian case.Comment: 22 pages, Extended version of a paper presented at WORDS 201

Innostava pianoryhmä : Kohti lasta motivoivaa pianoryhmäpedagogiikkaa

Opinnäytetyöni käsittelee 5–7-vuotiaiden lasten säännöllistä, viikoittaista alkeispianoryhmäopetusta innostumisen ja oppimisen ilon näkökulmasta. Pyrin selvittämään, millaisista asioista 5–7-vuotiaat pianoryhmätunneilla innostuvat, mihin innostavaa pianoryhmäopetusta järjestettäessä on syytä kiinnittää huomiota ja millaisia työtapoja innostavassa pianoryhmäpedagogiikassa voi hyödyntää. Tämän lisäksi tarkastelen, miten ilo ja innostuminen rakentavat oppimisen edellytyksiä. Havainnoimalla erilaisia alkeispianoryhmiä ja soitinmuskareita sekä haastattelemalla kyseisten ryhmien opettajia olen kerännyt tietoa vaihtoehtoisista alkeispianoryhmäopetuksen toiminta- ja lähestymistavoista ja päässyt tarkastelemaan läheltä lasten innostumista. Lähdeaineiston ja -kirjallisuuden avulla olen pyrkinyt selvittämään lasten yleisiä kiinnostuksen kohteita ja innostumisen tärkeyttä oppimisprosessissa. Olen teemoitellut työtapoja ja tärkeimpiä innostavan pianoryhmäopetuksen lähtökohtia opinnäytetyöni alaluvuiksi ja pohtinut, miten eri tavoin lasten innostumista ja oppimisen iloa voi tukea. 5–7-vuotiaiden alkeispianoryhmäopetuksessa sadun, leikin ja tarinan maailmaan uppoutumalla voidaan syventää sekä lasten oppimista että motivaatiota. Oppimisen iloa ja omakohtaisuutta voidaan lisätä erilaisten pelien, leikkien ja innostavien musiikillisten harjoitusten avulla. Monipuolisia työtapoja vaihtelemalla oppiminen pysyy mielenkiintoisena ja lasten vireys- ja aktivaatiotaso oppimisen kannalta otollisena. Säännöllinen, viikoittainen pianoryhmäopetus mahdollistaa pitkäkestoisen musiikkiharrastuksen kannalta tärkeiden toverisuhteiden syntymisen ja tarjoaa pienille pianisteille tärkeän yhteisöllisen musisoimisen kokemuksen. Pianoryhmän monipuoliset toimintatavat mahdollistavat yhteismusisointiin, esiintymiseen, laulamiseen ja musiikilliseen ilmaisuun liittyvien taitojen kehittymisen pianististen taitojen ohella. Opinnäytetyöni tarjoaa ryhmäopetuksesta kiinnostuneille musiikkipedagogeille välineitä suunnitella oppimisen iloon tähtäävää ja motivoivaa alkeispianoryhmäpedagogiikkaa ja alkeisryhmäopetuksesta kiinnostuneille näkökulmia kehittää opetusta.My Bachelor’s thesis focuses on piano group education for children between the ages 5–7 by the aspects of enthusiasm and the joy of learning. Especially, it considers weekly assembling groups in which children are not taking part in other piano education such as private lessons. The attempt is to research what sort of issues the children aged 5–7 are interested in at group piano lessons, to what the teacher should aim attention while arranging piano group lessons and what kind of practices and activities the teacher could engage to inspire piano group pedagogy. In addition to that, I clarify how joy and enthusiasm build the requirements of the learning process. By observing different beginners’ piano groups and by interviewing the teachers I have compiled knowledge on how piano education can be done in piano groups and what aspects can exist behind the teaching in piano groups. In addition, I had the opportunity to closely follow the children’s enthusiasm. By exploring source materials and source literature I have aimed to clarify what children between the ages 5–7 are generally interested in and why enthusiasm is important in the learning process. I have separated different practices and the most important aspects of teaching children’s piano group and arranged the structure of this thesis corresponding to them. Further, I have examined how the enthusiasm and the joy of learning can be promoted. By including fairy tales, stories and different kinds of play to the teaching of children’s piano group can the learning and motivation of children be improved. The joy of learning and the subjective experiences can be increased by different playing, role plays, games and inspiring musical exercises. By varying multiple practices the learning process remains engaging and the rate of energy and activation reasonable for learning. A regular, weekly assembling piano group enables important companionships that are relevant to long-lasting musical interest. It also offers the important experience of belonging to a musical society to the little pianists. The varying practices in piano group pedagogy enables the skills of playing together, performing, singing and expressing music as well as pianistic skills. My Bachelor’s thesis provides music teachers who are interested in group pedagogy advice and knowledge to organize and prepare beginners’ piano group pedagogy that aims to increase enthusiasm and motivation. It also provides different aspects to develop beginners’ piano group teaching

The effect of mode of breathing on craniofacial growth—revisited

It has been maintained that because of large adenoids, nasal breathing is obstructed leading to mouth breathing and an ‘adenoid face', characterized by an incompetent lip seal, a narrow upper dental arch, increased anterior face height, a steep mandibular plane angle, and a retrognathic mandible. This development has been explained as occurring by changes in head and tongue position and muscular balance. After adenoidectomy and change in head and tongue position, accelerated mandibular growth and closure of the mandibular plane angle have been reported. Children with obstructive sleep apnoea (OSA) have similar craniofacial characteristics as those with large adenoids and tonsils, and the first treatment of choice of OSA children is removal of adenoids and tonsils. It is probable that some children with an adenoid face would nowadays be diagnosed as having OSA. These children also have abnormal nocturnal growth hormone (GH) secretion and somatic growth impairment, which is normalized following adenotonsillectomy. It is hypothesized that decreased mandibular growth in adenoid face children is due to abnormal secretion of GH and its mediators. After normalization of hormonal status, ramus growth is enhanced by more intensive endochondral bone formation in the condylar cartilage and/or by appositional bone growth in the lower border of the mandible. This would, in part, explain the noted acceleration in the growth of the mandible and alteration in its growth direction, following the change in the mode of breathing after adenotonsillectom

Characterization of repetitions in Sturmian words: A new proof

We present a new, dynamical way to study powers (that is, repetitions) in Sturmian words based on results from Diophantine approximation theory. As a result, we provide an alternative and shorter proof of a result by Damanik and Lenz characterizing powers in Sturmian words [Powers in Sturmian sequences, Eur. J. Combin. 24 (2003), 377--390]. Further, as a consequence, we obtain a previously known formula for the fractional index of a Sturmian word based on the continued fraction expansion of its slope.Comment: 9 pages, 1 figur