DFAs and PFAs with Long Shortest Synchronizing Word Length

It was conjectured by \v{C}ern\'y in 1964, that a synchronizing DFA on $n$ states always has a shortest synchronizing word of length at most $(n-1)^2$, and he gave a sequence of DFAs for which this bound is reached. Until now a full analysis of all DFAs reaching this bound was only given for $n \leq 4$, and with bounds on the number of symbols for $n \leq 10$. Here we give the full analysis for $n \leq 6$, without bounds on the number of symbols. For PFAs the bound is much higher. For $n \leq 6$ we do a similar analysis as for DFAs and find the maximal shortest synchronizing word lengths, exceeding $(n-1)^2$ for $n =4,5,6$. For arbitrary n we give a construction of a PFA on three symbols with exponential shortest synchronizing word length, giving significantly better bounds than earlier exponential constructions. We give a transformation of this PFA to a PFA on two symbols keeping exponential shortest synchronizing word length, yielding a better bound than applying a similar known transformation.Comment: 16 pages, 2 figures source code adde

Caries Prevention in First Grade Schoolchildren in Zagreb

Suvremena istraživanja pokazuju da je učinkovitost zdravstvenog odgoja mnogo veća ako je orijentirana na održanje zdravlja a ne na bolesna stanja. Provedenim programom željeli smo obuhvatiti djecu pri polasku u prvi razred osnovne škole te ih nastavnim procesom obavijestiti o osnovama očuvanja oralnoga zdravlja. Programom je obuhvaćeno 7000 djece, učenika prvih razreda u zagrebačkim osnovnim školama. Pri obveznom pregledu za upis u školu njihov je stomatolog na unificiranom obrascu utvrdio stanje oralnoga zdravlja. U prvim danima nastave dodatno izobražena preventivna medicinska sestra održala im je zdravstveno-odgojno predavanje te su djeca neposredno nakon predavanja sudjelovala u igraonici radom na bojankama. Uključivanje roditelja provedeno je s pomoću Bojanki i za tu prigodu napisanog Biltena. Nakon dva mjeseca, predavanje i igraonica s ponešto proširenim sadržajem su ponovljeni. Za roditelje je pripremljen nov Bilten. Pregledom djece pri upisu u prvi razred osnovne škole utvrđeno je da samo 26,5% pregledane djece ima zdrave sve mliječne zube, i 76,2% djece ima zdrave sve trajne zube. Rezultati pregleda pri upisu u prvi razred potvrđuju da je potrebno provoditi stomatološki zdravstveni odgoj, a na taj način zdravstveno-odgojnim porukama obuhvaćena su ne samo djeca već i nastavno osoblje, medicinske sestre i roditelji. Loše stanje oralnoga zdravlja utvrđeno pri pregledu za upis u prvi razred te zanimanje s kojim su program prihvatili svi sudionici potvrđuje da ga je potrebno i dalje provoditi.Contemporary studies have shown that oral health education is more effective when it is prevention-oriented rather than disease oriented. The implemented program aimed at including all first grade aged children and making them familiar with the basies of oral health protection through the educational program. The program included 7.000 first grade aged children in primary schools in Zagreb. During the compulsory check-up required for enrolment in school, their dentist established the oral health status on a unified printed form. In the first school days, a nurse, previously additionally educated in oral health prevention, gave an oral health education lecture, and immediately after the lecture the children participated in a children`s workshop by colouring the colouring books. Parent’s involvement was achieved through colouring books and a special bulletin prepared and sent to them. Two months later, the lecture and the workshop with a slightly extended program were repeated. A new bulletin was prepared for the parents. The check-up of children for establishing their eligibility for enrolment in the first grade showed that only 26.5% of the children had caries-free primary teeth and 76.2% of the children had caries-free secondary teeth. The results obtained at such oral health check-ups speak for the need to implement oral healt education, and in this way the oral health educational messages reach not only the children but also their teachers, nurses and parents. A rather bad oral health situation, established at compulsory check-ups for enrolment in school, and the interest raised with those involved in the program indicate the need to continue its implementation

Towards a Robuster Interpretive Parsing

The input data to grammar learning algorithms often consist of overt forms that do not contain full structural descriptions. This lack of information may contribute to the failure of learning. Past work on Optimality Theory introduced Robust Interpretive Parsing (RIP) as a partial solution to this problem. We generalize RIP and suggest replacing the winner candidate with a weighted mean violation of the potential winner candidates. A Boltzmann distribution is introduced on the winner set, and the distribution’s parameter $T$ is gradually decreased. Finally, we show that GRIP, the Generalized Robust Interpretive Parsing Algorithm significantly improves the learning success rate in a model with standard constraints for metrical stress assignment

The random K-satisfiability problem: from an analytic solution to an efficient algorithm

We study the problem of satisfiability of randomly chosen clauses, each with K Boolean variables. Using the cavity method at zero temperature, we find the phase diagram for the K=3 case. We show the existence of an intermediate phase in the satisfiable region, where the proliferation of metastable states is at the origin of the slowdown of search algorithms. The fundamental order parameter introduced in the cavity method, which consists of surveys of local magnetic fields in the various possible states of the system, can be computed for one given sample. These surveys can be used to invent new types of algorithms for solving hard combinatorial optimizations problems. One such algorithm is shown here for the 3-sat problem, with very good performances.Comment: 38 pages, 13 figures; corrected typo

Charmonium in a weakly coupled quark-gluon plasma

We present a model of charmonium as two heavy quarks propagating classically in a weakly coupled quark-gluon plasma. The quarks interact via a static, color-dependent potential and also suffer collisions with the plasma particles. We calculate the radiation width of the color octet state (for fixed, classical $q\bar q$ separation) and find that it is long-lived provided a finite gluon mass is used to provide a threshold energy.Comment: 7 pages in plain LaTeX + 3 figures packed with uufiles; slight changes to comply with referees, added one referenc

Caries Prevention in First Grade Schoolchildren in Zagreb

Suvremena istraživanja pokazuju da je učinkovitost zdravstvenog odgoja mnogo veća ako je orijentirana na održanje zdravlja a ne na bolesna stanja. Provedenim programom željeli smo obuhvatiti djecu pri polasku u prvi razred osnovne škole te ih nastavnim procesom obavijestiti o osnovama očuvanja oralnoga zdravlja. Programom je obuhvaćeno 7000 djece, učenika prvih razreda u zagrebačkim osnovnim školama. Pri obveznom pregledu za upis u školu njihov je stomatolog na unificiranom obrascu utvrdio stanje oralnoga zdravlja. U prvim danima nastave dodatno izobražena preventivna medicinska sestra održala im je zdravstveno-odgojno predavanje te su djeca neposredno nakon predavanja sudjelovala u igraonici radom na bojankama. Uključivanje roditelja provedeno je s pomoću Bojanki i za tu prigodu napisanog Biltena. Nakon dva mjeseca, predavanje i igraonica s ponešto proširenim sadržajem su ponovljeni. Za roditelje je pripremljen nov Bilten. Pregledom djece pri upisu u prvi razred osnovne škole utvrđeno je da samo 26,5% pregledane djece ima zdrave sve mliječne zube, i 76,2% djece ima zdrave sve trajne zube. Rezultati pregleda pri upisu u prvi razred potvrđuju da je potrebno provoditi stomatološki zdravstveni odgoj, a na taj način zdravstveno-odgojnim porukama obuhvaćena su ne samo djeca već i nastavno osoblje, medicinske sestre i roditelji. Loše stanje oralnoga zdravlja utvrđeno pri pregledu za upis u prvi razred te zanimanje s kojim su program prihvatili svi sudionici potvrđuje da ga je potrebno i dalje provoditi.Contemporary studies have shown that oral health education is more effective when it is prevention-oriented rather than disease oriented. The implemented program aimed at including all first grade aged children and making them familiar with the basies of oral health protection through the educational program. The program included 7.000 first grade aged children in primary schools in Zagreb. During the compulsory check-up required for enrolment in school, their dentist established the oral health status on a unified printed form. In the first school days, a nurse, previously additionally educated in oral health prevention, gave an oral health education lecture, and immediately after the lecture the children participated in a children`s workshop by colouring the colouring books. Parent’s involvement was achieved through colouring books and a special bulletin prepared and sent to them. Two months later, the lecture and the workshop with a slightly extended program were repeated. A new bulletin was prepared for the parents. The check-up of children for establishing their eligibility for enrolment in the first grade showed that only 26.5% of the children had caries-free primary teeth and 76.2% of the children had caries-free secondary teeth. The results obtained at such oral health check-ups speak for the need to implement oral healt education, and in this way the oral health educational messages reach not only the children but also their teachers, nurses and parents. A rather bad oral health situation, established at compulsory check-ups for enrolment in school, and the interest raised with those involved in the program indicate the need to continue its implementation