2-сжимающие слова и проблема реконструкции последовательности

Для данного слова свойство быть 2-сжимаемым (2-синхронизирующим) существенно зависит от свойств некоторого специального множества S его факторов. Мы изучаем возможность реконструкции 2-сжимающего (2-синхронизирующего) слова по этому множеству. Переходя от множества S ко множеству Xs его факторов длины три, мы показываем, что 2-сжимающее (2-синхронизирующее) слово является накрывающим для Xs

Experiments on synchronizing automata

This work is motivated by the Černý Conjecture – an old unsolved problem in the automata theory. We describe the results of the experiments on synchronizing automata, which have led us to two interesting results. The first one is that the size of an automaton alphabet may play an important role in the issue of synchronization: we have found a 5-state automaton over a 3-letter alphabet which attains the upper bound from the Černý Conjecture, while there is no such automaton (except Černý automaton $C_5$) over a binary alphabet. The second result emerging from the experiments is a theorem describing the dependencies between the automaton structure $S$ expressed in terms of the so-called merging system and the maximal length of all minimal synchronizing words for automata of type $S$

Image reducing words and subgroups of free groups

AbstractA word w over a finite alphabet Σ is said to be n-collapsing if for an arbitrary finite automaton A=〈Q,Σ−·−〉, the inequality |Q·w|⩽|Q|−n holds provided that |Q·u|⩽|Q|−n for some word u (depending on A). We give an algorithm to test whether a word is 2-collapsing. To this aim we associate to every word w a finite family of finitely generated subgroups in finitely generated free groups and prove that the property of being 2-collapsing reflects in the property that each of these subgroups has index at most 2 in the corresponding free group. We also find a similar characterization for the closely related class of so-called 2-synchronizing words

Follower and Extender Sets in Symbolic Dynamics

Given a word w in the language of a one-dimensional shift space X, the follower set of w, denoted FX(w), is the set of all right-infinite sequences which follow w in some point of X. Extender sets are a generalization of follower sets and are defined similarly. To a given shift space X, then, we may associate a follower set sequence {|FX(n)|} which records the number of distinct follower sets in X corresponding to words of length n. Similarly, we may define an extender set sequence {|EX(n)|}. The complexity sequence {PhiX(n)} of a shift space X records the number of n-letter words in the language of X for each n. This thesis explores the relationship between the class of achievable follower and extender set sequences of one-dimensional shift spaces and the class of their complexity sequences. Some surprising similarities suggest a connection may exist, for instance, both the complexity sequence and the extender set sequence are bounded if and only if there exists some n such that the value of the nth term of the sequence is at most n. This thesis, however, also demonstrates important differences among complexity sequences and follower and extender set sequences of one-dimensional shifts. In particular, we show that unlike complexity sequences, follower and extender set sequences need not be monotone increasing. Finally, we use the classical beta-shifts to demonstrate that, while many follower set sequences may not be realized as complexity sequences, up to possible increase by 1, any complexity sequence may be realized as a follower set sequence of some shift space

Type IIP supernova 2008in: the explosion of a normal red supergiant

The explosion energy and the ejecta mass of a type IIP supernova make up the basis for the theory of explosion mechanism. So far, these parameters have only been determined for seven events. Type IIP supernova 2008in is another well-observed event for which a detailed hydrodynamic modeling can be used to derive the supernova parameters. Hydrodynamic modeling was employed to describe the bolometric light curve and the expansion velocities at the photosphere level. A time-dependent model for hydrogen ionization and excitation was applied to model the Halpha and Hbeta line profiles. We found an ejecta mass of 13.6 Msun, an explosion energy of 5.05x10^50 erg, a presupernova radius of 570 Rsun, and a radioactive Ni-56 mass of 0.015 Msun. The estimated progenitor mass is 15.5 Msun. We uncovered a problem of the Halpha and Hbeta description at the early phase, which cannot be resolved within a spherically symmetric model. The presupernova of SN 2008in was a normal red supergiant with the minimum mass of the progenitor among eight type IIP supernovae explored by means of the hydrodynamic modeling. The problem of the absence of type IIP supernovae with the progenitor masses <15 Msun in this sample remains open.Comment: 6 pages, 8 figures, 1 table, accepted for publication in A&

Words Guaranteeing Minimum Image

Given a positive integer n and a finite alphabet Σ, a word w over Σ is said to guarantee minimum image if, for every homomorphism φ from the free monoid Σ* over Σ into the monoid of all transformations of an n-element set, the range of the transformation wφ has the minimum cardinality among the ranges of all transformations of the form vφ where v runs over Σ*. Although the existence of words guaranteeing minimum image is pretty obvious, the problem of their explicit description is very far from being trivial. Sauer and Stone in 1991 gave a recursive construction for such a word w but the length of their word was doubly exponential (as a function of n). We first show that some known results of automata theory immediately lead to an alternative construction that yields a simpler word that guarantees minimum image: it has exponential length, more precisely, its length is O(|Σ|(n3-n)). Then with some more effort, we find a word guaranteeing minimum image similar to that of Sauer and Stone but of length O(|Σ|(n2-n)). On the other hand, we prove that the length of any word guaranteeing minimum image cannot be less than |Σ|n-1. © 2004 World Scientific Publishing Company