Computer Science

We present the investigation of a biogeochemical marine ecosystem model used as part of the climate change research focusing on the enhanced carbon dioxid concentration in the atmosphere. Numerical parameter optimization has been performed to improve represention of observational data using data assimilation techniques. Several local minima were found but no global optimum could be identified. To detect the actual capability of the model in simulating natural systems, a theoretical analysis of the model equations is conducted. Here, basic properties such as continuity and positivity of the model equations are investigated.

The paper approaches the classical combinatorial problem of freeness of words, in the more general case of partial words. First, we propose an algorithm that tests efficiently whether a partial word is kfree or not. Then, we show that there exist arbitrarily many cube-free infinite partial words containing an infinite number of holes, over binary alphabets; thus, there exist arbitrarily many k-free infinite partial words containing an infinite number of holes for k ≥ 3. Moreover, we present an efficient algorithm for the construction of a cube-free partial word with n holes. In the final section of the paper, we show that there exists an infinite word, over a four-symbol alphabet, in which we can substitute randomly one symbol with a hole, and still obtain a cube-free word; we show that such a word does not exist for alphabets with less symbols. Further, we prove that in this word we can replace arbitrarily many symbols with holes, such that each two consecutive holes are separated by at least two symbols, and obtain a cube-free partial word. This result seems interesting because any partial word containing two holes with less than two symbols between them is not cube-free. Finally, we modify the previously presented algorithm to construct, over a four-symbol alphabet, a cube-free partial word with exactly n holes, having minimal length, among all the possible cubefree partial words with at least n holes.

A well known result of Fraenkel and Simpson states that the number of distinct squares in a word of length n is bounded by 2n since at each position there are at most two distinct squares whose last occurrence start. In this paper, we investigate the problem of counting distinct squares in partial words, or sequences over a finite alphabet that may have some "do not know" symbols or "holes" (a (full) word is just a partial word without holes). A square in a partial word over a given alphabet has the form uu where u is compatible with u, and consequently, such square is compatible with a number of full words over the alphabet that are squares. We consider the number of distinct full squares compatible with factors in a partial word with h holes of length n over a k-letter alphabet, and show that this number increases polynomially with respect to k in constrast with full words, and give bounds in a number of cases. For partial words with one hole, it turns out that there may be more than two squares that have their last occurrence starting at the same position. We prove that if such is the case, then the hole is in the shortest square. We also construct a partial word with one hole over a k-letter alphabet that has more than k squares whose last occurrence start at position zero.

Partial words, or sequences over a finite alphabet that may have do-not-know symbols or holes, have been recently the subject of much investigation. Several interesting combinatorial properties have been studied such as the periodic behavior and the counting of distinct squares in partial words. In this paper, we extend the three-squares lemma on words to partial words with one hole.

El objeto de esta tesis está representado por las repeticiones de palabras parciales, palabras que, además de las letras regulares, pueden tener un número de símbolos desconocidos, llamados símbolos" agujeros" o" no sé qué". Más concretamente, se presenta y se resuelve una extensión de la noción de repetición establecida por Axel Thue.

We propose an algorithm that given as input a full word w of length n, and positive integers p and d, outputs, if any exists, a maximal p-periodic partial word contained in w with the property that no two holes are within distance d (so-called d-valid). Our algorithm runs in O (nd) time and is used for the study of repetition-freeness of partial words.

Abstract Extending the general undecidability result concerning the absoluteness of inequalities between subword histories, in this paper we show that the question whether such inequalities hold for all words is undecidable already over a binary alphabet and bounded number of blocks, and even in very simple cases an answer requires an intractable computation.

Abstract Pseudopalindromes are words that are fixed points for some antimorphic involution. In this paper we discuss a newer word operation, that of pseudopalindromic completion, in which symbols are added to either side of the word such that the new obtained words are pseudopalindromes. This notion represents a particular type of hairpin completion, where the length of the hairpin is at most one.

Abstract We consider here two formal operations on words inspired by the DNA biochemistry: hairpin lengthening introduced in [15] and its inverse called hairpin shortening. We study the closure of the class of regular languages under the non-iterated and iterated variants of the two operations. The main results are: although any finite number of applications of the hairpin lengthening to a regular language may lead to non-regular languages, the iterated hairpin lengthening of a regular language is always regular.

Abstract Pseudo-repetitions are a natural generalization of the classical notion of repetitions in sequences. We solve fundamental algorithmic questions on pseudo-repetitions by application of insightful combinatorial results on words. More precisely, we efficiently decide whether a word is a pseudo-repetition and find all the pseudo-repetitive factors of a word.

Abstract The problem of classifying all the avoidable binary patterns in (full) words has been completely solved (see Chapter 3 of M. Lothaire, Algebraic Combinatorics on Words, Cambridge University Press, 2002). In this paper, we classify all the avoidable binary patterns in partial words, or sequences that may have some undefined positions called holes.

Abstract A k-abelian cube is a word uvw, where u, v, w are either equal or have the same factors of length k with the same multiplicities and the same prefixes and suffixes of length k− 1. Previously it has been known that k-abelian cubes are avoidable over a binary alphabet for k≥ 8. Here it is proved that this holds for k≥ 5.

Abstract The notion of repetition of factors in words is central to considerations on sequences. One of the recent generalizations regarding this concept was introduced by Czeizler et al.(2010) and investigates a restricted version of that notion in the context of DNA computing and bioinformatics. It considers a word to be a pseudo-repetition if it is the iterated concatenation of one of its prefixes and the image of this prefix through an involution.

Erdős raised the question whether there exist infinite abelian square-free words over a given alphabet, that is, words in which no two adjacent subwords are permutations of each other. It can easily be checked that no such word exists over a three-letter alphabet. However, infinite abelian square-free words have been constructed over alphabets of sizes as small as four. In this paper, we investigate the problem of avoiding abelian squares in partial words, or sequences that may contain some holes.

Abstract We initiate a study of languages of partial words related to regular languages of full words. First, we study the possibility of expressing a regular language of full words as the image of a partial-words-language through a substitution that only replaces the hole symbols of the partial words with a finite set of letters. Results regarding the structure, uniqueness and succinctness of such a representation, as well as a series of related decidability and computational-hardness results, are presented.

Zusammenfassung Die projekt-orientierte Ausbildung im Software Engineering ist inzwischen etabliert, z.B. in Form von Softwarepraktika. Im vorliegenden Beitrag berichten wir über unsere Erfahrungen mit der Vermittlung von Entwurfsmustern in diesem Kontext. Wir konzentrieren uns dabei auf die Lehre im Grundstudium und erläutern wie wir hier ein nicht-triviales Softwaresystem vorlesungsbegleitend als Lehrbeispiel einsetzen. Ergänzt wird die Präsenzlehre durch ein auf Entwurfsmuster spezialisiertes E-Learning-und Assistenzsystem.