Covering arrays for words of length t over a d letter alphabet are k × n arrays with entries from the alphabet so that for each choice of t columns, each of the dt t-letter words appears at least once among the rows of the selected columns. We study two schemes in which all words are not considered to be different. In the first case, words are equivalent if they induce the same partition of a t element set. In the second case, words of the same weighted sum are equivalent. In both cases we produce logarithmic upper bounds on the minimum size k = k(n) of a covering array. Most definitive results are for t = 2, 3, 4.

[Formulae and special characters can only be approximated here. Please see the pdf version of the abstract for an accurate reproduction.] This thesis concerns the study of homogeneous factorisations of complete graphs with edge-transitive factors. A factorisation of a complete graph Kn is a partition of its edges into disjoint classes. Each class of edges in a factorisation of Kn corresponds to a spanning subgraph called a factor. If all the factors are isomorphic to one another, then a factorisation of Kn is called an isomorphic factorisation. A homogeneous factorisation of a complete graph is an isomorphic factorisation where there exists a group G which permutes the factors transitively, and a normal subgroup M of G such that each factor is M-vertex-transitive. If M also acts edge-transitively on each factor, then a homogeneous factorisation of Kn is called an edge-transitive homogeneous factorisation. The aim of this thesis is to study edge-transitive homogeneous factorisations of Kn. We achieve a nearly complete explicit classification except for the case where G is an affine 2-homogeneous group of the form ZR p x G0, where G0 is less than or equal to ΓL(1,p to the power of R). In this case, we obtain necessary and sufficient arithmetic conditions on certain parameters for such factorisations to exist, and give a generic construction that specifies the homogeneous factorisation completely, given that the conditions on the parameters hold. Moreover, we give two constructions of infinite families of examples where we specify the parameters explicitly. In the second infinite family, the arc-transitive factors are generalisations of certain arc-transitive, self-complementary graphs constructed by Peisert in 2001.

L'augmentation du nombre de génomes totalement séquencés rend de plus en plus efficace l'étude des mécanismes évolutifs à partir de la comparaison de génomes contemporains. L'un des principaux problèmes réside dans la reconstruction d'architectures de génomes ancestraux plausibles afin d'apporter des hypothèses à la fois sur l'histoire des génomes existants et sur les mécanismes de leur formation. Toutes les méthodes de reconstruction ancestrale ne convergent pas nécessairement vers les mêmes résultats mais sont toutes basées sur les trois mêmes étapes : l'identification de marqueurs commun dans les génomes contemporains, la construction de cartes comparatives des génomes, et la réconciliation de ces cartes en utilisant le critère de parcimonie maximum. La quantité importante des données à analyser nécessite l'automatisation des traitements et résoudre ces problèmes représente de formidables challenges computationnels. Affiner les modèles et outils mathématiques existants par l'ajout de contraintes biologiques fortes rend les hypothèses établies biologiquement plus réalistes. Dans cette thèse, nous proposons une nouvelle méthode permettant d'identifier des marqueurs communs pour des espèces évolutivement distantes. Ensuite, nous appliquons sur les cartes comparatives reconstituées une nouvelle méthode pour la reconstruction d'architectures ancestrales basée sur les adjacences entre les marqueurs calculés et les distances génomiques entre les génomes contemporains. Enfin, après avoir corrigé l'algorithme existant permettant de déterminer une séquence optimale de réarrangements qui se sont produits durant l'évolution des génomes existants depuis leur ancêtre commun, nous proposons un nouvel outil appelé VIRAGE qui permet la visualisation animée des scénarios de réarrangements entre les espèces.

Ce travail de thèse traite du maintien dynamique de représentations géométriques de graphes. Le manuscrit met en avant des connexions fortes entre trois types de représentation de graphes : les décompositions de graphes, les modèles géométriques et les représentations arborescentes à degrés de liberté (PQ-arbres, PC-arbres et autres structures du même type). De nouvelles relations entre ces objets sont mises en évidence et d'autres déjà connues sont approfondies. Notamment, il est établi une équivalence mathématique et algorithmique entre la décomposition modulaire des graphes d'intervalles et le PQ-arbre de leurs cliques maximales.<br /><br />Les connexions entre les trois types de représentation précités sont exploitées pour la conception d'algorithmes de reconnaissance entièrement dynamiques pour les cographes orientés, les graphes de permutation et les graphes d'intervalles. Pour les cographes orientés, l'algorithme présenté est de complexité optimale, il traite les modifications de sommet en temps O(d), où d est le degré du sommet en question, et les modifications d'arête en temps constant. Les algorithmes pour les graphes de permutation et les graphes d'intervalles ont la même complexité : les modifications d'arête et de sommet sont traitées en temps O(n), où n est le nombre de sommets du graphe. Une des contributions du mémoire est de mettre en lumière des similarités très fortes entre les opérations d'ajout d'un sommet dans un graphe de permutation et dans un graphe d'intervalles. <br />L'approche mise en oeuvre dans ce mémoire est assez générale pour laisser entrevoir les mêmes possibilités algorithmiques pour d'autres classes de graphes définies géométriquement.

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Eulerian quasisymmetric functions were introduced by Shareshian and Wachs in order to obtain a q-analog of Euler's exponential generating function formula for the Eulerian polynomials. They are defined via the symmetric group, and applying the stable and nonstable principal specializations yields formulas for joint distributions of permutation statistics. We consider the wreath product of the cyclic group with the symmetric group, also known as the group of colored permutations. We use this group to introduce colored Eulerian quasisymmetric functions, which are a generalization of Eulerian quasisymmetric functions. We derive a formula for the generating function of these colored Eulerian quasisymmetric functions, which reduces to a formula of Shareshian and Wachs for the Eulerian quasisymmetric functions. We show that applying the stable and nonstable principal specializations yields formulas for joint distributions of colored permutation statistics. The family of colored permutation groups includes the family of symmetric groups and the family of hyperoctahedral groups, also called the type A Coxeter groups and type B Coxeter groups, respectively. By specializing our formulas to these cases, they reduce to the Shareshian-Wachs q-analog of Euler's formula, formulas of Foata and Han, and a new generalization of a formula of Chow and Gessel.

The dissertation covers two problems motivated by computational biology: genome rearrangements, and haplotype networks. Genome rearrangement problems are a particular case of edit distance problems, where one seeks to transform two given objects into one another using as few operations as possible, with the additional constraint that the set of allowed operations is fixed beforehand; we are also interested in computing the corresponding distances between those objects, i.e. merely computing the minimum number of operations rather than an optimal sequence. Genome rearrangement problems can often be formulated as sorting problems on permutations (viewed as linear orderings of {1,2,...,n}) using as few (allowed) operations as possible. In this thesis, we focus among other operations on ``transpositions', which displace intervals of a permutation. Many questions related to sorting by transpositions are open, related in particular to its computational complexity. We use the disjoint cycle decomposition of permutations, rather than the ``standard tools' used in genome rearrangements, to prove new upper bounds on the transposition distance, as well as formulae for computing the exact distance in polynomial time in many cases. This decomposition also allows us to solve a counting problem related to the ``cycle graph' of Bafna and Pevzner, and to construct a general framework for obtaining lower bounds on any edit distance between permutations by recasting their computation as factorisation problems on related even permutations. Haplotype networks are graphs in which a subset of vertices is labelled, used in comparative genomics as an alternative to trees. We formalise a new method due to Cassens, Mardulyn and Milinkovitch, which consists in building a graph containing a given set of partially labelled trees and with as few edges as possible. We give exact algorithms for solving the problem on two graphs, with an exponential running time in the general case but with a polynomial running time if at least one of the graphs belong to a particular class. / La thèse couvre deux problèmes motivés par la biologie: l'étude des réarrangements génomiques, et celle des réseaux d'haplotypes. Les problèmes de réarrangements génomiques sont un cas particulier des problèmes de distances d'édition, où l'on cherche à transformer un objet en un autre en utilisant le plus petit nombre possible d'opérations, les opérations autorisées étant fixées au préalable; on s'intéresse également à la distance entre les deux objets, c'est-à-dire au calcul du nombre d'opérations dans une séquence optimale plutôt qu'à la recherche d'une telle séquence. Les problèmes de réarrangements génomiques peuvent souvent s'exprimer comme des problèmes de tri de permutations (vues comme des arrangements linéaires de {1,2,...,n}) en utilisant le plus petit nombre d'opérations (autorisées) possible. Nous examinons en particulier les ``transpositions', qui déplacent un intervalle de la permutation. Beaucoup de problèmes liés au tri par transpositions sont ouverts, en particulier sa complexité algorithmique. Nous nous écartons des ``outils standards' utilisés dans le domaine des réarrangements génomiques, et utilisons la décomposition en cycles disjoints des permutations pour prouver de nouvelles majorations sur la distance des transpositions ainsi que des formules permettant de calculer cette distance en temps polynomial dans de nombreux cas. Cette décomposition nous sert également à résoudre un problème d'énumération concernant le ``graphe des cycles' de Bafna et Pevzner, et à construire une technique générale permettant d'obtenir de nouvelles minorations en reformulant tous les problèmes de distances d'édition sur les permutations en termes de factorisations de permutations paires associées. Les réseaux d'haplotypes sont des graphes dont une partie des sommets porte des étiquettes, utilisés en génomique comparative quand les arbres sont trop restrictifs, ou quand l'on ne peut choisir une ``meilleure' topologie parmi un ensemble donné d'arbres. Nous formalisons une nouvelle méthode due à Cassens, Mardulyn et Milinkovitch, qui consiste à construire un graphe contenant tous les arbres partiellement étiquetés donnés et possédant le moins d'arêtes possible, et donnons des algorithmes résolvant le problème de manière optimale sur deux graphes, dont le temps d'exécution est exponentiel en général mais polynomial dans quelques cas que nous caractérisons.

One of the most challenging tasks remaining in the field of biochemistry is the one of understanding how the information within the amino acid sequence of proteins translates into a unique structure. Solving this problem would lead to endless possibilities for application in the medical and biotechnology industry. Many decades ago scientists realized that the process that facilitates the folding of a polypeptide chain could not be random and happen by chance; there needs to be direction in the folding free energy landscape. This landscape is defined by the thermodynamic factors entropy and enthalpy. The contribution made by enthalpy i.e. the contact energies from intra- and intermolecular interactions have been extensively investigated by various mutational studies. The influence of entropy on the other hand, is less well understood. My work focuses on the effect of altering the entropic components of forming the various parts of a known protein scaffold. This is done by genetic engineering in combination with biophysical characterisation and analysis. The results show effects on protein folding rates as well as on the pathway for nucleation and emphasis the ability of the folding landscape to readjust to entropic variations. Proteins are therefore not required to fold along a unique route to their final structure but can do so in several ways. The folding pathways we observe today have hence likely evolved as an adaptation to biological demands.

Functional magnetic resonance imaging (fMRI) is a prime example of multi-disciplinary research. Without the beautiful physics of MRI, there wouldnot be any images to look at in the first place. To obtain images of goodquality, it is necessary to fully understand the concepts of the frequencydomain. The analysis of fMRI data requires understanding of signal pro-cessing, statistics and knowledge about the anatomy and function of thehuman brain. The resulting brain activity maps are used by physicians,neurologists, psychologists and behaviourists, in order to plan surgery andto increase their understanding of how the brain works. This thesis presents methods for real-time fMRI and non-parametric fMRIanalysis. Real-time fMRI places high demands on the signal processing,as all the calculations have to be made in real-time in complex situations.Real-time fMRI can, for example, be used for interactive brain mapping.Another possibility is to change the stimulus that is given to the subject, inreal-time, such that the brain and the computer can work together to solvea given task, yielding a brain computer interface (BCI). Non-parametricfMRI analysis, for example, concerns the problem of calculating signifi-cance thresholds and p-values for test statistics without a parametric nulldistribution. Two BCIs are presented in this thesis. In the first BCI, the subject wasable to balance a virtual inverted pendulum by thinking of activating theleft or right hand or resting. In the second BCI, the subject in the MRscanner was able to communicate with a person outside the MR scanner,through a virtual keyboard. A graphics processing unit (GPU) implementation of a random permuta-tion test for single subject fMRI analysis is also presented. The randompermutation test is used to calculate significance thresholds and p-values forfMRI analysis by canonical correlation analysis (CCA), and to investigatethe correctness of standard parametric approaches. The random permuta-tion test was verified by using 10 000 noise datasets and 1484 resting statefMRI datasets. The random permutation test is also used for a non-localCCA approach to fMRI analysis.

The problem of counting ramified covers of a Riemann surface up to homeomorphism was proposed by Hurwitz in the late 1800's. This problem translates combinatorially into factoring a permutation with a specified cycle type, with certain conditions on the cycle types of the factors, such as minimality and transitivity. Goulden and Jackson have given a proof for the number of minimal, transitive factorizations of a permutation into transpositions. This proof involves a partial differential equation for the generating series, called the Join-Cut equation. Furthermore, this argument is generalized to surfaces of higher genus. Recently, Bousquet-M&eacute;lou and Schaeffer have found the number of minimal, transitive factorizations of a permutation into arbitrary unspecified factors. This was proved by a purely combinatorial argument, based on a direct bijection between factorizations and certain objects called <em>m</em>-Eulerian trees. In this thesis, we will give a new proof of the result by Bousquet-M&eacute;lou and Schaeffer, introducing a simple partial differential equation. We apply algebraic methods based on Lagrange's theorem, and combinatorial methods based on a new use of Bousquet-M&eacute;lou and Schaeffer's <em>m</em>-Eulerian trees. Some partial results are also given for a refinement of this problem, in which the number of cycles in each factor is specified. This involves Lagrange's theorem in many variables.