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The motivation for the work in this thesis is the study of models for extreme values that have clear practical benefit. Specific emphasis is placed on the modelling of extremes of environmental phenomena, which often exhibit spatial or temporal dependence, or both, or are forced by external factors. The peaks-over-threshold approach to modelling extremes combats temporal dependence, providing a way in which likelihood-based methods may he used reliably. The method is widely used and has sound asymptotic foundations. However its performance in practical situations is less well understand. The essence of the method is to identify clusters of extremes and estimate the required extremal properties based only on the cluster peaks. A simulation study is used here to assess the performance of the method. This study shows that while not robust to some of its arbitrary choices, such as cluster identification procedure, if clusters are identified using Ferro and Segers' (2003) automatic procedure then the peaks-over-threshold method typically gives accurate estimates of extremal properties. It is often common for extreme values to be affected by external factors. For example environmental extremes may be expected to behave differently at different times of the year. Incorporating beliefs about external factors was recognised early on in the development of extremal models as an important consideration, and a simple way in which this can be achieved is by allowing parameters of extremal distributions to depend on covariates. The work here considers whether choosing logical covariate forms for variation in parameters leads to improved estimation of extremal properties. It is found that a degree of improved accuracy in estimates can be achieved upon choice of a suitable model. but that the uncertainty in estimates, which is important to report, is poorly quantified.

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This thesis is concerned mainly with the interplay between identities involving power series (which are called q-series) and combinatorics, in particular the theory of partitions. The thesis includes new proofs of some q-series identities and some ideas about the generating functions for the rank and crank, a new proof of the triple product identity and a combinatorial proof of a q-elliptic identity.

The topic of this thesis is the hat problem. In this problem, a team of n players enters a room, and a blue or red hat is randomly placed on the head of each player. Every player can see the hats of all of the other players but not his own. Then each player must simultaneously guess the color of his own hat or pass. The team wins if at least one player guesses his hat color correctly and no one guesses his hat color wrong, otherwise the team loses. The aim is to maximize the probability of winning. This thesis is based on publications, which form the second chapter. In the first chapter we give an overview of the published results. In Section 1.1 we introduce to the hat problem and the hat problem on a graph, where vertices correspond to players, and a player can see the adjacent players. To the hat problem on a graph we devote the next few sections. First, we give some fundamental theorems about the problem. Then we solve the hat problem on trees, cycles, and unicyclic graphs. Next we consider the hat problem on graphs with a universal vertex. We also investigate the problem on graphs with a neighborhood-dominated vertex. In addition, we consider the hat problem on disconnected graphs. Next we investigate the problem on graphs such that the only known information are degrees of vertices. We also present Nordhaus-Gaddum type inequalities for the hat problem on a graph. In Section 1.6 we investigate the hat problem on directed graphs. The topic of Section 1.7 is the generalized hat problem with q >= 2 colors. A modified hat problem is considered in Section 1.8. In this problem there are n >= 3 players and two colors. The players do not have to guess their hat colors simultaneously and we modify the way of making a guess. We give an optimal strategy for this problem which guarantees the win. Applications of the hat problem and its connections to different areas of science are presented in Section 1.9. We also give there a comprehensive list of variations of the hat problem considered in the literature.

We consider different ways of describing a matroid to a Turing machine by listing the members of various families of subsets, and we construct an order on these different methods of description. We show that, under this scheme, several natural matroid problems are complete in classes thought not to be equal to P. We list various results linking parameters of basis graphs to parameters of their associated matroids. For small values of k we determine which matroids have the clique number, chromatic number, or maximum degree of their basis graphs bounded above by k. If P is a class of graphs that is closed under isomorphism and induced subgraphs, then the set of matroids whose basis graphs belong to P is closed under minors. We characterise the minor-closed classes that arise in this way, and exhibit several examples. One way of choosing a basis of a matroid at random is to select a total ordering of the ground set uniformly at random and use the greedy algorithm. We consider the class of matroids having the property that this procedure chooses a basis uniformly at random. Finally we consider a problem mentioned by Oxley. He asked if, for every two elements and n - 2 cocircuits in an n-connected matroid, there is a circuit that contains both elements and that meets every cocircuit. We show that a slightly stronger property holds for regular matroids.

Cette thèse concerne la combinatoire et l'algorithmique des cartes. En utilisant la théorie des espèces de Joyal, on parvient à des résultats énumératifs concernant les cartes non-étiquetées et étiquetées, enracinées ou non, en genre quelconque suivant leur nombre de faces et d'arêtes. Nous relions la combinatoire des cartes à l'asymptotique de la fonction de Airy par un rapprochement inattendu entre la série génératrice du nombre de cartes triangulaires et le développement asymptotique de la fonction de Airy. Nous donnons également un algorithme permettant de dresser une liste exhaustive des cartes triangulaires, en temps amorti constant pour le cas enraciné. / This thesis is about combinatoric and algorithmic aspects of maps. Using the species theory of Joyal, we get enumerative results concerning labeled and unlabeled maps both rooted or not, of any genus, by the number of their edges and faces. We relate the combinatorics of maps to the asymptotics of the Airy function by a unexpected matching of the generating series of triangular maps and the asymptotic development of the Airy function.We also give an algorithm able to produce an exhaustive list of triangular maps, in constant amortized time in the rooted case.