Memoir on the theory of the partitions of numbers /

Mode of access: Internet.

Weak hypergraph regularity and linear hypergraphs

We consider conditions which allow the embedding of linear hypergraphs of fixed size. In particular, we prove that any k-uniform hypergraph H of positive uniform density contains all linear k-uniform hypergraphs of a given size. More precisely, we show that for all integers l >= k >= 2 and every d > 0 there exists Q > 0 for which the following holds: if His a sufficiently large k-uniform hypergraph with the property that the density of H induced on every vertex subset of size on is at least d, then H contains every linear k-uniform hypergraph F with l vertices. The main ingredient in the proof of this result is a counting lemma for linear hypergraphs, which establishes that the straightforward extension of graph epsilon-regularity to hypergraphs suffices for counting linear hypergraphs. We also consider some related problems. (C) 2009 Elsevier Inc. All rights reserved.

Researches on Waring's problem,

Mode of access: Internet.

The expectation of parts into which a magnitude is divided at random, investigated mainly by algebraical methods,

A chapter supplementary to the author's Choice and Chance.

Choice and chance with 1000 exercises,

Mode of access: Internet.

Choice and chance, two chapters of arithmetic, with an appendix containing the algebraical treatment of permutations and combinations newly set forth.

Mode of access: Internet.

Choice and chance, with 1000 exercises.

Mode of access: Internet.

Splittings of free groups, normal forms and partitions of ends

Splittings of a free group correspond to embedded spheres in the 3-manifold M = # (k) S (2) x S (1). These can be represented in a normal form due to Hatcher. In this paper, we determine the normal form in terms of crossings of partitions of ends corresponding to normal spheres, using a graph of trees representation for normal forms. In particular, we give a constructive proof of a criterion determining when a conjugacy class in pi (2)(M) can be represented by an embedded sphere.

Partitions spectrales optimales pour les problèmes aux valeurs propres de Dirichlet et de Neumann

Les façons d'aborder l'étude du spectre du laplacien sont multiples. Ce mémoire se concentre sur les partitions spectrales optimales de domaines planaires. Plus précisément, lorsque nous imposons des conditions aux limites de Dirichlet, nous cherchons à trouver la ou les partitions qui réalisent l'infimum (sur l'ensemble des partitions à un certain nombre de composantes) du maximum de la première valeur propre du laplacien sur tous ses sous-domaines. Dans les dernières années, cette question a été activement étudiée par B. Helffer, T. Hoffmann-Ostenhof, S. Terracini et leurs collaborateurs, qui ont obtenu plusieurs résultats analytiques et numériques importants. Dans ce mémoire, nous proposons un problème analogue, mais pour des conditions aux limites de Neumann cette fois. Dans ce contexte, nous nous intéressons aux partitions spectrales maximales plutôt que minimales. Nous cherchons alors à vérifier le maximum sur toutes les $k$-partitions possibles du minimum de la première valeur propre non nulle de chacune des composantes. Cette question s'avère plus difficile que sa semblable dans la mesure où plusieurs propriétés des valeurs propres de Dirichlet, telles que la monotonicité par rapport au domaine, ne tiennent plus. Néanmoins, quelques résultats sont obtenus pour des 2-partitions de domaines symétriques et des partitions spécifiques sont trouvées analytiquement pour des domaines rectangulaires. En outre, des propriétés générales des partitions spectrales optimales et des problèmes ouverts sont abordés.

Probabilistic models over ordered partitions with applications in document ranking and collaborative filtering

Ranking is an important task for handling a large amount of content. Ideally, training data for supervised ranking would include a complete rank of documents (or other objects such as images or videos) for a particular query. However, this is only possible for small sets of documents. In practice, one often resorts to document rating, in that a subset of documents is assigned with a small number indicating the degree of relevance. This poses a general problem of modelling and learning rank data with ties. In this paper, we propose a probabilistic generative model, that models the process as permutations over partitions. This results in super-exponential combinatorial state space with unknown numbers of partitions and unknown ordering among them. We approach the problem from the discrete choice theory, where subsets are chosen in a stagewise manner, reducing the state space per each stage significantly. Further, we show that with suitable parameterisation, we can still learn the models in linear time. We evaluate the proposed models on two application areas: (i) document ranking with the data from the recently held Yahoo! challenge, and (ii) collaborative filtering with movie data. The results demonstrate that the models are competitive against well-known rivals.

Mixed partitions of sets of triples into small planes

We continue our study of partitions of the set of all ((v)(3)) triples chosen from a v-set into pairwise disjoint planes with three points per line. We develop further necessary conditions for the existence of partitions of such sets into copies of PG(2, 2) and copies of AG(2, 3), and deal with the cases v = 13, 14, 15 and 17. These partitions, together with those already known for v = 12, 16 and 18, then become starters for recursive constructions of further infinite families of partitions. (C) 2004 Elsevier B.V. All rights reserved.