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Published in: Annales Mathématiques Blaise Pascal, 24(2) (2017), 235--291.; Let k be a totally real number field ant let k∞ be its cyclotomic Zp-extension for a prime p>2. We give (Theorem 3.2) a sufficient condition of nullity of the Iwasawa invariants lambda, mu, when p totally splits in k, and we obtain important tables of quadratic fields and p for which we can conclude that lambda = mu=0.We show that the number of ambiguous p-classes of kn (nth stage in k∞) is equal to the order of the torsion group T, of the Galois group of the maximal Abelian p-ramified pro-p-extension of k (Theorem 4.2), for all n >> e, where p^e is the exponent of U*/ adh(E) (in terms of local and global units of k). Then we establish analogs of Chevalley's formula using a family (Lambda_i^n)_{0≤i≤m_n} of subgroups of k* containing E, in which any x is norm of an ideal of kn. This family is attached to the classical filtration of the p-class group of kn defining the algorithm of computation of its order in m_n steps. From this, we prove (Theorem 6.1) that m_n ≥ (lambda.n + mu.p^n + nu)/v_p(T_k) and that the condition m_n = O(1) (i.e., lambda = mu=0) essentially depends on the P-adic valuations of the (x^(p-1)-1)/p, x in Lambda_i^n, for P I p, so that Greenberg's conjecture is strongly related to ``Fermat quotients'' in k*. Heuristics and statistical analysis of these Fermat quotients (Sections 6, 7, 8) show that they follow natural probabilities, linked to T_k whatever n, suggesting that lambda = mu=0 (Heuristics 7.1, 7.2, 7.3). This would imply that, for a proof of Greenberg's conjecture, some deep p-adic results (probably out of reach now), having some analogy with Leopoldt's conjecture, are necessary before referring to the sole algebraic Iwasawa theory.

Dans cet article, nous nous intéressons aux dimensions de certaines variétés qui ont été introduites récemment par Kisin pour démontrer la modularité de certaines représentations galoisiennes. Nous étudions plus spécialement un cas particulier pour lequel nous donnons une estimation de la dimension en question, puis, en nous basant sur ce résultat, nous énonçons une conjecture dans le cas général. In this paper, we study dimensions of some varieties, that were introduced recently by Kisin in order to prove modularity of some Galois representations. In fact, we mainly consider a special case for which we obtain an estimation of the dimension we are interested in. Then, based on this result, we state a conjecture for the general case.

On R d , we consider the differential operator −�+jj xjj 2 +R(x)+M. Here R is in the Schwartz class and M is a bounded operator of L 2 (R). Under a localization hypothesis on the spectrum of −� + jj xjj 2 + R(x), which is automatically satisfied if d = 1, one proves that there are arbitrarily small operator M, i.e. jj M jj � 1, such that for all r � 3, for small Cauchy data of order " in high Sobolev norms, the nonlinear Schrodinger equation admits a solution which is bounded by 2" on a time existence interval of length � " −r . The case R = 0 has been proved by Grebert-Imekraz-Paturel. Here we do not need any explicit spectral analysis (eigenvalues and eigenfunctions) of −� + jj xjj 2 + R(x). The role of M is making the spectrum of −�+ jj xjj 2 + R(x)+ M nonresonant. We also give a partial answer in the case M = 0 : there are some potentials R such that the spectrum of −@2 x + x2 + R(x) is nonresonant. For this, we use some Chelkak-Kargaev-Korotyaev's results about the inverse spectral problem of harmonic oscillator and construct explicit dual basis of the squared Hermite functions. Sur R d , nous considerons l'operateur differentiel −� + jj xjj 2 + R(x) +

Nous donnons une interprétation des fonctions ordinales invariantes comme fonctions indépendantes des échelles pour l’agrégation sur des échelles ordinales finies. Plus précisément, nous montrons comment les fonctions ordinales invariantes agissent, à travers des représentants discrets, sur des échelles ordinales représentées par des chaînes finies. En particulier, cette interprétation permet de justifier d’une façon naturelle la propriété de continuité pour certaines fonctions ordinales invariantes. We give an interpretation of order invariant functions as scale independent functions for the aggregation on finite ordinal scales. More precisely, we show how order invariant functions can act, through discrete representatives, on ordinal scales represented by finite chains. In particular, this interpretation allows us to justify the continuity property for certain order invariant functions in a natural way.

RésuméOn the set of minimax trees of a given order there can be defined two families of operations, the complements and the reverses. We study the actions of those operations and show that their orbits are enumerated by combinatorial objects previously introduced, such as the Hetyei–Reiner trees, the increasing trees, and the André trees. Various generating functions for those trees by several statistics are also derived.

The determination of roots of a function is a problem that manifests itself in almost every branch of acoustics. The usual way to determine them involves using Newton-Raphson iteration to refine their positions from approximate initial values which often leads to problems. This paper deals with an entirely different method based on winding number integral. This method has already been used but not when the function has branch points. In this paper, we give the strategy to use in order to deal with branch points arising from square root functions

In a recent paper (Hairer-Wanner [1]) we have given a theory with which it is easy to calculate the order conditions for Nystrom methods. Here we show how it is possible to solve this system of non-linear algebraic equations. Moreover we present all methods of orders fors=2, 3, 4 (s?1 indicates the number of function evaluations per step); methods with one parameter of orders fors=5, 6 and some special methods of orders?1 fors=8, 9. Dans un recent article (Hairer-Wanner [1]) nous avons donne une theorie a l'aide de laquelle on peut facilement calculer les conditions d'ordre pour une methode de Nystrom. Ici nous montrons comment on peut resoudre ce systeme d'equations non-lineaires. Nous donnons de plus toutes les methodes d'ordres pours=2, 3, 4 (ous?1 indique le nombre d'evaluations de la fonction a chaque pas); des methodes avec un parametre d'ordres pours=5, 6 et des methodes particulieres d'ordres?1 pours=8, 9.