We prove a conjecture of Conrad, Diamond, and Taylor on the size of certain deformation rings parametrizing potentially Barsotti-Tate Galois representations. To achieve this, we extend results of Breuil and Mézard (classifying Galois lattices in semistable representations in terms of “strongly divisible modules”) to the potentially crystalline case in Hodge-Tate weights (0, 1). We then use thes...

Abstract: Let A be a separable, unital, approximately divisible C∗-algebra. We show that A is generated by two self-adjoint elements and the topological free entropy dimension of any finite generating set of A is less than or equal to 1. In addition, we show that the similarity degree of A is at most 5. Thus an approximately divisible C∗-algebra has an affirmative answer to Kadison’s similarity...

Discrete multiplicative turbulent cascades are described using a formalism involving infinitely divisible random measures. This permits to consider the continuous limit of a cascade developed on a continuum of scales, and to provide the stochastic equations defining such processes, involving infinitely divisible stochastic integrals. Causal evolution laws are also given. This gives the first ge...

The object of this paper is to study the asymptotic dependence structure of the linear time series models with infinitely divisible innovations by the use of their characteristic functions. Autoregressive moving-average (ARMA) models and fractional autoregressive integrated moving-average (FARIMA) models are analyzed. As examples of infinitely divisible innovations, the class of radially absolu...

A resolvable modified group divisible design (RMGD) is a modified group divisible design whose blocks can be partitioned into parallel classes. We show that the necessary conditions for the existence of a 3-RMGDDλ of type g, namely g ≥ 3, u ≥ 3, gu ≡ 0 mod 3 and λ(g−1)(u−1) ≡ 0 mod 2, are sufficient with the two exceptions of (g, u, λ) ∈ {(6, 3, 1), (3, 6, 1)}.

LetX be a smooth complex variety of dimension at most two, and let F be its function ¢eld. We prove that the K -groups of F are divisible above the dimension of X, and that the K-groups of X are divisible-by-¢nite.We also describe the torsion in the K -groups of F and X. Mathematics Subject Classi¢cations (2000). Primary: 19E08; secondary: 19E20, 14C35, 14F20.

For every prime p we give infinitely many examples of torsors under abelian varieties over Q that are locally trivial but not divisible by p in the Weil-Châtelet group. We also give an example of a locally trivial torsor under an elliptic curve over Q which is not divisible by 4 in the Weil-Châtelet group. This gives a negative answer to a question of Cassels.

A multigraph G is divisible by t if its edge set can be partitioned into / subsets, such that the subgraphs (called factors) induced by the subsets are all isomorphic. If G has e(G) edges, then it is t-rational if it is divisible by t or if / does not divide e{G). A short proof is given that any graph G is /-rational for all t ^ x'(G) ( t n e chromatic index of G), and thus any r-regular graph ...

The notion of exponential Dowling structures is introduced, generalizing Stanley’s original theory of exponential structures. Enumerative theory is developed to determine the Möbius function of exponential Dowling structures, including a restriction of these structures to elements whose types satisfy a semigroup condition. Stanley’s study of permutations associated with exponential structures l...