Soient k un corps de nombres et f: X → P1k un k-morphisme surjectif de k-variétés projectives lisses, à fibre générique géométriquement intègre. Supposons que les fibres de f au-dessus d'un sous-ensemble Hilbertien de P1(k) satisfont le principe de Hasse (resp. le principe de Hasse et l'approximation faible). Supposons que toutes les fibres géométriques de f ont au moins une composante de multiplicité un. Le rang de f est la somme des degrés des points fermés P dont la fibre XP = f−1(P) ne possède pas de composante de multiplicité un géométriquement intègre. Supposons le rang au plus égal à 2. Alors l'obstruction de Brauer–Manin au principe de Hasse (resp. à l'approximation faible) sur X est la seule. Dans des articles antérieurs nous n'avions obtenu ce résultat que sous des hypothèses plus fortes sur la nature des fibres. Le présent énoncé, obtenu grâce à une descente sur des variétés ouvertes, permet d'étudier des variétés données par des équations affines simples, sans calcul explicite d'un modè le projectif et lisse.

In this paper, we construct the Ext-quivers of the principal blocks of F[Sfr ]11 and F[Sfr ]12, where F is an algebraically closed field of characteristic 3. We also obtain the Loewy structures of three principal indecomposable modules of the principal block of F[Sfr ]11.

In analogy with the setting of homogeneous, locally finite trees we introduce the Radon transforms on affine, locally finite buildings with diagram of type Ãn. We study the problem of the inversion of these transforms and in some cases we show how inversion algorithms and formulas are available.

In the study of highest weight categories, the class of Weyl modules Δ(λ) and their duals ∇(λ) are of central interest; this is for example motivated by the problem of finding the characters of the simple modules. Weyl modules form the building blocks for the category [Fscr ](Δ), whose objects have a filtration 0 = M0 [les ] M1 [les ] … [les ] Mi−1 [les ] Mi = M with quotients isomorphic to Δ(λ) for various λ. Knowing Extr(Δ(λ), Δ(μ)) is essential for the understanding of this category.

In [3, 13], we determined Ext1 for Weyl modules of SL(2, k) and q-GL(2, k) over an infinite field k of characteristic p > 0. Here we are able to extend these results to determine Ext2 for Weyl modules in both these cases (see Theorem 4 · 6). Moreover, this also gives Ext2 between any pair of Weyl modules Δ(λ), Δ(μ) for q-GL(n, k) (where n [ges ] 2) such that both λ and μ have at most two rows or two columns, or where they differ by some multiple of a simple root (see Section 7).

Consider (for simplicity) polynomial representations of degree d for GL(n, k). A partition of d which has at most two rows is uniquely determined by the difference in the row lengths, which we use as a label for the partition. In this case our main result is

Modify the usual percolation process on the infinite binary tree by forbidding infinite clusters to grow further. The ultimate configuration will consist of both infinite and finite clusters. We give a rigorous construction of a version of this process and show that one can do explicit calculations of various quantities, for instance the law of the time (if any) that the cluster containing a fixed edge becomes infinite. Surprisingly, the distribution of the shape of a cluster which becomes infinite at time t > ½ does not depend on t; it is always distributed as the incipient infinite percolation cluster on the tree. Similarly, a typical finite cluster at each time t > ½ has the distribution of a critical percolation cluster. This elaborates an observation of Stockmayer [12].

We use the theory of the integral closure of an ideal to study the equivalence of map-germs under C0 coordinate changes in the target. We also derive a formula for the number of double points of a map germ from Cn → C2n in terms of the Segre number of dimension 0 of an ideal associated to the double point locus of f, and the number of Whitney umbrellas of the composition of f with a generic projection to C2n−1.

In order to model biological reactions, distances between knots/links based on tangle replacement are defined. Given a tangle R, an R-move is defined as the replacement of the zero tangle in a link L with the tangle R. The R-distance between the link L and the link M is defined to be the minimum number of R-moves required to change L into M where the minimum is taken over all diagrams of the link L. A formula is given to determine when one 4-plat knot/link can be obtained from another 4-plat via one R-move when R is a rational tangle and not equal to the 1/n tangle. The formula can also be used to find all rational tangles R for which the R-distance between two given 4-plats is 1 except in the case R = 1/n tangle. These results are also generalized to the case when any rational tangle P is replaced with any rational tangle R, P not necessarily the zero tangle.

Roughly speaking, an ideal immersion of a Riemannian manifold into a space form is an isometric immersion which produces the least possible amount of tension from the ambient space at each point of the submanifold. In this paper we study Lagrangian immersions in complex space forms which are ideal. We prove that all Lagrangian ideal immersions in a complex space form are minimal. We also determine ideal Lagrangian submanifolds in complex space forms.

An orthonormal basis of analytic functions in the Bergman space is a doubly orthogonal system if it simultaneously is an orthogonal system in a larger domain. The case when one of the basis elements is a monomial is characterized as belonging to a pair of domains of concentric discs.

We prove that in a separable infinite dimensional Hilbert space the Dynkin system generated by the family of all open balls (that is, the smallest collection containing the open balls and closed under complements and countable disjoint unions) does not contain all Borel sets.

Consider an (L, α)-superdiffusion X, 1 < α [les ] 2, in a smooth cylinder Q = ℝ+ × D. Where L is a uniformly elliptic operator on ℝ+ × ℝd and D is a bounded smooth domain in ℝd. Criteria for determining which (internal) subsets of Q are not hit by the graph [Gscr ] of X were established by Dynkin [5] in terms of Bessel capacity and according to Sheu [14] in terms of restricted Hausdorff dimension (partial results were also obtained by Barlow, Evans and Perkins [3]). While using Poisson capacity on the lateral boundary ∂Q of Q, Kuznetsov [10] recently characterized complete subsets of ∂Q which have no intersection with [Gscr ]. In this work, we examine the relations between Poisson capacity and restricted Hausdorff measure. According to our results, the critical restricted Hausdorff dimension for the lateral [Gscr ]-polarity is d − (3 − α)/(α − 1). (A similar result also holds for the case d = (3 − α)/(α − 1)). This investigation provides a different proof for the critical dimension of the boundary polarity for the range of X (as established earlier by Le Gall [12] for L = Δ, α = 2 and by Dynkin and Kuznetsov [7] for the general case).

Sanov's Theorem states that the sequence of empirical measures associated with a sequence of i.d.d. random variables satisfies the large deviation principle (LDP) in the weak topology with rate function given by a relative entropy. We present a derivative which allows one to establish LDPs for symmetric functions of many i.d.d. random variables under the condition that (i) a law of large numbers holds whatever the underlying distribution and (ii) the functions are uniformly Lipschitz. The heuristic (of the title) is that the LDP follows from (i) provided the functions are ‘sufficiently smooth’. As an application, we obtain large deviations results for the stochastic bin-packing problem.