Dwork cohomology and algebraic ${\Cal D}$-modules

Using local cohomology and algebraic -Modules, we generalize a comparison theorem between relative de Rham cohomology and Dwork cohomology due to N. Katz, P. Monsky, A. Adolphson and S. Sperber. Received June 10, 1999 / Published online July 20, 2000     

Monotonicity of uniqueness for percolation on Cayley graphs: all infinite clusters are born simultaneously

. Consider site or bond percolation with retention parameter p on an infinite Cayley graph. In response to questions raised by Grimmett and Newman (1990) and Benjamini and Schramm (1996), we show that the property of having (almost surely) a unique infinite open cluster is increasing in p. Moreover, in the standard coupling of the percolation models for all parameters, a.s. for all p ₂>p ₁>p _c , each infinite p ₂-cluster contains an infinite p ₁-cluster; this yields an extension of Alexander's (1995) “simultaneous uniqueness” theorem. As a corollary, we obtain that the probability θ _v (p) that a given vertex v belongs to an infinite cluster is depends continuously on p throughout the supercritical phase p>p _c . All our results extend to quasi-transitive infinite graphs with a unimodular automorphism group. Received: 22 December 1997 / Revised version: 1 July 1998     

Rational points on some PEL-stacks

This work studies polarized abelian varieties with endomorphism ring a maximal order in a definite quaternion algebra. The result is that over small fields of characteristic 0 the degree of the polarization can only attain certain values. Received: 2 Febrauary 1999     

Milnor number at infinity, topology and Newton boundary of a polynomial function

Abstract. In this paper we show that the Euler characteristic of the generic fibre of a complex polynomial function can be easily computed using the Newton number of f. We apply this result to study polynomials with a finite number of critical points. Received May 25, 1998; in final form January 28, 1999     

Non-Archimedean integrals and stringy Euler numbers of log-terminal pairs

Using non-Archimedian integration over spaces of arcs of algebraic varieties, we define stringy Euler numbers associated with arbitrary Kawamata log-terminal pairs. There is a natural Kawamata log-terminal pair corresponding to an algebraic variety V having a regular action of a finite group G. In this situation we show that the stringy Euler number of this pair coincides with the physicists’ orbifold Euler number defined by the Dixon-Harvey-Vafa-Witten formula. As an application, we prove a conjecture of Miles Reid on the Euler numbers of crepant desingularizations of Gorenstein quotient singularities. Received March 19, 1998     

A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem

Summary. The Monge-Kantorovich mass transfer problem [31] is reset in a fluid mechanics framework and numerically solved by an augmented Lagrangian method. Received August 30, 1998 / Published online September 24, 1999     

Convergence of approximations to the primal problem in plasticity under conditions of minimal regularity

Summary. This work considers semi- and fully discrete approximations to the primal problem in elastoplasticity. The unknowns are displacement and internal variables, and the problem takes the form of an evolution variational inequality. Strong convergence of time-discrete, as well as spatially and fully discrete approximations, is established without making any assumptions of regularity over and above those established in the proof of well-posedness of this problem. Received June 8, 1998 / Published online July 12, 2000     

Exit time moments, boundary value problems, and the geometry of domains in Euclidean space

Let X _t be a diffusion in Euclidean space. We initiate a study of the geometry of smoothly bounded domains in Euclidean space using the moments of the exit time for particles driven by X _t , as functionals on the space of smoothly bounded domains. We provide a characterization of critical points for each functional in terms of an overdetermined boundary value problem. For Brownian motion we prove that, for each functional, the boundary value problem which characterizes critical points admits solutions if and only if the critical point is a ball, and that all critical points are maxima. Received: 23 January 1997 / Revised version: 21 January 1998     

Limiting angle of Brownian motion on certain manifolds

Summary. Suppose that M is a complete, simply connected Riemannian manifold of non-positive sectional curvature with dimension m ≧ 3. If, outside a fixed compact set, the sectional curvatures are bounded above by a negative constant multiple of the inverse of the square of the geodesic distance from a fixed point and below by another negative constant multiple of the square of the geodesic distance, then the angular part of Brownian motion on M tends to a limit as time tends to infinity, and the closure of the support of the distribution of this limit is the entire S ^m−1 . This improves a result of Hsu and March. Received: 7 December 1994/In revised form: 2 September 1995     

Hopf bifurcation of reaction-diffusion and Navier-Stokes equations under discretization

Summary. The long-time behaviour of numerical approximations to the solutions of a semilinear parabolic equation undergoing a Hopf bifurcation is studied in this paper. The framework includes reaction-diffusion and incompressible Navier-Stokes equations. It is shown that the phase portrait of a supercritical Hopf bifurcation is correctly represented by Runge-Kutta time discretization. In particular, the bifurcation point and the Hopf orbits are approximated with higher order. A basic tool in the analysis is the reduction of the dynamics to a two-dimensional center manifold. A large portion of the paper is therefore concerned with studying center manifolds of the discretization. Received March 18, 1997 / Revised version received February 19, 1998     

Differential operators and the Frobenius pull back of the tangent bundle on G/P

In this paper we study the sheaf of differential operators on a flag manifold X in characteristic p>0. We generalize the non-vanishing theorem of Haastert on the associated filtration of . We make use of the lifting of X to characteristic zero and the complex geometry of X. Received: 27 March 1998     

Schubert polynomials of types A--D

Schubert polynomials of type B, C, and D have been described first by S. Billey and M. Haiman [BH] using a combinatorial method. In this paper we give a unified algebraic treatment of Schubert polynomials of types A–D in the style of the Lascoux–Schützenberger theory in type A, i.e. Schubert polynomials are generated by the application of sequences of divided difference operators to “top polynomials”. The use of the creation operators for Q-Schur and P-Schur functions allows us to give: (1) simple and natural forms of the “top polynomials”, (2) formulas for the easy computation with all divided differences, (3) recursive structures, and (4) simplified derivations of basic properties. Received: 23 July 1998     

Dolbeault cohomology of G /( P,P )

Let G be a complex connected semi-simple Lie group, with parabolic subgroup P. Let (P,P) be its commutator subgroup. The generalized Borel-Weil theorem on flag manifolds has an analogous result on the Dolbeault cohomology . Consequently, the dimension of is either 0 or . In this paper, we show that the Dolbeault operator has closed image, and apply the Peter-Weyl theorem to show how q determines the value 0 or . For the case when P is maximal, we apply our result to compute the Dolbeault cohomology of certain examples, such as the punctured determinant bundle over the Grassmannian. Received: September 2, 1997; in final form February 9, 1998     

Sur l'effectivité numérique des images inverses de fibrés en droites

Sans résumé

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Re?u le 7 Octobre 1996