Showing papers in "Duke Mathematical Journal in 1987"

Stable bundles and integrable systems

Abstract: On considere la geometrie symplectique des fibres cotangents aux espaces de modules de fibres vectoriels stables sur une surface de Riemann. On montre que ce sont des systemes dynamiques hamiltoniens algebriquement completement integrables

Uniform distribution of eigenfunctions on compact hyperbolic surfaces

Abstract: On demontre que les fonctions propres {φ k } du laplacien sur une surface hyperbolique compacte X devient uniformement distribuee sur X quand k→∞

Regularity of solutions to the Schrödinger equation

Abstract: Soit f dans l'espace de Schwartz S(R n ) et soit S t f(x)=u(x,t)=∫ Rn exp(ix•ξ)•exp(it|ξ| 2 )•f^(ξ)dξ, x∈R n , t∈R. f^ est la transformee de Fourier de f. On considere u solution de l'equation de Schrodinger Δu=i(δu/δt). On pose S*f(x)=sup 0

Perron’s method for Hamilton-Jacobi equations

Abstract: On considere l'existence des solutions d'equations aux derivees partielles non lineaires scalaires d'ordre 1: F(x, u, Du)=0 dans Ω, ou Ω est un sous-ensemble ouvert de R N , F:Ω×R×R N →R est continue, u:Ω→R est l'inconnue

A theorem on refining division orders by the reverse lexicographic order

Abstract: Let k be an infinite field of any characteristic, and let S = k[xl9..., xn] be a graded polynomial ring, where each xf has degree one. Let / c S be a homogeneous ideal. Let Sd denote the finite vector space of all homogeneous, degree d polynomials in S, so S = SQ © #! © • • • © Sd © • • • . Writing / in the same manner as / = 70 © /a © • • • ©7rf © • • •, we have Id c Sd for each d. An order > on the monomials of Sd for each d is compatible with the monoid structure on the monomials of S if whenever X > X for two monomials X, X, then XX > XX for all monomials x. We shall only consider orders satisfying this compatibility condition. If an order > is a strict order on the monomials of each degree, one can use > in applying the division algorithm to constructing a standard (Grobner) basis for /. The standard basis for /, and its properties, will vary in a crucial way with the choice of order > . The subject of computing standard or Grobner bases has a long history; see [Bay85] for a recent survey. One can generalize the necessary definitions to nonstrict orders > , which fail to distinguish between all monomials of a given degree: For each polynomial / e S9 define in(/) to be the sum of those terms CX A of / which are greatest with respect to the order > . Define in(7) to be the ideal generated by {in(/)|/ e /}. Define fl9..., fr to be a standard basis for / with respect to the order > if in(/1),..., in(/r) generate the ideal in(7). If > is a strict order, in(J) will be a monomial ideal; if > is not strict, in(/) may fail to be a monomial ideal. A nonstrict order >l can be refined to a strict order by breaking any ties with a fixed strict order >2 ; the resulting order >3 is then a compatible order, so the usual division algorithm can be applied to compute standard bases with respect to >3 . Let ml9in29m3 correspond to >x, >2 , >3 . We shall see that in3(/) = in2(in1(/)), so a standard basis with respect to >3 is already a standard basis with respect to >x. Call >3 the refinement of >x by >2 . Thus, refinements provide a mechanism for computing with nonstrict orders. This has been observed for example in [MoM683], where in the affine setting, homogenizing bases (in the above sense, standard bases with respect to the total degree order) are computed via standard bases with respect to a strict order. We recall two frequently used strict orders: The lexicographic order is defined by X > X if the first nonzero entry in A-B is positive. The reverse lexico-

Weak (1,1) bounds for oscillatory singular integrals

Abstract: Pour un polynome P:R n ×R n →R et un noyau de Calderon-Zygmund K, l'operateur T f(x)=pv∫ exp [iP(x, y)], K (x-y) F (y) dy est de type faible sur L 1 , avec une borne dependant seulement de ||K|| CZ et le degre de P

Cm approximation by solutions of elliptic equations, and Calderón-Zygmund operators

Abstract: Soit XCR n compact et soit f une fonction deux fois continument differentiable dans un voisinage de X, dont le laplacien s'annule en chaque point de X. Alors il y a une suite (f j ) telle que chaque f j est une fonction harmonique sur un voisinage (dependant de j) de X, et ∂αf j →∂αf uniformement sur X, pour o≤|α|≤2

On the monodromy groups attached to certain families of exponential sums

Abstract: Introduction. The main theme of this paper is that very innocuous looking one-parameter families of exponential sums over finite fields can have quite strong variation as the parameter moves. Even when the parameter variety is the affine line A or the multiplicative group G over a finite field, the algebraic group