Showing papers in "Inventiones Mathematicae in 1971"

Undecidability and Nonperiodicity for Tilings of the Plane.

Abstract: This paper is related to the work of Hao Wang and others growing out of a problem which he proposed in [8], w 4.1. Suppose that we are given a finite set of unit squares with colored edges, placed with their edges horizontal and vertical. We are interested in tiling the plane with copies of these tiles obtained by translation only. The tiles are to be placed with their vertices at lattice points, and abutting edges must have the same color. Wang raised the question whether there is a general method of deciding which finite sets of colored squares can be used to tile the plane in this way. He also discussed the relation of this problem to the decision problem for certain classes of formulas of the predicate calculus, but we shall consider only the geometrical problem here. Suppose that we have a tiling of the plane of this type which has a horizontal period. That is, we assume that the tiling remains invariant under a certain horizontal translation. There will then be a vertical strip which can be repeated to cover the plane. This strip has only a finite number of different horizontal cross sections, and hence has two which are alike. Thus the same tiles may be used to construct a tiling which has a vertical period as well as a horizontal period. A similar argument can be used even when the given period is not horizontal. That is, if a set of tiles permits a periodic tiling, then it also permits a doubly periodic tiling. In any such tiling, we can find equal horizontal and vertical periods, and hence can find a square of some size which repeats to cover the plane. Wang made the conjecture, since proved false, that any set of tiles which permits a tiling of the plane also permits a periodic tiling. He pointed out that if this conjecture were true, then we would have a decision method for an arbitrary set of tiles. Indeed, it would be sufficient to form all possible squares from the given set of tiles, starting with the smaller squares and working up, until we either reach a square which can be repeated periodically, or we find a square of a certain size which cannot be tiled at all. The latter will always happen if tiling of the whole plane

La conjecture de Weil pour les surfaces K3.

Abstract: 1. Enonc6 du th~or~me Soient Fq un corps ~t q ~l~ments, Fq une cl6ture alg6brique de Fq, r la substitution de Frobenius xv-~x q et F= tp -1 le << Frobenius g6om6trique >>. Soit X un sch6ma (s6par6 de type fini) su r Fq, et soit X le sch6ma sur Fq qui s'en d6duit par extension des scalaires. Pour tout point ferm6 x de X, soit deg(x)=[k(x): Fq] le degr6 sur Fq de l'extension r6siduelle. La fonction zSta Z(X, t)~Z [[t]] est d6finie par

Régularité de processus gaussiens

Abstract: On etudie la continuite presque sure ou la majoration presque sure des trajectoires de certains processus gaussiens en fonction de la regularite de leur covariance: on utilise la notion d'espace d'Orlicz pour obtenir des resultats plus fins que les resultats classiques. En appendice, on donne une demonstration elementaire de l'integrabilite des vecteurs aleatoires gaussiens a valeurs dans les espaces vectoriels generaux.

Two Theorems on Extensions of Holomorphic Mappings.

Abstract: In this paper we shall prove two theorems about extending holo-morphic mappings between complex manifolds. Both results involve extending such mappings across pseudo-concave boundaries. The first is a removable singularities statement for meromorphic mappings into compact K~ihler manifolds. The precise result and several illustrative examples are given in Section 1. The second theorem is a Hartogs'-type result for holomorphic mappings into a complex manifold which has a complete Hermitian metric with non-positive holomorphic sectional curvatures. This theorem answers one of Chern's problems posed at the Nice Congress [3]. The precise statement and further discussion is given in Section 4. The proofs of both theorems use the class of pluri-sub-harmonic (p. s.h.) functions, which is intrinsically defined on any complex mani-fold [9]. The second proof is rather elementary and essentially relates the p.s.h, functions on the domain off to the curvature assumption on the image manifold. The first theorem is technically a little more delicate and makes use of the removable singularity theorems for analytic sets due to Bishop-Stoll [14] together with the strong estimates available for the amount of singularity which the Levi form of a p. s. h. function may have at an isolated singularity of such a function. At the end of this paper there are two appendices. The first contains a brief survey of some removable singularity theorems for holomorphic mappings between complex manifolds. In the second appendix we give an informal discussion of the general problem of defining the \"order of growth\" of a holomorphic mapping and using this notion to study such maps. The basic open question here is what might be termed \"Bezout's theorem for holomorphic functions of several variables,\" and this problem is discussed and precisely formulated there. It is my pleasure to acknowledge many helpful discussions with H. Wu concerning the material presented below. In particular, several of the ideas and results in Appendix 2 were communicated to me by him.

Duality and the deRham cohomology of infinitesimal neighborhoods

Abstract: Our principal application of the techniques developed in this note appears in w 7 where we establish the following result. Let X be a nonsingular, compact, connected, analytic space (resp. a complete algebraic variety over C) of dimension m and let Y be an arbitrary closed subvariety of X, U = X E Then the standard exact sequences 1) ..---+ HI(U, C ) , HP(X, C)-* He(Y,, C)--~Hy + '(U, C)-+ . . . 2) . . . < H e m p ( g , C ) ~ ~ H 2 m p ( x , C ) , ~ H 2 m p ( x , C )

Construction of Non-Linear Local Quantum Processes.

Abstract: : A species of singular perturbation theory applicable to a general class of nonlinear local quantum fields is developed. The theory is applied in detail to the arbitrary scalar relativistic field in two space time dimensions with positive-energy self-interaction. (Author)