In this paper we define some lifts of tensor fields of types (1, k) and (0, k) as well as connections to a product preserving functor . We study algebraic properties of introduced lifts and we apply these lifts to prolongation of geometric structures from a manifold M to (M). In particular cases of the tangent bundle of pr -velocities and the tangent bundle of infinitesimal near points our constructions contain all constructions due to Morimoto (see [20]-[23]). In the cases of the tangent bundle our definitions coincide with the definitions of Yano and Kobayashi (see [31]). To construct our lifts and to study its properties we use only general properties of product preserving functors. All lifts verify so-called the naturality condition. It means that for a smooth mapping φ:M → N and for two φ-related geometric objects defined on M and N its lifts to (M) and (N) respectively are (φ)-related. We explain later the term φ-related for considered geometric objects.

All rings are commutative with unit and modules are unital. With one exception all rings are Noetherian. We consider only rings of positive characteristic p. Section 0 contains background, definitions and terminology.

Recent development in hypergeometric functions in several variables has made the importance of studying twisted rational de Rham cohomology clear to many specialists. Roughly speaking, a hypergeometric function in our sense is the integral of a product of complex powers of polynomials Pj(u1, . . . . ,un) : ∫ U du1 ∧ · · · ∧ dun, U = Π , integration being taken over some cycle. So we are led naturally to consider the twisted rational de Rham cohomology, which is a direct generalization of the usual de Rham cohomology to multivalued case.

This paper deals with a decomposition problem for some classes of distributions. Let D be a given class of distribution on R1, which we are interested in. After showing that the class D is closed under convolution, our purpose is to give an answer to the inverse problem: if the convolution of two distributions μ1 and (μ2 belongs to D, then do μ1 and μ2 belong to D?

The evaluation of the integral

(1.1)

represents one of the fundamental problems of the theory of the Riemann zeta-function (see [4] for a comprehensive account). In view of the functional equation

it is clear that one has to distinguish between the following three principal cases:

• a) σ = 1/2 (“the critical line”),

• b) 1/2 < σ < 1 (“the critical strip”),

• c) σ = 1.

1.1. Thickness. Let E be a real inner product space. For a finite sequence of points a0, . . . ,ak in E we let a0. . . ,ak denote the convex hull of the set {a0, . . . , ak}. If these points are affinely independent, the set Δ = a0. . .ak is a k-simplex with vertices a0. . . ,ak. It has a well-defined k-volume written as mk(Δ) or briefly as m(Δ). We are interested in sets A ⊂ E which are “nowhere too flat in dimension k”. More precisely, suppose that A ⊂ E, q > 0 and that k: is a positive integer. We let denote the closed ball with center x and radius r. We say that A is (q, k)-thick if for each x ∈ A and r> 0 such that A\ ≠ there is a k-simplex Δ with vertices in A ∩ such that mk(Δ) ≥ qr.

As is well known, there exists a canonical transversal vector field on a non-degenerate affine hypersurface M. This vector field is called the affine normal. The second fundamental form associated to this affine normal is called the affine metric. If M is locally strongly convex, then this affine metric is a Riemannian metric. And also, using the affine normal and the Gauss formula one can introduce an affine connection ∇ on M which is called the induced affine connection. Thus there are in general two different connections on M: one is the induced connection ∇ and the other is the Levi Civita connection of the affine metric h. The difference tensor K is defined by K(X, Y) = KXY — ∇XY — XY. The cubic form C is defined by C = ∇h and is related to the difference tensor by

.

In the study of ordinary differential equations, Malgrange ([Ma]) and Ramis ([R1], [R2]) established index theorem in (formal) Gevrey spaces, and the notion of irregularity was nicely defined for the study of irregular points. In their studies, a Newton polygon has a great advantage to describe and understand the results in visual form. From this point of view, Miyake ([M2], [M3], [MH]) studied linear partial differential operators on (formal) Gevrey spaces and proved analogous results, and showed the validity of Newton polygon in the study of partial differential equations (see also [Yn]).

Soit φ un polynome symétrique de q variables (q ≥ 1), homogène de degré p (p entier ≥ 2), à coefficients complexes. Soient (α1, α2, . . . , p) et (β1, β2, . . . , βp) deux familles de p nombres complexes, tous non nuls, et telles que toutes les différences soient également non nulles pour i ≠ j. Soient enfin (λ1, λ2, . . . , λq) et (μ1, μ2, . . . , μq) deux familles de q nombres complexes. On a alors:

THÉORÈME 1.

.