Dans ce qui suit f(X) désigne un polynöme irréductible de degré n de et un polynöme de tel que a≠0 et r désigne un entier non nul.

On montre dans cet article que les polynÔmes ne sont pas irréductibles sur pour m≥3.

Many problems in mathematical physics can be formulated as differential equations of second order in time:

with V a convex functional. This is the Euler equation for the Lagrangian

which is convex with respect to x, and concave with respect to x.

A space Z is said to have the complete invariance property (CIP) provided that every nonempty closed subset of Z is the fixed point set of some continuous self-mapping of Z. In this paper it is shown that there exists a one-dimensional contractible planar continuum having CIP whose wedge with itself at a specified point is contractible, planar, and does not have CIP.

For l≤p<∞ μ∈ℝ, let ℒμ.p denote the collection of functions f, measurable on (0, ∞) and such that

Let C0 be the collection of functions continuous and compactly supported on (0, ∞); it is known that C0 is dense in ℒμ.p—see [2; Lemma 2.2]. If X and Y are Banach spaces, denote by [X, Y] the collection of bounded linear operators from X into Y, abbreviating [X, X] to [X].

We study here the smoothness of solutions of the Dirichlet problem for elliptic equations in a region G with a piece-wise smooth boundary. The smoothness of the solution given depends on the smoothness of the coefficients of the equation, the boundary, the boundary function and the values of the angles on the boundary and the values of the coefficients of the second derivatives at the corners.

Let L be a countable first-order language and T a fixed complete theory in L. If is a model of T, is an n-sequence of variables, and ā=〈a1,…, an〉 is an n-sequence of elements of M, the universe of , we let where ranges over formulas of L containing freely at most the variables υ1,…υn. ā is said to realize in We let be where is the sequence of the first n variables of L.

The classical rings of number theory, Dedekind domains, are characterized by the property that every ideal is a product of prime ideals. More generally, a commutative ring R with identity has the property that every ideal is a product of prime ideals if and only if R is a finite direct sum of Dedekind domains and special principal ideal rings. These rings, called general Z.P.I. rings, are also characterized by the property that every (prime) ideal is finitely generated and locally principal.

A variety of applications depend on the number of solutions of polynomial equations over finite fields. Here the usual situation is reversed and we show how to use geometrical methods to estimate the number of solutions of a non-homogeneous symmetric equation in three variables.

Let T be a Sylow 2-subgroup of the finite group G and let 1<S≤T with S strongly closed in T with respect to G. Let Γ=〈CG(s), NG(Ω1(S))| s∊I(S)〉. Suppose that Γ≤H<G for some subgroup H of G. Then H∩〈SG〉 is strongly embedded in 〈SG〉 and S=T∩〈SG〉 or S=Ω1(T∩〈SG〉). Also if m(S) = 1, then S=T∩〈SG〉.

Let X denote a non-vanishing infinitesimal isometry on a compact Riemannian manifold Mn. Let denote the deRham complex of M. We write i(X) for the operator of interior product, and L(X) the Lie derivative on the elements of A(M). We define E(M) = {u ∈ A(M)| i(X)u = 0, L(X)u= 0}.

A von Neumann algebra is said to be approximately finite-dimensional if it is of the form

where Mn⊆Mn+1 for each n and each Mn is a finite-dimensional matrix algebra. A factor is said to be ITPFI if it is of the form

In this paper we show how to compute the probability that a sequence Zn = X0 + … + Xn of partial sums of dependent random variables, each taking on the values ±1, will first leave an interval (a, b) at a; and how to compute the expected time it takes for the partial sums to leave the interval (a, b).

A theorem on the barycentre of a measure is proven which leads to generalization of Carathéodory′s theorem and to extension of various results. A mathematical programming problem is examined in application.

A non-zero module M having a minimal generator set contains a maximal submodule. If M is Artinian and all submodules of M have minimal generator sets then M is Noetherian; it follows that every left Artinian module of a left perfect ring is Noetherian. Every right Noetherian module of a left perfect ring is Artinian. It follows that a module over a left and right perfect ring (in particular, commutative) is Artinian if and only if it is Noetherian. We prove that a local ring is left perfect if and only if each left module has a minimal generator set.

It is shown that the boundary of the set of accumulation points of a metrizable space X is compact iff X has a compatible metric d such that d(A, B)>0 whenever A and B are disjoint closed subsets of X, each of which is disjoint from the set of accumulation points.

A partially ordered set P has the fixed point property if every orderpreserving mapping f of P to P has a fixed point, that is, f(a) = a for some aϵP; call P fixed point free if P does not have the fixed point property.

A regular Borel measure μ is said to be r*-invariant on a locally compact semigroup if μ(Ba-1) = μ(B) for all Borel sets B and points a of S, where Ba-1 ={xϵS, xaϵB}. In [1] Argabright conjectured that the support of an r*-invariant measure on a locally compact semigroup is a left group, Mukherjea and Tserpes [4] proved this conjecture in the case that the measure is finite; however their method of proof fails when the measure is infinite.

Let τ be a transformation from a compact metric space X into itself. Conditions are presented which ensure that there exists an orbit which is dense in X. An example is given.

It is proved that a regular piecewise n-convex function differs from an n-convex function only by a polynomial spline of degree n - 1. The argument is given in terms of Peano and de la Vallée Poussin derivatives.

The geometric notions of a gap and gap points between “concentric” spheres in a normed linear space are introduced and studied. The existence of gap points characterizes finitedimensional spaces. General conditions are given under which an infinite-dimensional normed linear space admits concentric spheres such that both these spheres and their dual spheres fail to have gap points.