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We will consider global problems in the ringK[X 1, …,X n] on the polynomials with coefficients in a subfieldK ofC. LetP=(P 1, …,P n):K n →K n be a polynomial map such that (P 1,…,P n) is a quasi-regular sequence generating a proper ideal, the main thing we do is to use the algebraic residues theory (as described in [5]) as a computational tool to give some result to test when a map (P 1, …,P n) is a proper map by computing a finite number of residue symbols.

LetK be a compact subset of the complex plane and 0<α≤1. In this article we determine the maximal ideal space ofLi pA(K, α).

A generalization is given of the work of J. Bokowski [1] referring to the product of the volumes of the two parts into which a convex body is divided by a plane. The proof uses formulas of Integral Geometry and a conjecture of L. A. Santaló [2], and holds for the two parts determined by any (n−1)-dimensional surface in the euclideann-space and for dimensionsn=2, 3 in the hyperbolic space.

In this paper we consider nonlinear-dependent systems with multivalued perturbations in the framework of an evolution triple of spaces. First we prove a surjectivity result for generalized pseudomonotone operators and then we establish two existence theorems: the first for a periodic problem and the second for a Cauchy problem. As applications we work out in detail a periodic nonlinear parabolic partial differential equation and an optimal control problem for a system driven by a nonlinear parabolic equation.

Etant donné un opérateur fermé dans un espace de BanachX, nous définissons le spectre régulier (respectivement essentiellement régulier) σ r (A)) (respectivement σ e,r (A)) deA. Nous montrerons que σ r (A) et σ e,r (A) sont des sous-ensembles fermés du spectre classique σ(A) deA. Si en plusA est borné, nous montrons que σ r (A) et σ e,r (A) verifient le théorème de l’application spectrale.

Theorems are proved on the strong summability of orthonormal series and an error in a recent paper of Sunouchi is corrected.

Let ℬ be a Banach space of analytic functions defined on the open unit disk. We characterize the commutant ofM Z 2 (the operator of multiplication by the square of independent variable defined on ℬ) and show that for an operatorS in the commutantM Z 2 ifSM Z 2k+1−M Z 2k+1 S is compact for some nonnegative integerk, thenS=M ϕ whereϕ is a multiplier of ℬ. Letn be a positive integer andS be an operator in the commutant ofM Z n defined on a functional Hilbert spaces of analytic functions. We show that under certain conditionsS has the formM ϕ.