Remarques sur les m\'ethodes de projection pour l'approximation des \'equations de Navier--Stokes

Summary. L'objet de cet article est de montrer que les estimations de convergence sur la pression pour les m\'ethodes de projection d\'ecrites dans \cite{Shen1} et \cite{Shen2} ne sont pas obtenues correctement car elles sont toutes bas\'ees sur une in\'egalit\'e fausse. Il semble qu'on ait besoin d'une convergence en de la vitesse dans pour d\'emontrer les estimations de convergence sur la pression en . La question de savoir si la m\'ethode de projection a un taux de convergence pour la pression plus \'elev\'e que reste ouverte. Received June 1, 1993     

Facial reduction in partially finite convex programming

We consider the problem of minimizing an extended-valued convex function on a locally convex space subject to a finite number of linear (in)equalities. When the standard constraint qualification fails a reduction technique is needed to derive necessary optimality conditions. Facial reduction is usually applied in the range of the constraints. In this paper it is applied in the domain space, thus maintaining any structure (and in particular lattice properties) of the underlying domain. Applications include constrained approximation and best entropy estimation.Research partially supported by the Natural Sciences and Engineering Research Council of Canada.     

Cores and retracts

Theretracts (idempotent, isotone self-maps) of an ordered set are naturally ordered as functions. In this note we characterize the possible ways that one retract can cover another one. This gives some insight into the structure of the ordered set of retracts and leads to a natural generalization of the core of an ordered set.Supported by NSERC Operating Grant 41702.     

Dimension and entropy of a non-ergodic Markovian process and its relation to Rademacher Riesz Products

Given a Markov chain (not necessarily stationary or homogeneous) with finite state space and an initial distribution, we can construct a measure on the unit interval [0, 1]. In this work we examine the equality (up to a constant) of the Hausdorff dimension of and of a suitably defined entropy for the Markovian process. The results are applied to the so-called Rademacher-Riesz Products.     

On the Durrmeyer-type modification of some discrete approximation operators

In [10], for continuous functionsf from the domain of certain discrete operatorsL _n the inequalities are proved concerning the modulus of continuity ofL _nf. Here we present analogues of the results obtained for the Durrmeyer-type modification $\tilde L_n $ ofL _n. Moreover, we give the estimates of the rate of convergence of $\tilde L_n f$ in Hölder-type norms     

Another quadrature rule of highest algebraic degree of precision

Summary. We consider certain quadrature rules of highest algebraic degree of precision that involve strong Stieltjes distributions (i.e., strong distributions on the positive real axis). The behavior of the parameters of these quadrature rules, when the distributions are strong -inversive Stieltjes distributions, is given. A quadrature rule whose parameters have explicit expressions for their determination is presented. An application of this quadrature rule for the evaluation of a certain type of integrals is also given. Received April 17, 1991 / Revised version received July 16, 1993     

Nonstationary curves in Hilbert spaces and their correlation functions II

Partially supportted by Grant MM-22/1991 of MESC     

Optimal impulse control problems for degenerate diffusions with jumps

Optimal stopping and impulse control problems for degenerate diffusion with jumps are studied in this paper. Lipschitzian coefficients for the diffusion process, data with polynomial growth, and evolution in the whole space are the main assumptions on the models. Several characterizations of the optimal cost functions are given. Existence of optimal policies is obtained.This research has been supported in part by Army Research Office Contract DAAG29-83-K-0014 and by National Science Foundation Grant DMS-8601998.