K. Nikodem and the present author proved in [3] a theorem concerning separation by affine functions. Our purpose is to generalize that result for polynomials. As a consequence we obtain two theorems on separation of an n-convex function from an n-concave function by a polynomial of degree at most n and a stability result of Hyers-Ulam type for polynomials.
The aim of this paper is to give a necessary and sufficient condition for a set-valued function to be a polynomial s.v. function of order at most 2.
It is quite natural to conjecture that a positively homogeneous function with degree d ≥ 2 on satisfies the Łojasiewicz gradient inequality with exponent θ = 1/d without any need for an analyticity assumption. We show that this property is true under some additional hypotheses, but not always, even for N = 2.
We generalize the Malgrange preparation theorem to matrix valued functions satisfying the condition that vanishes to finite order at . Then we can factor near (0,0), where is inversible and is polynomial function of depending on . The preparation is (essentially) unique, up to functions vanishing to infinite order at , if we impose some additional conditions on . We also have a generalization of the division theorem, and analytic versions generalizing the Weierstrass preparation...
This note brings a complement to the study of genericity of functions which are nowhere analytic mainly in a measure-theoretic sense. We extend this study to Gevrey classes of functions.
On donne une variante du principe de la phase stationnaire, où l’intégrale est remplacée par une sommation sur le réseau cubique de maille égale à l’unité de phase.
Considering jets, or functions, belonging to some strongly non-quasianalytic Carleman class on compact subsets of , we extend them to the whole space with a loss of Carleman regularity. This loss is related to geometric conditions refining Łojasiewicz’s “regular separation” or Whitney’s “property (P)”.
The paper is a continuation of an earlier one where we developed a theory of active and non-active infinitesimals and intended to establish quantifier elimination in quasianalytic structures. That article, however, did not attain full generality, which refers to one of its results, namely the theorem on an active infinitesimal, playing an essential role in our non-standard analysis. The general case was covered in our subsequent preprint, which constitutes a basis for the approach presented here....
This paper investigates the geometry of the expansion of the real field ℝ by restricted quasianalytic functions. The main purpose is to establish quantifier elimination, description of definable functions by terms, the valuation property and preparation theorem (in the sense of Parusiński-Lion-Rolin). To this end, we study non-standard models of the universal diagram T of in the language ℒ augmented by the names of rational powers. Our approach makes no appeal to the Weierstrass preparation...
This paper investigates hyperbolic polynomials with quasianalytic coefficients. Our main purpose is to prove factorization theorems for such polynomials, and next to generalize the results of K. Kurdyka and L. Paunescu about perturbation of analytic families of symmetric matrices to the quasianalytic setting.
The main results presented in this paper concern multivalued maps. We consider the cliquishness, quasicontinuity, almost continuity and almost quasicontinuity; these properties of multivalued maps are characterized by the analogous properties of some real functions. The connections obtained are used to prove decomposition theorems for upper and lower quasicontinuity.
Nous précisons la classe de différentiabilité de où désigne une fonction positive de classe , -plate sur l’ensemble de ses zéros, et un réel, ; de plus, nous étudions l’existence locale d’une racine -ième de classe , pour une fonction de classe admettant une racine -ième formelle en chaque point.