Quasi-convexity, strictly quasi-convexity and pseudo-convexity of composite objective functions
We characterize the family of quotients of peripherally continuous functions. Moreover, we study cardinal invariants related to quotients in the case of peripherally continuous functions and the complement of this family.
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.
We investigate the completely Ramsey, Lebesgue, and Marczewski σ-algebras and their relations to the Baire property in the Ellentuck and density topologies. Two theorems concerning the Marczewski σ-algebra (s) are presented. THEOREM. In the density topology D, (s) coincides with the σ-algebra of Lebesgue measurable sets. THEOREM. In the Ellentuck topology on , is a proper subset of the hereditary ideal associated with (s). We construct an example in the Ellentuck topology of a set which is...
Various ordinal ranks for Baire-1 real-valued functions, which have been used in the literature, are adapted to provide ranks for Baire-1 multifunctions. A new rank is also introduced which, roughly speaking, gives an estimate of how far a Baire-1 multifunction is from being upper semicontinuous.
The existence of a unique solution φ of equation (1) is proved under the condition that f: I → I is convex or concave and of class in I, 0 < f(x) < x in I*, and f’(x) > 0 in I. Here I = [0, a] or [0, a), 0 < a ≤ ∞, and I* = I 0.