A new Löwenheim-Skolem theorem

This paper establishes a refinement of the classical Löwenheim-Skolem theorem. The main result shows that any first order structure has a countable elementary substructure with strong second order properties. Several consequences for Singular Cardinals Combinatorics are deduced from this.

     

Solvability of a nonlinear fifth‐order difference equation

Some formulas for well‐defined solutions to four very special cases of a nonlinear fifth‐order difference equation have been presented recently in this journal, where some of them were proved by the method of induction, some are only quoted, and no any theory behind the formulas was given. Here, we show in an elegant constructive way how the general solution to the difference equation can be obtained, from which the special cases very easily follow, which is also demonstrated here. We also give some comments on the local stability results on the special cases of the nonlinear fifth‐order difference equation previously publish in this journal.     

A Littlewood-Paley type inequality

In this note we prove the following theorem: Let u be a harmonic function in the unit ball and . Then there is a constant C = C(p, n) such that

Global stability and asymptotics of some classes of rational difference equations

In this note we prove that the positive solutions of some classes of rational difference equations are globally asymptotically stable. Using a Berg's result, we also find asymptotics of some solutions of these equations.     

A class of Banach spaces with no unconditional basic sequence

We give a construction of a reflexive Banach space X_ω1 with a transfinite basis of length ω₁ and with no unconditional basic sequence. In addition every bounded operator from a subspace of X_ω1 into the space X_ω1 is a sum of a simple diagonal operator and a strictly singular one. To cite this article: S.A. Argyros et al., C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Essential norm of products of multiplication composition and differentiation operators on weighted Bergman spaces

Let ψ be a holomorphic function on the open unit disk D and φ a holomorphic self-map of D. Let C_φ,M_ψ and D denote the composition, multiplication and differentiation operator, respectively. We find an asymptotic expression for the essential norm of products of these operators on weighted Bergman spaces on the unit disk. This paper is a continuation of our recent paper concerning the boundedness of these operators on weighted Bergman spaces.     

Solving a class of nonautonomous difference equations by generalized invariants

By using generalized invariants, we describe a method for solving a class of higher‐order nonautonomous difference equations. Solvability of the equations of order two, three, and four are studied in detail. Our results explain and extend some problems in the literature. As far as we know, the case when the order is four is considered for the first time in this paper. For the equations of second order, we also give an explanation how they can be obtained in a natural way.     

Local sentences and Mahlo cardinals

Local sentences were introduced by Ressayre in [6] who proved certain remarkable stretching theorems establishing the equivalence between the existence of finite models for these sentences and the existence of some infinite well ordered models. Two of these stretching theorems were only proved under certain large cardinal axioms but the question of their exact (consistency) strength was left open in [4]. Here we solve this problem, using a combinatorial result of J. H. Schmerl [7]. In fact, we show that the stretching principles are equivalent to the existence of n ‐Mahlo cardinals for appropriate integers n. This is done by proving first that for every integer n, there is a local sentence φ_n having well ordered models of order type τ, for every infinite ordinal τ > ω which is not an n ‐Mahlo cardinal. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)