Some properties and applications of weakly equicompact sets

Let X and Y be Banach spaces. A set (the space of all weakly compact operators from X into Y) is weakly equicompact if, for every bounded sequence (x _n) in X, there exists a subsequence (x _k(n)) so that (Tx_k(n)) is uniformly weakly convergent for T ∈ M. In this paper, the notion of weakly equicompact set is used to obtain characterizations of spaces X such that X ↩̸ ℓ₁, of spaces X such that B _X* is weak* sequentially compact and also to obtain several results concerning to the weak operator and the strong operator topologies. As another application of weak equicompactness, we conclude a characterization of relatively compact sets in when this space is endowed with the topology of uniform convergence on the class of all weakly null sequences. Finally, we show that similar arguments can be applied to the study of uniformly completely continuous sets. Received: 5 July 2006     

Finsler metrics with positive constant flag curvature

We showed that any reversible Finsler metric with positive constant flag curvature must be Riemannian. Received: 18 August 2008     

“Setting” n-Opposition

Our aim is to show that translating the modal graphs of Moretti’s “n-opposition theory” (2004) into set theory by a suited device, through identifying logical modal formulas with appropriate subsets of a characteristic set, one can, in a constructive and exhaustive way, by means of a simple recurring combinatory, exhibit all so-called “logical bi-simplexes of dimension n” (or n-oppositional figures, that is the logical squares, logical hexagons, logical cubes, etc.) contained in the logic produced by any given modal graph (an exhaustiveness which was not possible before). In this paper we shall handle explicitly the classical case of the so-called 3(3)-modal graph (which is, among others, the one of S5), getting to a very elegant tetraicosahedronal geometrisation of this logic.      

A Characterization of Some Classes of Harmonic Functions

In this paper we investigate harmonic Hardy-Orlicz and Bergman-Orlicz b _φ,α (B) spaces, using an identity of Hardy-Stein type. We also extend the notion of the Lusin property by introducing (φ, α)-Lusin property with respect to a Stoltz domain. The main result in the paper is as follows: Let be a nonnegative increasing convex function twice differentiable on (0, ∞), and u a harmonic function on the unit ball B in . Then the following statements are equivalent:

On orbits of automorphism groups II

Let G be a finite group, let A be a group of automorphisms of G and let C_G(A) denote the subgroup of fixed points of A in G. If the order of C_G(A) is coprime to the number of orbits of A in G, then C_G(A) is contained in the autocommutator subgroup [G, A]. The notion of class-avoiding automorphism is used to extend theorems of J. Thompson and P. Rowley. Received: 3 November 2008, Revised: 1 December 2008     

On the Wu metric in unbounded domains

We discuss the properties of the Wu pseudometric and present counterexamples for its upper semicontinuity that answers the question posed by Jarnicki and Pflug. We also give formulae for the Wu pseudometric in elementary Reinhardt domains. Received: 12 September 2007     

Some subordination results for classes of analytic functions defined by the Al-Oboudi–Al-Amoudi operator

In this paper, we investigate several interesting subordination results of some classes of analytic functions defined by means of the Al-Oboudi–Al-Amoudi operator. Received: 15 August 2008, Revised: 19 November 2008     

A nondegeneracy result for a nonlinear elliptic equation

Let Ω be a smooth bounded domain of with N ≥ 5. In this paper we prove, for ɛ > 0 small, the nondegeneracy of the solution of the problem

Smoluchowski-Kramers approximation for a general class of SPDEs

We prove that the so-called Smoluchowski-Kramers approximation holds for a class of partial differential equations perturbed by a non-Gaussian noisy term. Namely, we show that the solution of the one-dimensional semi-linear stochastic damped wave equations , u(0) = u₀, u_t (0) = v₀, endowed with Dirichlet boundary conditions, converges as the parameter μ goes to zero to the solution of the semi-linear stochastic heat equation , u(0) = u₀, endowed with Dirichlet boundary conditions. Dedicated to Giuseppe Da Prato on the occasion of his 70th birthday     

Bifurcation problems for superlinear elliptic indefinite equations with exponential growth

This paper deals with the existence and the behaviour of global connected branches of positive solutions of the problem

Field induced aggregation in electrorheological suspension: kernel form and self similar solutions

Within the framework of the study of the fibrillation mechanism in an electrorheological (ER) suspension, this work presents a comparison between the self similar solutions when the kernel is K_i,j ~ (i⁻¹ + j⁻¹) and the behaviour of the chains growth. Till now, the field induced chains formation has only been studied by numerical or experimental methods. The work of Fournier and Lauren?ot (Communications in Mathematical Physics 256 2005) on the Smoluchowski’s equation allows us to present an analytical solution for the field induced pearl chains in a colloidal ER suspension. René Limage: Chercheur indépendant, dipl?mé de l’Université de Liége.     

Definable groups of partial automorphisms

The motivation for this paper is to extend the known model-theoretic treatment of differential Galois theory to the case of linear difference equations (where the derivative is replaced by an automorphism). The model-theoretic difficulties in this case arise from the fact that the corresponding theory ACFA does not eliminate quantifiers. We therefore study groups of restricted automorphisms, preserving only part of the structure. We give conditions for such a group to be (infinitely) definable, and when these conditions are satisfied we describe the definition of the group and the action explicitly. We then examine the special case when the theory in question is obtained by enriching a stable theory with a generic automorphism. Finally, we interpret the results in the case of ACFA, and explain the connection of our construction with the algebraic theory of Picard–Vessiot extensions. The only model-theoretic background assumed is the notion of a definable set.      

Agmon’s condition for incompressible elasticity: A formulation of Mielke–Sprenger type

Agmon’s Condition for incompressible elasticity is recast in a tractable variational form analogous to that developed by Mielke and Sprenger for the compressible case.

“A teacher affects eternity; he can never tell where his influence will stop.” – Henry Adams, The Education of Henry Adams, 1918.

To Professor Michael Hayes on the Occasion of his Seventieth Birthday     

Compact AC(σ) Operators

All compact AC(σ) operators have a representation analogous to that for compact normal operators. As a partial converse we obtain conditions which allow one to construct a large number of such operators. Using the results in the paper, we answer a number of questions about the decomposition of a compact AC(σ) operator into real and imaginary parts.      

The Aumann-Gould Integral

We introduce and study a multi-valued integral of Aumann-type where the selectors are Gould integrals of real functions with respect to finitely additive vector-valued measures. We also discuss the relationship between this kind of integral and the M-Gould set-valued integral with respect to a finitely additive multimeasure M. Received: November 17, 2006, Revised: September 17, 2007 and January, 8, 2008, Accepted: March 13, 2008.     

Universal Laurent series in finitely connected domains

Universal Laurent series where the universal approximation is valid on the boundary of the multiple connected open Ω appears for the first time in [2]. We show that it is possible to demand universal approximation only to a part of the boundary while on the remaining part the universal function can be smooth. Received: 29 May 2007     

Partitions and functional Santaló inequalities

We give a direct proof of a functional Santaló inequality due to Fradelizi and Meyer. This provides a new proof of the Blaschke-Santaló inequality. The argument combines a logarithmic form of the Prékopa-Leindler inequality and a partition theorem of Yao and Yao. Received: 21 August 2008