We provide uniqueness results for holomorphic functions in the Nevanlinna class which bridge those previously obtained by Hayman and by Lyubarskii and Seip. In particular, we propose certain classes of hyperbolically separated sequences in the disc, in terms of the rate of non-tangential accumulation to the boundary (the outer limits of this spectrum of classes being, respectively, the sequences with a non-tangential cluster set of positive measure, and the sequences satisfying the Blaschke condition). For each of those classes, we give a critical condition of radial decrease on the modulus which will force a Nevanlinna class function to vanish identically.

AMS 2000 Mathematics subject classification: Primary 30D50; 30D55

We show that the representations of the Cuntz $C^*$-algebras $\mathcal{O}_n$ which arise in wavelet analysis and dilation theory can be classified through a simple analysis of completely positive maps on finite-dimensional space. Based on this analysis, we find an application in quantum information theory; namely, a structure theorem for the fixed-point set of a unital quantum channel. We also include some open problems motivated by this work.

AMS 2000 Mathematics subject classification: Primary 46L45; 47A20; 46L60; 42C40; 81P15

In this paper, we use the theorem of Burchnall and Shaundy to give the capacity of the spectrum $\sigma(A)$ of a periodic tridiagonal and symmetric matrix. A special family of Chebyshev polynomials of $\sigma(A)$ is also given. In addition, the inverse problem is considered: given a finite union $E$ of closed intervals, we study the conditions for a Jacobi matrix $A$ to exist satisfying $\sigma(A)=E$. We relate this question to Carathéodory theorems on conformal mappings.

AMS 2000 Mathematics subject classification: Primary 31B15; 30C20; 39A70

We show that the metric version of Pansu’s differentiability result for Lipschitz maps fails; this illustrates an interesting difference between Euclidean domains and domains that are non-abelian stratified groups.

AMS 2000 Mathematics subject classification: Primary 26B05. Secondary 22E25

The topology of certain weighted inductive limits of Fréchet spaces of holomorphic functions on the unit disc can be described by means of weighted sup-seminorms in case the weights are radial and satisfy certain natural assumptions due to Lusky; in the sense of Shields and Williams the weights have to be normal. It turns out that no assumption on the (double) sequence of normal weights is necessary for the topological projective description in the case of o-growth conditions. For O-growth conditions, we give a necessary and sufficient condition (in terms of associated weights) for projective description in the case of (LB)-spaces and normal weights. This last result is related to a theorem of Mattila, Saksman and Taskinen.

AMS 2000 Mathematics subject classification: Primary 46E10. Secondary 30H05; 46A13; 46M40

In this paper we introduce two classes of operators on spaces of continuous functions with values in $F$-spaces under the action of which many functions behave chaotically near the boundary. Several examples—including onto linear operators, left and right composition operators, multiplication operators, and operators with pointwise dense range or with some stability property—are given. This new theory extends one recently developed on spaces of holomorphic functions.

AMS 2000 Mathematics subject classification: Primary 47B38. Secondary 30D40; 46E10; 54D45

In this paper we examine periodic problems driven by the scalar $p$-Laplacian. Using non-smooth critical-point theory and a recent multiplicity result based on local linking (the original smooth version is due to Brezis and Nirenberg), we prove three multiplicity results, the third for semilinear problems with resonance at zero. We also study a quasilinear periodic eigenvalue problem with the parameter near resonance. We prove the existence of three distinct solutions, extending in this way a semilinear and smooth result of Mawhin and Schmitt.

AMS 2000 Mathematics subject classification: Primary 34C25

A symplectic connection on a symplectic manifold, unlike the Levi-Civita connection on a Riemannian manifold, is not unique. However, some spaces admit a canonical connection (symmetric symplectic spaces, Kähler manifolds, etc.); besides, some ‘preferred’ symplectic connections can be defined in some situations. These facts motivate a study of the symplectic connections, inducing a parallel Ricci tensor. This paper gives the possible forms of the Ricci curvature on such manifolds and gives a decomposition theorem (linked with the holonomy decomposition) for them.

AMS 2000 Mathematics subject classification: Primary 53B05; 53B30; 53B35; 53C25; 53C55

In this paper we develop general Minkowski-type formulae for compact spacelike hypersurfaces immersed into conformally stationary spacetimes, that is, Lorentzian manifolds admitting a timelike conformal field. We apply them to the study of the umbilicity of compact spacelike hypersurfaces in terms of their $r$-mean curvatures. We derive several uniqueness results, for instance, compact spacelike hypersurfaces are umbilical if either some of their $r$-mean curvatures are linearly related or one of them is constant.

AMS 2000 Mathematics subject classification: Primary 53C42. Secondary 53B30; 53C50; 53Z05; 83C99

An analogue of the Stein embedding theorem for $C^\infty$ manifolds endowed with two equidimensional supplementary foliations is proved.

AMS 2000 Mathematics subject classification: Primary 30G35. Secondary 16P10; 26E05; 32E10; 53C12; 57R30.

We compute the (generalized) Poincaré series of the multi-index filtration defined by a finite collection of divisorial valuations on the ring $\mathcal{O}_{\mathbb{C}^2,0}$ of germs of functions of two variables. We use the method initially elaborated by the authors and Campillo for computing the similar Poincaré series for the valuations defined by the irreducible components of a plane curve singularity. The method is essentially based on the notions of the so-called extended semigroup and of the integral with respect to the Euler characteristic over the projectivization of the space of germs of functions of two variables. The last notion is similar to (and inspired by) the notion of the motivic integration.

AMS 2000 Mathematics subject classification: Primary 14B05; 16W70