Cinzia Bisi

The Schwarz-Pick lemma for slice regular functions

The celebrated Schwarz-Pick lemma for the complex unit disk is the basis for the study of hyperbolic geometry in one and in several complex variables. In the present paper, we turn our attention to the quaternionic unit ball B. We prove a version of the Schwarz-Pick lemma for self-maps of B that are slice regular, according to the definition of Gentili and Struppa. The lemma has interesting applications in the fixed-point case, and it generalizes to the case of vanishing higher order derivatives.

Micro and macro models of granular computing induced by the indiscernibility relation

In rough set theory (RST), and more generally in granular computing on information tables (GRC-IT), a central tool is the Pawlaks indiscernibility relation between objects of a universe set with respect to a fixed attribute subset. Let us observe that Pawlaks relation induces in a natural way an equivalence relation ź on the attribute power set that identifies two attribute subsets yielding the same indiscernibility partition. We call indistinguishability relation of a given information table I the equivalence relation ź, that can be considered as a kind of global indiscernibility. In this paper we investigate the mathematical foundations of indistinguishability relation through the introduction of two new structures that are, respectively, a complete lattice and an abstract simplicial complex. We show that these structures can be studied at both a micro granular and a macro granular level and that are naturally related to the core and the reducts of I . We first discuss the role of these structures in GrC-IT by providing some interpretations, then we prove several mathematical results concerning the fundamental properties of such structures.

Dominance order on signed integer partitions

Abstract In 1973 Brylawski introduced and studied in detail the dominance partial order on the set Par(m) of all integer partitions of a fixed positive integer m. As it is well known, the dominance order is one of the most important partial orders on the finite set Par(m). Therefore it is very natural to ask how it changes if we allow the summands of an integer partition to take also negative values. In such a case, m can be an arbitrary integer and Par(m) becomes an infinite set. In this paperwe extend the classical dominance order in this more general case. In particular, we consider the resulting lattice Par(m) as an infinite increasing union on n of a sequence of finite lattices O(m, n). The lattice O(m, n) can be considered a generalization of the Brylawski lattice. We study in detail the lattice structure of O(m, n).

Regular vs. Classical Möbius Transformations of the Quaternionic Unit Ball

The regular fractional transformations of the extended quaternionic space have been recently introduced as variants of the classical linear fractional transformations. These variants have the advantage of being included in the class of slice regular functions, introduced by Gentili and Struppa in 2006, so that they can be studied with the useful tools available in this theory. We first consider their general properties, then focus on the regular Mobius transformations of the quaternionic unit ball \(\mathbb{B}\), comparing the latter with their classical analogs. In particular we study the relation between the regular Mobius transformations and the Poincare metric of \(\mathbb{B}\), which is preserved by the classical Mobius transformations. Furthermore, we announce a result that is a quaternionic analog of the Schwarz-Pick lemma.

Commuting holomorphic maps and linear fractional models

Let f be a holomorphic map of the open unit disc Δ of into itself, having no fixed points in Δ and Wolff point τε∂Δ. In the open case in which f ′(τ) = 1 we study the centralizer of f i.e., the family Gf of all holomorphic maps of Δ into itself which commute with f under composition. We prove that if the sequence of iterates {fn } converges to τ non tangentially, then Gf coincides with the set of all elements of the pseudo-iteration semigroup of f (in the sense of Cowen, see [5,6]) whose Wolff point is τ. In the same hypotheses we give a representation of the centralizer Gf in Aut(Δ) or Aut , study its main features and generalize a result due to Pranger ([15]).

On the Geometry of the Quaternionic Unit Disc

In the space ℍ of quaternions, the natural invariant geometry of the open unit disc ΔH, diffeomorphic to the open half-space H+ via a Cayleytype transformation, has been investigated extensively. This was accomplished by constructing, in a natural geometrical manner.

On Proper Polynomial Maps of ℂ2

Two proper polynomial maps f1, f2:ℂ2⟶ℂ2 are said to be equivalent if there exist Φ1, Φ2∈Aut(ℂ2) such that f2=Φ2○f1○Φ1. We investigate proper polynomial maps of topological degree d≥2 up to equivalence. Under the further assumption that the maps are Galois coverings, we also provide the complete description of equivalence classes. This widely extends previous results obtained by Lamy in the case d=2.

A Landau’s theorem in several complex variables

In one complex variable, it is well known that if we consider the family of all holomorphic functions on the unit disc that fix the origin and with first derivative equal to 1 at the origin, then there exists a constant

Landau’s theorem for slice regular functions on the quaternionic unit ball

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