229 research outputs found
Linear Fractional Maps of the unit ball: a geometric study.
No full textWe classify up to conjugation with automorphisms the linear fractional self-maps of the unit ball of (n>1). Then we give some applications of these normal forms to the study of composition operators
The pluricomplex Poisson kernel for strongly pseudoconvex domains
No full textIn this paper we introduce, via a Phragmén-Lindelöf type theorem, a maximal plurisubharmonic function in a strongly pseudoconvex domain. We call such a function the pluricomplex Poisson kernel because it shares many properties with the classical Poisson kernel of the unit disc. In particular, we show that such a function is continuous, it is zero on the boundary except at one boundary point where it has a non-tangential simple pole, and reproduces pluriharmonic functions. We also use such a function to obtain a new “intrinsic” version of the classical Julia's Lemma and Julia-Wolff-Carathéodory's Theorem
Exploring structure-function relationships in neocortical networks by means of neuromodelling techniques
No full textDetermining the neuronal architecture underlying certain visual functions is of fundamental importance for understanding how sensory processing is implemented in the brain. The wealth of anatomical, physiological and biophysical data that is being currently acquired on the neocortex could be used to constrain its functional architecture. However, given the intrinsic complexity and diversity of the data, it is difficult to provide a comprehensive framework to use these data in order to characterize structure-function relationships. Here, we discuss the use of biophysically plausible models of dynamics of neuronal networks, constructed to reflect the known properties of neocortical connectivity and modularity, as a tool to bring together anatomy and physiology. We illustrate the utility and rationale of the neuro-dynamics modelling approach by considering recent studies on the relationship between functional structure of the visual cortex and its response timing, and on the cellular and network origin of neuronal oscillations in the gamma frequency range. We also critically discuss how an interaction between theory and experiments could help this approach to become directly relevant for clinical applications
Semigroup-fication of univalent self-maps of the unit disc
No full textLet f be a univalent self-map of the unit disc. We introduce a
technique, that we call semigroup-fication, which allows to construct a continuous
semigroup (φt) of holomorphic self-maps of the unit disc whose time one map φ1
is, in a sense, very close to f. The semigroup-fication of f is of the same type as f
(elliptic, hyperbolic, parabolic of positive step or parabolic of zero step) and there
is a one-to-one correspondence between the set of boundary regular fixed points
of f with a given multiplier and the corresponding set for φ1. Moreover, in case
f (and hence φ1) has no interior fixed points, the slope of the orbits converging
to the Denjoy–Wolff point is the same. The construction is based on holomorphic
models, localization techniques and Gromov hyperbolicity. As an application of
this construction, we prove that in the non-elliptic case, the orbits of f converge
non-tangentially to the Denjoy–Wolff point if and only if the Koenigs domain of f
is “almost symmetric” with respect to vertical lines
Shearing maps and a Runge map of the unit ball which does not embed into a Loewner chain with range C^n
No full textIn this paper we study the class of "shearing" holomorphic maps of the unit ball of the form (z ,w) bar right arrow (z + g(w), w). Besides general properties, we use such maps to construct an example of a normalized univalent map of the ball onto a Runge domain in C-n which however cannot be embedded into a Loewner chain whose range is C-n
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