72 research outputs found
Landau-Toeplitz theorems for slice regular functions over quaternions
The theory of slice regular functions of a quaternionic variable extends the
notion of holomorphic function to the quaternionic setting. This theory,
already rich of results, is sometimes surprisingly different from the theory of
holomorphic functions of a complex variable. However, several fundamental
results in the two environments are similar, even if their proofs for the case
of quaternions need new technical tools. In this paper we prove the
Landau-Toeplitz Theorem for slice regular functions, in a formulation that
involves an appropriate notion of regular -diameter. We then show that the
Landau-Toeplitz inequalities hold in the case of the regular -diameter, for
all . Finally, a -diameter version of the Landau-Toeplitz Theorem
is proved using the notion of slice -diameter.Comment: 20 page
The Mittag-Leffler Theorem for regular functions of a quaternionic variable
We prove a version of the classical Mittag-Leffler Theorem for regular
functions over quaternions. Our result relies upon an appropriate notion of
principal part, that is inspired by the recent definition of spherical
analyticity.Comment: 10 page
A direct approach to quaternionic manifolds
The recent definition of slice regular function of several quaternionic
variables suggests a new notion of quaternionic manifold. We give the
definition of quaternionic regular manifold, as a space locally modeled on
, in a slice regular sense. We exhibit some significant classes
of examples, including manifolds which carry a quaternionic affine structure.Comment: 13 page
Ideals of regular functions of a quaternionic variable
In this paper we prove that, for any , the ideal generated by
slice regular functions having no common zeros concides
with the entire ring of slice regular functions. The proof required the study
of the non-commutative syzygies of a vector of regular functions, that manifest
a different character when compared with their complex counterparts.Comment: 13 page
The interactive effect of herbivory, nutrient enrichment and mucilage on shallow rocky macroalgal communities
This paper focuses on the interactive short and long-term effect of three different stressors on a macroalgal assemblage. Three stressors are considered: herbivory, nutrients and mucilage. The experiment was conducted in Tavolara Punta Coda Cavallo Marine Protected Area (Mediterranean Sea) during a bloom of the benthic mucilage-producing microalga Chrysophaeum taylorii (Pelagophyceae); this microalga is recently spreading in the Mediterranean Sea. On a rocky substratum, 36 plots 20 × 20 cm in size were prepared. Factorial combinations of three experimental treatments were applied in triplicate, including three grazing levels crossed with two nutrient enrichment and two mucilage removal treatments. Significant differences were observed among treatments 8 weeks later, at the end of summer. In particular, dark filamentous algae were more abundant in all enriched plots, especially where mucilage and macroalgae had been removed; a higher percent cover of crustose coralline algae was instead observed where nutrients had been increased and no grazing pressure acted. Furthermore, the abundance of Dictyota spp. and Laurencia spp. was significantly higher in enriched mucilage-free plots where the grazing pressure was null or low. However, the effects of the treatments on the overall assemblage of the macroalgal community were not long persistent (36 weeks later). These results illustrate the capacity of a shallow-water macroalgal community to quickly recover from the simultaneous impacts of herbivory, nutrient enrichment, and mucilage
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