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 $2$-diameter. We then show that the Landau-Toeplitz inequalities hold in the case of the regular $n$-diameter, for all $n\geq 2$. Finally, a $3$-diameter version of the Landau-Toeplitz Theorem is proved using the notion of slice $3$-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 $\mathbb{H}^n$, 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 $n\in \mathbb N$, the ideal generated by $n$ slice regular functions $f_1,\ldots,f_n$ 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