Regular Moebius transformations of the space of quaternions

Let H be the real algebra of quaternions. The notion of regular function of a quaternionic variable recently presented by G. Gentili and D. C. Struppa developed into a quite rich theory. Several properties of regular quaternionic functions are analogous to those of holomorphic functions of one complex variable, although the diversity of the quaternionic setting introduces new phenomena. This paper studies regular quaternionic transformations. We first find a quaternionic analog to the Casorati-Weierstrass theorem and prove that all regular injective functions from H to itself are affine. In particular, the group Aut(H) of biregular functions on H coincides with the group of regular affine transformations. Inspired by the classical quaternionic linear fractional transformations, we define the regular fractional transformations. We then show that each regular injective function from the Alexandroff compactification of H to itself is a regular fractional transformation. Finally, we study regular Moebius transformations, which map the unit ball B onto itself. All regular bijections from B to itself prove to be regular Moebius transformations.Comment: 12 page

Regular vs. classical M\"obius 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 M\"obius transformations of the quaternionic unit ball B, comparing the latter with their classical analogs. In particular we study the relation between the regular M\"obius transformations and the Poincar\'e metric of B, which is preserved by the classical M\"obius transformations. Furthermore, we announce a result that is a quaternionic analog of the Schwarz-Pick lemma.Comment: 14 page

The dissolved yellow substance and the shades of blue in the Mediterranean Sea

When the nominal algorithms commonly in use in Space Agencies are applied to satellite Ocean Color data, the retrieved chlorophyll concentrations in the Mediterranean Sea are recurrently notable overestimates of the field values. Accordingly, several regionally tuned algorithms have been proposed in the past to correct for this deviation. Actually, the blueness of the Mediterranean waters is not as deep as expected from the actual (low) chlorophyll content, and the modified algorithms account for this peculiarity. Among the possible causes for such a deviation, an excessive amount of yellow substance (or of chromophoric dissolved organic matter, CDOM) has been frequently cited. This conjecture is presently tested, by using a new technique simply based on the simultaneous consideration of marine reflectance determined at four spectral bands, namely at 412, 443, 490, and 555 nm, available on the NASA-SeaWiFS sensor (Sea–viewing Wide Field-of-view Sensor). It results from this test that the concentration in yellow colored material (quantified as <i>a<sub>y</sub></i>, the absorption coefficient of this material at 443 nm) is about twice that one observed in the nearby Atlantic Ocean at the same latitude. There is a strong seasonal signal, with maximal <i>a<sub>y</sub></i> values in late fall and winter, an abrupt decrease beginning in spring, and then a flat minimum during the summer months, which plausibly results from the intense photo-bleaching process favored by the high level of sunshine in these areas. Systematically, the <i>a<sub>y</sub></i> values, reproducible from year to year, are higher in the western basin compared with those in the eastern basin (by about 50%). The relative importance of the river discharges into this semi-enclosed sea, as well as the winter deep vertical mixing occurring in the northern parts of the basins may explain the high yellow substance background. The regionally tuned [Chl] algorithms, actually reflect the presence of an excess of CDOM with respect to its standard (Chl-related) values. When corrected for the presence of the actual CDOM content, the [Chl] values as derived via the nominal algorithms are restored to more realistic values, i.e., approximately divided by about two; the strong autumnal increase is smoothed whereas the spring bloom remains as an isolated feature

Characterization of anomalous diffusion classical statistics powered by deep learning (CONDOR)

Diffusion processes are important in several physical, chemical, biological and human phenomena. Examples include molecular encounters in reactions, cellular signalling, the foraging of animals, the spread of diseases, as well as trends in financial markets and climate records. Deviations from Brownian diffusion, known as anomalous diffusion (AnDi), can often be observed in these processes, when the growth of the mean square displacement in time is not linear. An ever-increasing number of methods has thus appeared to characterize anomalous diffusion trajectories based on classical statistics or machine learning approaches. Yet, characterization of anomalous diffusion remains challenging to date as testified by the launch of the AnDi challenge in March 2020 to assess and compare new and pre-existing methods on three different aspects of the problem: the inference of the anomalous diffusion exponent, the classification of the diffusion model, and the segmentation of trajectories. Here, we introduce a novel method (CONDOR) which combines feature engineering based on classical statistics with supervised deep learning to efficiently identify the underlying anomalous diffusion model with high accuracy and infer its exponent with a small mean absolute error in single 1D, 2D and 3D trajectories corrupted by localization noise. Finally, we extend our method to the segmentation of trajectories where the diffusion model and/or its anomalous exponent vary in time

Poles of regular quaternionic functions

This paper studies the singularities of Cullen-regular functions of one quaternionic variable. The quaternionic Laurent series prove to be Cullen-regular. The singularities of Cullen-regular functions are thus classified as removable, essential or poles. The quaternionic analogues of meromorphic complex functions, called semiregular functions, turn out to be quotients of Cullen-regular functions with respect to an appropriate division operation. This allows a detailed study of the poles and their distribution.Comment: 14 page

Quaternionic Toric Manifolds

In the present paper we introduce and study a new notion of toric manifold in the quaternionic setting. We develop a construction with which, starting from appropriate m-dimensional Delzant polytopes, we obtain manifolds of real dimension 4m, acted on by m copies of the group Sp(1) of unit quaternions. These manifolds are quaternionic regular and can be endowed with a 4-plectic structure and a generalized moment map. Convexity properties of the image of the moment map are studied. Quaternionic toric manifolds appear to be a large enough class of examples where one can test and study new results in quaternionic geometry

Verso una "cultura" del disinvestimento: Efficienza, Superiorit\ue0 e Conformit\ue0

Strategic management research has always been committed to portray divestitures just as a reaction to strategic mistakes or a change to earlier decisions. Yet, scholars have recently gauged that divestiture operations represent a keystone in firm value creation (e.g. Moschieri and Mair, 2011) propelling change process. This article reviews existing research on divestiture classifying it into three schools of thought - firm efficiency, firm superiority and firm conformity \u2013 based on different assumptions and conceptualizations of firms and firm objectives. Moreover, it introduces a taxonomy of divestiture operations that accounts for their heterogeneous determinants, thus proposing to read divestiture no longer and not only as a sign of failure, but rather as a tool to create and preserve shareholders\u2019 wealth

On Compact Affine Quaternionic Curves and Surfaces

This paper is devoted to the study of affine quaternionic manifolds and to a possible classification of all compact affine quater- nionic curves and surfaces. It is established that on an affine quater- nionic manifold there is one and only one affine quaternionic structure. A direct result, based on the celebrated Kodaira Theorem that classifies all compact complex manifolds in complex dimension 2, states that the only compact affine quaternionic curves are the quaternionic tori. As for compact affine quaternionic surfaces, the study of their fundamental groups, together with the inspection of all nilpotent hypercomplex sim- ply connected 8-dimensional Lie Groups, identifies a path towards their classification

Zeros of regular functions of quaternionic and octonionic variable: a division lemma and the camshaft effect

We study in detail the zero set of a regular function of a quaternionic or octonionic variable. By means of a division lemma for convergent power series, we find the exact relation existing between the zeros of two octonionic regular functions and those of their product. In the case of octonionic polynomials, we get a strong form of the fundamental theorem of algebra. We prove that the sum of the multiplicities of zeros equals the degree of the polynomial and obtain a factorization in linear polynomials.Comment: Proof of Lemma 7 rewritten (thanks to an anonymous reviewer