On the real differential of a slice regular function

In this paper we show that the real differential of any injective slice regular function is everywhere invertible. The result is a generalization of a theorem proved by G. Gentili, S. Salamon and C. Stoppato, and it is obtained thanks, in particular, to some new information regarding the first coefficients of a certain polynomial expansion for slice regular functions (called \textit{spherical expansion}), and to a new general result which says that the slice derivative of any injective slice regular function is different from zero. A useful tool proven in this paper is a new formula that relates slice and spherical derivatives of a slice regular function. Given a slice regular function, part of its singular set is described as the union of surfaces on which it results to be constant.Comment: 23 pages, some adjustment in the structure of the sections, some typos removed, last example reviewe

Twistor interpretation of slice regular functions

Given a slice regular function $f:\Omega\subset\mathbb{H}\to \mathbb{H}$, with $\Omega\cap\mathbb{R}\neq \emptyset$, it is possible to lift it to a surface in the twistor space $\mathbb{CP}^{3}$ of $\mathbb{S}^4\simeq \mathbb{H}\cup \{\infty\}$ (see~\cite{gensalsto}). In this paper we show that the same result is true if one removes the hypothesis $\Omega\cap\mathbb{R}\neq \emptyset$ on the domain of the function $f$. Moreover we find that if a surface $\mathcal{S}\subset\mathbb{CP}^{3}$ contains the image of the twistor lift of a slice regular function, then $\mathcal{S}$ has to be ruled by lines. Starting from these results we find all the projective classes of algebraic surfaces up to degree 3 in $\mathbb{CP}^{3}$ that contain the lift of a slice regular function. In addition we extend and further explore the so-called twistor transform, that is a curve in $\mathbb{G}r_2(\mathbb{C}^4)$ which, given a slice regular function, returns the arrangement of lines whose lift carries on. With the explicit expression of the twistor lift and of the twistor transform of a slice regular function we exhibit the set of slice regular functions whose twistor transform describes a rational line inside $\mathbb{G}r_2(\mathbb{C}^4)$, showing the role of slice regular functions not defined on $\mathbb{R}$. At the end we study the twistor lift of a particular slice regular function not defined over the reals. This example shows the effectiveness of our approach and opens some questions.Comment: 29 page

Comparing Fiscal (De)Centralization and Multilevel Governments in Different Institutional Settings: A comparative study of Argentina and Denmark (2000-2010). European Diversity and Autonomy Papers EDAP 02/2020

The magnitude and complexity of the different processes of decentralization that took place around the world in the last five decades, involving all types of states (unitary and federal, as well), has challenged the concepts and the traditional distinction among the forms of the States. Therefore, to get a more complete and comprehensive idea of the whole phenomenon it is necessary to return to a theoretical discussion about decentralization and this requires also comparative studies between federal countries and unitary countries. With this background, the aim of this paper is twofold: first, it discusses some concepts surrounding the idea of decentralization and the different aspect it encompasses; second, it measures and compares institutional and fiscal decentralization in two countries with very different institutional settings, Argentina and Denmark, through six indicators, in order to explore some causal explanations of the role of subnational units in the process of decentralization

Algebraic surfaces with infinitely many twistor lines

We prove that a reduced and irreducible algebraic surface in $\mathbb{CP}^{3}$ containing infinitely many twistor lines cannot have odd degree. Then, exploiting the theory of quaternionic slice regularity and the normalization map of a surface, we give constructive existence results for even degrees.Comment: 7 pages. arXiv admin note: substantial text overlap with arXiv:1802.06697. This paper was extracted from the last section of arXiv:1802.06697v2, where a question was left open. For this reason most of the material is the sam

The (Un-) Stable Relationship between The Exchange rate and its Fundamentals

This study investigates the relationship between the euro-dollar exchange rate and its underlying fundamentals by adopting non-linear time series modelling. We found that this relationship is episodically unstable. We also found that an equilibrium-distorting shock is likely to have a greater effect on the exchange rate during periods when the deviation between exchange rate and fundamentals is large; as a consequence, when the exchange rate is close to its equilibrium value it tends to be less sensitive to any shocks in the fundamentals.Non-linearity, Markov-switching Model, Fundamentals

Log-biharmonicity and a Jensen formula in the space of quaternions

Given a complex meromorphic function, it is well defined its Riesz measure in terms of the laplacian of the logarithm of its modulus. Moreover, related to this tool, it is possible to prove the celebrated Jensen formula. In the present paper, using among the other things the fundamental solution for the bilaplacian, we introduce a possible generalization of these two concepts in the space of quaternions, obtaining new interesting Riesz measures and global (i.e. four dimensional), Jensen formulas.Comment: Final Version. To appear on Annales Academiae Scientiarum Fennicae Mathematica, Volume 44 (2019

S-regular functions which preserve a complex slice

We study global properties of quaternionic slice regular functions (also called s-regular) defined on symmetric slice domains. In particular, thanks to new techniques and points of view, we can characterize the property of being one-slice preserving in terms of the projectivization of the vectorial part of the function. We also define a "Hermitian" product on slice regular functions which gives us the possibility to express the $*$-product of two s-regular functions in terms of the scalar product of suitable functions constructed starting from $f$ and $g$. Afterwards we are able to determine, under different assumptions, when the sum, the $*$-product and the $*$-conjugation of two slice regular functions preserve a complex slice. We also study when the $*$-power of a slice regular function has this property or when it preserves all complex slices. To obtain these results we prove two factorization theorems: in the first one, we are able to split a slice regular function into the product of two functions: one keeping track of the zeroes and the other which is never-vanishing; in the other one we give necessary and sufficient conditions for a slice regular function (which preserves all complex slices) to be the symmetrized of a suitable slice regular one.Comment: 23 pages, to appear in Annali di Matematica Pura e Applicat

Information combination and forecast (st)ability evidence from vintages of time-series data

This paper explores the role of model and vintage combination in forecasting, with a novel approach that exploits the information contained in the revision history of a given variable. We analyse the forecast performance of eleven widely used models to predict inflation and GDP growth, in the three dimensions of accuracy, uncertainty and stability by using the real-time data set for macroeconomists developed at the Federal Reserve Bank of Philadelphia. Instead of following the common practice of investigating only therelationship between first available and fully revised data, we analyse the entire revision history for each variable and extract a signal from the entire distribution of vintages of a given variable to improve forecast accuracy and precision. The novelty of our study relies on the interpretation of the vintages of a real time data base as related realizations or units of a panel data set. The results suggest that imposing appropriate weights on competing models of inflation forecasts and output growth — reflecting the relative ability each model has over different sub-sample periods — substantially increases the forecast performance. More interestingly, our results indicate that augmenting the information set with a signal extracted from all available vintages of time-series consistently leads to a substantial improvement in forecast accuracy, precision and stability. JEL Classification: C32, C33, C53data and model uncertainty, forecast combination, real-time data