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

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

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

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

∗*-exponential of slice-regular functions

According to [5] we define the $*$-exponential of a slice-regular function, which can be seen as a generalization of the complex exponential to quaternions. Explicit formulas for $\exp_*(f)$ are provided, also in terms of suitable sine and cosine functions. We completely classify under which conditions the $*$-exponential of a function is either slice-preserving or $\mathbb{C}_J$-preserving for some $J\in\mathbb{S}$ and show that $\exp_*(f)$ is never-vanishing. Sharp necessary and sufficient conditions are given in order that $\exp_*(f+g)=\exp_*(f)*\exp_*(g)$, finding an exceptional and unexpected case in which equality holds even if $f$ and $g$ do not commute. We also discuss the existence of a square root of a slice-preserving regular function, characterizing slice-preserving functions (defined on the circularization of simply connected domains) which admit square roots. Square roots of this kind of functions are used to provide a further formula for $\exp_{*}(f)$. A number of examples is given throughout the paper.Comment: 15 pages; to appear in Proceedings of the American Mathematical Societ

The Effects of Monetary Policy on Unemployment Dynamics Under Model Uncertainty. Evidence from the US and the Euro Area

This paper explores the role that the imperfect knowledge of the structure of the economy plays in the uncertainty surrounding the effects of rule-based monetary policy on unemployment dynamics in the euro area and the US. We employ a Bayesian model averaging procedure on a wide range of models which differ in several dimensions to account for the uncertainty that the policymaker faces when setting the monetary policy and evaluating its effect on real economy. We find evidence of a high degree of dispersion across models in both policy rule parameters and impulse response functions. Moreover, monetary policy shocks have very similar recessionary effects on the two economies with a different role played by the participation rate in the transmission mechanism. Finally, we show that a policy maker who does not take model uncertainty into account and selects the results on the basis of a single model may come to misleading conclusions not only about the transmission mechanism, but also about the differences between the euro area and the US, which are on average essentially small.Monetary policy, Model uncertainty, Bayesian model averaging, Unemployment gap, Taylor rule

Inflation Forecasts, monetary policy and unemployment dynamics: evidence from the US and the euro area

This paper explores the role that inflation forecasts play in the uncertainty surrounding the estimated effects of alternative monetary rules on unemployment dynamics in the euro area and the US. We use the inflation forecasts of 8 competing models in a standard Bayesian VAR to analyse the size and the timing of these effects, as well as to quantify the uncertainty relative to the different inflation models under two rules. The results suggest that model uncertainty can be a serious issue and strengthen the case for a policy strategy that takes into account several sources of information. We find that combining inflation forecasts from many models not only yields more accurate forecasts than those of any specific model, but also reduces the uncertainty associated with the real effects of policy decisions. These results are in line with the model-combination approach that central banks already follow when conceiving their strategy. JEL Classification: C53, E24, E37E24, E37, Inflation forecasts, JEL Classification: C53, Model uncertainty, Unemployment

Monetary Policy Analysis in Real-Time. Vintage combination from a real-time dataset

This paper provides a general strategy for analyzing monetary policy in real time which accounts for data uncertainty without explicitly modelling the revision process. The strategy makes use of all the data available from a real-time data matrix and averages model estimates across all data releases. Using standard forecasting and policy models to analyze monetary authorities’ reaction functions, we show that this simple method can improve forecasting performance and provide reliable estimates of the policy model coe¢cients associated with small central bank losses, in particular during periods of high macroeconomic uncertainty.Monetary policy, Taylor rule, Real-time data, Great Moderation, Forecasting.

Inflation models, optimal monetary policy and uncertain unemployment dynamics: Evidence from the US and the euro area

This paper explores the role that model uncertainty plays in determining the effect of monetary policy shocks on unemployment dynamics in the euro area and the US. We specify a range of BVARs that differ in terms of variables, lag structure, and the way the inflation process is modelled. For each model the central bank sets the interest rate minimizing a loss function. Given this solution, we quantify the impact of a monetary policy shock on unemployment for each model, and measure the degree of uncertainty as represented by the dispersion of both the policy rule parameters and the impulse response functions between models. The comparative evidence from the US and the euro area data indicates that model uncertainty is indeed an important feature, and that a model combination strategy might be a valuable advise to policymakers.Inflation models, Unemployment, Model uncertainty, Taylor rule, Impulse response analysis