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

The Impact of the Temporal Distribution of Communicating Civilizations on their Detectability

We use a statistical model to investigate the detectability (defined by the requirement that they are in causal contact with us) of communicating civilizations within a volume of the universe surrounding our location. If the civilizations are located in our Galaxy, the detectability requirement imposes a strict constraint on their epoch of appearance and their communicating lifespan. This, in turn, implies that the fraction of civilizations of which we can find any empirical evidence strongly depends on the specific features of their temporal distribution. Our approach shed light on aspects of the problem that can escape the standard treatment based on the Drake equation. Therefore, it might provide the appropriate framework for future studies dealing with the evolutionary aspects of the search for extraterrestrial intelligence (SETI).Comment: 17 pages, 1 figure. Accepted for publication in Astrobiolog

Cosmology from Planck

I briefly review some of the main scientific outputs expected from the upcoming Planck mission. Planck will map the CMB sky with 5' resolution and $\mu$K sensitivity, with minimal foreground contribution and superb control on systematics. It will collect the entire information enclosed in the temperature primary anisotropy signal and will also get a good measurement of the polarized component of the CMB. This will have profound implications on our knowledge of the physics of the early universe and on the determination of cosmological parameters.Comment: Proceedings of the Francesco Melchiorri Memorial Conference (Rome, Italy, April 12-14 2006). To appear in New Astron. Re

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