Abstract:
This work proposes a 3D Virtual World environment and a didactic experience for training young students in an environment capable of supporting the engineering practices based on technical drawing. The main difficulty of technical drawing consists in representing a 3D object on a 2D medium. This restriction imposes to human mind to be able to summarize the spatial properties of objects on the paper. The proposed system trains these capabilities by requiring students to build, in the simulated environment, simple objects represented with 2D drawings. In this way, the students are not only pushed to move themselves between different dimensionality spaces, but also they benefit of the 3D spaces for moving and exploring the models they are building. An empirical evaluation, conducted as a controlled experiment, has provided enthusiastic results in terms of user performances and impressions.

Abstract:
Let n be an integer. A group G is said to be n-abelian if the map phi_n that sends g to g^n is an endomorphism of G. Then (xy)^n=x^ny^n for all x,y in G, from which it follows [x^n,y]=[x,y]^n=[x,y^n]. It is also easy to see that a group G is n-abelian if and only if it is (1-n)-abelian. If nneq 0,1 and G is an n-abelian group, then the quotient group G/Z(G) has finite exponent dividing n(n-1). This implies that every torsion-free n-abelian group is abelian. We denote by B_n and C_n the classes of all groups G for which phi_n is a monomorphism and an epimorphism of G, respectively. Then B_0=C_0 contains only the trivial group, B_1=C_1 is the class of all groups, and B_-1=C_-1 is the class of all abelian groups. Furthermore, with |n|>1, G is in B_n if and only if G is an n-abelian group having no elements of order dividing |n|. Similarly, G is in C_n if and only if G is n-abelian and for every g in G there exists an element x in G such that g=x^n. We also set A_n=B_ncap C_n. In this paper we give a characterization for groups in B_n and for groups in C_n. We also obtain an arithmetic description of the set of all integers n such that a group G is in A_n.

Abstract:
The thermodynamic and dynamical properties of a variable dark energy model with density scaling as rho_x \propto (1+z)^m, z being the redshift, are discussed following the outline of Jetzer et al. This kind of models are proven to lead to the creation/disruption of matter and radiation, which affect the cosmic evolution of both matter and radiation components in the Universe. In particular, we have concentrated on the temperature-redshift relation of radiation, which has been constrained using a very recent collection of cosmic microwave background (CMB) temperature measurements up to z ~ 3. For the first time, we have combined this observational probe with a set of independent measurements (Supernovae Ia distance moduli, CMB anisotropy, large-scale structure and observational data for the Hubble parameter), which are commonly adopted to constrain dark energy models. We find that, within the uncertainties, the model is indistinguishable from a cosmological constant which does not exchange any particles with other components. Anyway, while temperature measurements and Supernovae Ia tend to predict slightly decaying models, the contrary happens if CMB data are included. Future observations, in particular measurements of CMB temperature at large redshift, will allow to give firmer bounds on the effective equation of state parameter w_eff of this kind of dark energy models.

Abstract:
We discuss the thermodynamic and dynamical properties of a variable dark energy model with density scaling as $\rho_x \propto (1+z)^{m}$, z being the redshift. These models lead to the creation/disruption of matter and radiation, which affect the cosmic evolution of both matter and radiation components in the Universe. In particular, we have studied the temperature-redshift relation of radiation, which has been constrained using a recent collection of cosmic microwave background (CMB) temperature measurements up to $z \sim 3$. We find that, within the uncertainties, the model is indistinguishable from a cosmological constant which does not exchange any particles with other components. Future observations, in particular measurements of CMB temperature at large redshift, will allow to give firmer bounds on the effective equation of state parameter $w_{eff}$ for such types of dark energy models.

Abstract:
The present work describes the investigation of the navigation anomaly of Pioneer 10 and 11 probes which became known as the Pioneer Anomaly. It appeared as a linear drift in the Doppler data received by the spacecraft, which has been ascribed to an approximately constant sunward acceleration of about $8.5 \times 10^{-13} km/s^2$. Since then, the existence of the anomaly has been confirmed independently by several groups and a large effort was devoted to find its origin. The present study consists of two main parts: thermal modeling of the spacecraft throughout its trajectory, and orbit determination analysis. Based on existing documentation and published telemetry data we built a thermal finite element model of the spacecraft, whose complexity has been constrained to a degree allowing for sensitivity analysis, leading to the computation of its formal uncertainty. The trajectory analysis and orbit determination was carried out using NASA/JPL's ODP (Orbit Determination Program) and our results show that orbital solutions may be achieved that do not require the addition of any "unknown" acceleration other than the one of thermal origin.

Abstract:
We define the monomial invariants of a projective variety $Z$; they are invariants coming from the generic initial ideal of $Z$. Using this notion, we generalize a result of Cook: If $Z$ is an integral variety of codimension two, satisfying the additional hypothesis $s_Z=s_\Gamma,$ then its monomial invariants are connected.

Abstract:
We generalize to FC*, the class of generalized FC-groups introduced in [F. de Giovanni, A. Russo, G. Vincenzi, Groups with restricted conjugacy classes, Serdica Math. J. 28 (2002), 241-254], a result of Baer on Engel elements. More precisely, we prove that the sets of left Engel elements and bounded left Engel elements of an FC*-group G coincide with the Fitting subgroup; whereas the sets of right Engel elements and bounded right Engel elements of G are subgroups and the former coincides with the hypercentre. We also give an example of an FC*-group for which the set of right Engel elements contains properly the set of bounded right Engel elements.

Abstract:
We investigate scaling relations between the dark matter (DM) halo model parameters for a sample of intermediate redshift early - type galaxies (ETGs) resorting to a combined analysis of Einstein radii and aperture velocity dispersions. Modeling the dark halo with a Navarro - Frenk - White profile and assuming a Salpeter initial mass function (IMF) to estimate stellar masses, we find that the column density ${\cal{S}}$ and the Newtonian acceleration within the halo characteristic radius $r_s$ and effective radius $R_{eff}$ are not universal quantities, but correlate with the luminosity $L_V$, the stellar mass $M_{\star}$ and the halo mass $M_{200}$, contrary to recent claims in the literature. We finally discuss a tight correlation among the DM mass $M_{DM}(R_{eff})$ within the effective radius $R_{eff}$, the stellar mass $M_{\star}(R_{eff})$ and $R_{eff}$ itself. The slopes of the scaling relations discussed here strongly depend, however, on the DM halo model and the IMF adopted so that these ingredients have to be better constrained in order to draw definitive conclusions on the DM scaling relations for ETGs.