Towards a theory of area in homogeneous groups

A general approach to compute the spherical measure of submanifolds in homogeneous groups is provided. We focus our attention on the homogeneous tangent space, that is a suitable weighted algebraic expansion of the submanifold. This space plays a central role for the existence of blow-ups. Main applications are area-type formulae for new classes of $C^1$ smooth submanifolds. We also study various classes of distances, showing how their symmetries lead to simpler area and coarea formulas. Finally, we establish the equality between spherical measure and Hausdorff measure on all horizontal submanifolds.Comment: 60 page

Study of Crystal-field Effects in Rare-earth (RE) - Transition-metal Intermetallic Compounds and in RE-based Laser Crystals

Rare-earth (RE) based compounds and alloys are of great interest both for their fundamental physical properties and for applications. In order to tailor the required compounds for a specific task, one must be able to predict the energy level structure and transition intensities for any magnetic ion in any crystalline environment. The crystal-field (CF) analysis is one of the most powerful theoretical methods to deal with the physics of magnetic ions. In the present work, this technique is used to analyze peculiar physical properties of some materials employed in the production of new-generation solid-state laser and high-performance permanent magnets.Comment: 6 pages, 2 figures; extended abstract of PhD thesis (final version with updated references

Contact equations, Lipschitz extensions and isoperimetric inequalities

We characterize locally Lipschitz mappings and existence of Lipschitz extensions through a first order nonlinear system of PDEs. We extend this study to graded group-valued Lipschitz mappings defined on compact Riemannian manifolds. Through a simple application, we emphasize the connection between these PDEs and the Rumin complex. We introduce a class of 2-step groups, satisfying some abstract geometric conditions and we show that Lipschitz mappings taking values in these groups and defined on subsets of the plane admit Lipschitz extensions. We present several examples of these groups, called Allcock groups, observing that their horizontal distribution may have any codimesion. Finally, we show how these Lipschitz extensions theorems lead us to quadratic isoperimetric inequalities in all Allcock groups.Comment: This version has additional references and a revisited introductio

Nonexistence of horizontal Sobolev surfaces in the Heisenberg group

Involutivity is a well known necessary condition for integrability of smooth tangent distributions. We show that this condition is still necessary for integrability with Sobolev surfaces. We specialize our study to the left invariant horizontal distribution of the first Heisenberg group \H^1. Here we answer a question raised in a paper by Z.M.Balogh, R.Hoefer-Isenegger, J.T.Tyson

Note on a Two-Player All-pay Auction with Asymmetrical Bidders and Incomplete Information

The present paper analyzes a general class of first-price all-pay auctions where two players have different "bidding technologies" and one bidder has a head start advantage over his/her opponent. Equilibria are characterized for the complete information setting and for the case where there is incomplete asymmetrical information. In particular, the handicapped player is uncertain about the size of the opponent’s advantage.All-pay auctions and Auction theory and Games with asymmetrical players and Incomplete information games

Labour market regulation and retirement age

Pension system and labor market reforms are widely debated issues in all industrialized countries and especially in Europe; any change over these two aspects of the Social Security System indeed, can affect heavily the functioning of the whole economy.A preminent role in this sense is played by employment protection regulation and by the mandatory retirement age; in this paper I focus on the political economy of such social policies jointly and consider the interaction between the choice over the protection of the employees in the labour market and that over retirement age. In particular, I look at the effects of the turnover generated either by temporary, selective exits due to the dynamic of the labour market or by permanent, non-selective exits due to retirements. The degree of employment protection and the mandatory retirement age emerge as a result of the political bargaining between three social groups: young, high and low prductivity old. Workforce composition in this setting defines the efficiency of the economy and determine the rise of a social consensus towards different assets of the Social Security Systemsocial security, turnover on the labor market, political equilibria, employment protection, retirement age