On the Integrability of Tonelli Hamiltonians

In this article we discuss a weaker version of Liouville's theorem on the integrability of Hamiltonian systems. We show that in the case of Tonelli Hamiltonians the involution hypothesis on the integrals of motion can be completely dropped and still interesting information on the dynamics of the system can be deduced. Moreover, we prove that on the n-dimensional torus this weaker condition implies classical integrability in the sense of Liouville. The main idea of the proof consists in relating the existence of independent integrals of motion of a Tonelli Hamiltonian to the size of its Mather and Aubry sets. As a byproduct we point out the existence of non-trivial common invariant sets for all Hamiltonians that Poisson-commute with a Tonelli one.Comment: 19 pages. Version accepted by Trans. Amer. Math. So

The Racial Oppression in America’s Mass Incarceration

This paper seeks to expose the racial oppression embedded within the United States\u27 practice of mass incarceration and will provide recommendations to ameliorate this discriminatory practice that harshly and inequitably impacts people of color. Many minority communities are stuck in a continuous cycle of poverty and incarceration, in part because they are targeted and oppressed by the criminal justice system more frequently than middle class white communities. Consequently, incarcerated people of color exhibit high rates of recidivism because of being stripped of resources and being sent back to impoverished, drug-ridden neighborhoods. The War on Drugs in the 1980s and the continuance of poor relations between law enforcement and minority communities are significant contributing factors that have led to the mass incarceration of racial minority groups. The economic, political, and societal oppression of minority communities that unquestionably contributes to mass incarceration will be highlighted throughout this paper. Creating policies that involve transforming the U.S. legal system and providing communal support will be crucial in eradicating this systemic racial oppression

A variational approach to the study of the existence of invariant Lagrangian graphs

This paper surveys some results by the author and collaborators on the existence of invariant Lagrangian graphs for Tonelli Hamiltonian systems. The presentation is based on an invited talk by the author at XIX Congresso Unione Matematica Italiana (Bologna, 12-17 Sept. 2011).Comment: 28 page

Adaptive synchronization of dynamics on evolving complex networks

We study the problem of synchronizing a general complex network by means of an adaptive strategy in the case where the network topology is slowly time varying and every node receives at each time only one aggregate signal from the set of its neighbors. We introduce an appropriately defined potential that each node seeks to minimize in order to reach/maintain synchronization. We show that our strategy is effective in tracking synchronization as well as in achieving synchronization when appropriate conditions are met.Comment: Accepted for publication on Physical Review Letter

Sequential Monte Carlo samplers for semilinear inverse problems and application to magnetoencephalography

We discuss the use of a recent class of sequential Monte Carlo methods for solving inverse problems characterized by a semi-linear structure, i.e. where the data depend linearly on a subset of variables and nonlinearly on the remaining ones. In this type of problems, under proper Gaussian assumptions one can marginalize the linear variables. This means that the Monte Carlo procedure needs only to be applied to the nonlinear variables, while the linear ones can be treated analytically; as a result, the Monte Carlo variance and/or the computational cost decrease. We use this approach to solve the inverse problem of magnetoencephalography, with a multi-dipole model for the sources. Here, data depend nonlinearly on the number of sources and their locations, and depend linearly on their current vectors. The semi-analytic approach enables us to estimate the number of dipoles and their location from a whole time-series, rather than a single time point, while keeping a low computational cost.Comment: 26 pages, 6 figure