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Control Theory and applications to Quantum Mechanics and Geometry of Vision

Ugo Boscain (CNRS, Université de Dijon) - 27 Novembre 2007

In this seminar I will speak about some modern applications of Control Theory to problems of quantum mechanics, including optimal control problems in finite dimension and controllability problems in infinite dimensions. I will also present a model of Geometry of Vision (due to Petitot, Citti and Sarti) in which the visual cortex V1 is seen as a sub-Riemannian manifold. Problems of nonisotropic diffusion (i.e. with the hypoelliptic Laplacian) are also considered in this context.

Long Term Risk

José SCHEINKMAN (Princeton University)

We create an analytical structure that reveals the long-run risk-return relationship for nonlinear continuous time Markov environments. We do so by studying an eigenvalue problem associated with a positive eigenfunction for a conveniently chosen semigroup of valuation operators. We represent the semigroup using a positive process with three components : an exponential term constructed from the eigenvalue, a martingale and a transient eigenfunction term. The eigenvalue encodes the risk adjustment, the martingale alters the probability measure to capture long-run approximation, and the eigenfunction gives the long-run dependence on the Markov state. We discuss sufficient conditions for the existence and uniqueness of the relevant eigenvalue and eigenfunction. By showing how changes in the stochastic growth components of cash flows induce changes in the corresponding eigenvalues and eigenfunctions, we reveal a long-run risk-return tradeoff.

Threshold based quasi-static evolution for damage

Adriana Garroni (Univeristé de Rome, La Sapienza)

We consider a variational model for elastic damage proposed by Francfort and Marigo. This energy based model is nonconvex since only to extreme states (damaged and undameged material) are possible, and in the minimization procedure microstructures can be produced. A relaxed incremental problem that accounts for irreversibility can be defined and, by means of time discretization, a relaxed quasi-static evolution can be obtained. This relaxed quasi-static evolution accounts for a damage process that in principle in completely predictive and does not require any a priori assumption on the damage path. We give an alternative model for damage based on a threshold criterion. We prove that an ’energy based’ solution is also a ’threshold’ solution. As a byproduct we also obtain that local minimizers for the energy based model are actually global minimizers.

Scale Spaces on Lie Groups and their Application to Image Processing

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Scale Spaces on Lie Groups and their Application to Image Processing


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