arXiv subjects
Publications and source records attributed to L. Viana.
In this work, we prove a Carleman estimate for a parabolic problem which has a dissipative degenerate term. The prove relies on choose a suitable weight function that change of sign inside the control domain.
In this paper, we are concerned with the internal control of a class of one-dimensional nonlinear parabolic systems with nonlocal and weakly degenerate diffusion coefficients. Our main theorem establishes a local null controllability result with only one internal control for a system of two equations. The proof, based on the ideias developed by Fursikov and Imanivilov, is obtained from the global null controllability of the linearized system provided by \textit{Lyusternik's Inverse Mapping Theorem}. This work extends the results previously treated by the authors for just one equation. For the system, the main issue is to obtain similar results with just one internal control, which requires a new Carleman estimate with the local term just depending on one of the state function.
We perform a systematic analytical study of finite size effects in separable recurrent neural network models with sequential dynamics, away from saturation. We find two types of finite size effects: thermal fluctuations, and disorder-induced `frozen' corrections to the mean-field laws. The finite size effects are described by equations that correspond to a time-dependent Ornstein-Uhlenbeck process. We show how the theory can be used to understand and quantify various finite size phenomena in recurrent neural networks, with and without detailed balance.
We perform a stationary state replica analysis for a layered network of Ising spin neurons, with recurrent Hebbian interactions within each layer, in combination with strictly feed-forward Hebbian interactions between successive layers. This model interpolates between the fully recurrent and symmetric attractor network studied by Amit el al, and the strictly feed-forward attractor network studied by Domany et al. Due to the absence of detailed balance, it is as yet solvable only in the zero temperature limit. The built-in competition between two qualitatively different modes of operation, feed-forward (ergodic within layers) versus recurrent (non- ergodic within layers), is found to induce interesting phase transitions.