4,270 research outputs found
Stochastic nonlinear wave dynamics on compact surfaces
We study the Cauchy problem for the nonlinear wave equations (NLW) with
random data and/or stochastic forcing on a two-dimensional compact Riemannian
manifold without boundary. (i) We first study the defocusing stochastic damped
NLW driven by additive space-time white-noise, and with initial data
distributed according to the Gibbs measure. By introducing a suitable
space-dependent renormalization, we prove local well-posedness of the
renormalized equation. Bourgain's invariant measure argument then allows us to
establish almost sure global well-posedness and invariance of the Gibbs measure
for the renormalized stochastic damped NLW. (ii) Similarly, we study the random
data defocusing NLW (without stochastic forcing), and establish the same
results as in the previous setting. (iii) Lastly, we study the stochastic NLW
without damping. By introducing a space-time dependent renormalization, we
prove its local well-posedness with deterministic initial data in all
subcritical spaces.
These results extend the corresponding recent results on the two-dimensional
torus obtained by (i) Gubinelli-Koch-Oh-Tolomeo (2018), (ii) Oh-Thomann (2017),
and (iii) Gubinelli-Koch-Oh (2018), to a general class of compact manifolds.
The main ingredient is the Green's function estimate for the Laplace-Beltrami
operator in this setting to study regularity properties of stochastic terms
appearing in each of the problems.Comment: 55 pages, some typos corrected and details adde
Optimal Schedules in Multitask Motor Learning
Although scheduling multiple tasks in motor learning to maximize long-term retention of performance is of great practical importance in sports training and motor rehabilitation after brain injury, it is unclear how to do so. We propose here a novel theoretical approach that uses optimal control theory and computational models of motor adaptation to determine schedules that maximize long-term retention predictively. Using Pontryagin’s maximum principle, we derived a control law that determines the trial-by-trial task choice that maximizes overall delayed retention for all tasks, as predicted by the state-space model. Simulations of a single session of adaptation with two tasks show that when task interference is high, there exists a threshold in relative task difficulty below which the alternating schedule is optimal. Only for large differences in task difficulties do optimal schedules assign more trials to the harder task. However, over the parameter range tested, alternating schedules yield long-term retention performance that is only slightly inferior to performance given by the true optimal schedules. Our results thus predict that in a large number of learning situations wherein tasks interfere, intermixing tasks with an equal number of trials is an effective strategy in enhancing long-term retention
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