Some stable homology calculations and Occam's razor for Hodge structures

Motivated by motivic zeta function calculations, Vakil and Wood in [10] made several conjectures regarding the topology of subspaces of symmetric products. The purpose of this note is to prove two of these conjectures and disprove a strengthening of one of them.     

A Brownian Particle in a Microscopic Periodic Potential

We study a model for a massive test particle in a microscopic periodic potential and interacting with a reservoir of light particles. In the regime considered, the fluctuations in the test particle’s momentum resulting from collisions typically outweigh the shifts in momentum generated by the periodic force, so the force is effectively a perturbative contribution. The mathematical starting point is an idealized reduced dynamics for the test particle given by a linear Boltzmann equation. In the limit that the mass ratio of a single reservoir particle to the test particle tends to zero, we show that there is convergence to the Ornstein–Uhlenbeck process under the standard normalizations for the test particle variables. Our analysis is primarily directed towards bounding the perturbative effect of the periodic potential on the particle’s momentum.     

Transcranial direct current stimulation of the prefrontal cortex modulates working memory performance: combined behavioural and electrophysiological evidence

Background  

Transcranial direct current stimulation (tDCS) is a technique that can systematically modify behaviour by inducing changes in the underlying brain function. In order to better understand the neuromodulatory effect of tDCS, the present study examined the impact of tDCS on performance in a working memory (WM) task and its underlying neural activity. In two experimental sessions, participants performed a letter two-back WM task after sham and either anodal or cathodal tDCS over the left dorsolateral prefrontal cortex (DLPFC).