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Abstract We prove matrix Li–Yau–Hamilton estimates for positive solutions to the heat equation and the backward conjugate heat equation, both coupled with the Ricci flow. We then apply these estimates to establish the monotonicity of parabolic frequencies up to correction factors. As applications, we obtain some unique continuation results under the nonnegativity of sectional or complex sectional curvature. 

Contexts and discrete stimuli often hierarchically influence the association between a stimulus and outcome. This phenomenon, called occasion setting, is central to modulation-based Pavlovian learning. We conducted two experiments with humans in fear and appetitive conditioning paradigms, training stimuli in differential conditioning, feature-positive discriminations, and feature-negative discriminations. We also investigated the effects of trait anxiety and trait depression on these forms of learning. Results from both experiments showed that participants were able to successfully learn which stimuli predicted the electric shock and monetary reward outcomes. Additionally, as hypothesized, the stimuli trained as occasion setters had little-to-no effect on simple reinforced or non-reinforced stimuli, suggesting the former were indeed occasion setters. Lastly, in fear conditioning, trait anxiety was associated with increases in fear of occasion setter/conditional stimulus compounds; in appetitive conditioning, trait depression was associated with lower expectations of monetary reward for the trained negative occasion setting compound and transfer of the negative occasion setter to the simple reinforced stimulus. These results suggest that clinically anxious individuals may have enhanced fear of occasion setting compounds, and clinically depressed individuals may expect less reward with compounds involving the negative occasion setter.