Optimal extension to Sobolev rough paths

We show that every $\mathbb{R}^d$-valued Sobolev path with regularity $\alpha$ and integrability $p$ can be lifted to a Sobolev rough path in the sense of T. Lyons provided $\alpha >1/p>0$. Moreover, we prove the existence of unique rough path lifts which are optimal w.r.t. strictly convex functionals among all possible rough path lifts given a Sobolev path. As examples, we consider the rough path lift with minimal Sobolev norm and characterize the Stratonovich rough path lift of a Brownian motion as optimal lift w.r.t. to a suitable convex functional. Generalizations of the results to Besov spaces are briefly discussed.Comment: Typos fixed. To appear in Potential Analysi

Genetic insights on sleep schedules: this time, it's PERsonal.

The study of circadian rhythms is emerging as a fruitful opportunity for understanding cellular mechanisms that govern human physiology and behavior, fueled by evidence directly linking sleep disorders to genetic mutations affecting circadian molecular pathways. Familial advanced sleep-phase disorder (FASPD) is the first recognized Mendelian circadian rhythm trait, and affected individuals exhibit exceptionally early sleep-wake onset due to altered post-translational regulation of period homolog 2 (PER2). Behavioral and cellular circadian rhythms are analogously affected because the circadian period length of behavior is reduced in the absence of environmental time cues, and cycle duration of the molecular clock is likewise shortened. In light of these findings, we review the PER2 dynamics in the context of circadian regulation to reveal the mechanism of sleep-schedule modulation. Understanding PER2 regulation and functionality may shed new light on how our genetic composition can influence our sleep-wake behaviors

Laser Mode Bifurcations Induced by PT\mathcal{PT}-Breaking Exceptional Points

A laser consisting of two independently-pumped resonators can exhibit mode bifurcations that evolve out of the exceptional points (EPs) of the linear system at threshold. The EPs are non-Hermitian degeneracies occurring at the parity/time-reversal ($\mathcal{PT}$) symmetry breaking points of the threshold system. Above threshold, the EPs become bifurcations of the nonlinear zero-detuned laser modes, which can be most easily observed by making the gain saturation intensities in the two resonators substantially different. Small pump variations can then switch abruptly between different laser behaviors, e.g. between below-threshold and $\mathcal{PT}$-broken single-mode operation.Comment: 4 pages, 3 figure