Quantum frieze patterns in quantum cluster algebras of type A

We introduce a quantisation of the Coxeter-Conway frieze patterns and prove that they realise quantum cluster variables in quantum cluster algebras associated with linearly oriented Dynkin quivers of type A. As an application, we obtain the explicit polynomials arising from the lower bound phenomenon in these quantum cluster algebras.Comment: 10 page

A Mitochondrial Health Index Sensitive to Mood and Caregiving Stress.

BACKGROUND:Chronic life stress, such as the stress of caregiving, can promote pathophysiology, but the underlying cellular mechanisms are not well understood. Chronic stress may induce recalibrations in mitochondria leading to changes either in mitochondrial content per cell, or in mitochondrial functional capacity (i.e., quality). METHODS:Here we present a functional index of mitochondrial health (MHI) for human leukocytes that can distinguish between these two possibilities. The MHI integrates nuclear and mitochondrial DNA-encoded respiratory chain enzymatic activities and mitochondrial DNA copy number. We then use the MHI to test the hypothesis that daily emotional states and caregiving stress influence mitochondrial function by comparing healthy mothers of a child with an autism spectrum disorder (high-stress caregivers, n = 46) with mothers of a neurotypical child (control group, n = 45). RESULTS:The MHI outperformed individual mitochondrial function measures. Elevated positive mood at night was associated with higher MHI, and nightly positive mood was also a mediator of the association between caregiving and MHI. Moreover, MHI was correlated to positive mood on the days preceding, but not following the blood draw, suggesting for the first time in humans that mitochondria may respond to proximate emotional states within days. Correspondingly, the caregiver group, which had higher perceived stress and lower positive and greater negative daily affect, exhibited lower MHI. This effect was not explained by a mismatch between nuclear and mitochondrial genomes. CONCLUSIONS:Daily mood and chronic caregiving stress are associated with mitochondrial functional capacity. Mitochondrial health may represent a nexus between psychological stress and health

Dynamics on nilpotent character varieties

Let R(N,G) be the connected component of the identity of the variety of representations of a finitely generated nilpotent group N into a connected compact Lie group G, and let X(N,G) be the corresponding moduli space. We show that there exists a natural Out(N)-invariant measure on X(N,G) and that whenever Out(N) has at least one hyperbolic element, the action of Out(N) on X(N,G) is mixing with respect to this measure.Comment: 17 pages, Version 2 has minor updates and corrections, accepted for publication in Conformal Geometry and Dynamic

Proper affine deformations of positive representations

We define for every positive Anosov representation of a nonabelian free group into $\mathrm{SO}(2n,2n-1)$ a family of $\mathbb{R}^{4n-1}$-valued cocycles which induce proper affine actions on $\mathbb{R}^{4n-1}$. We construct fundamental domains in $\mathbb{R}^{4n-1}$ bounded by generalized crooked planes for these affine actions, and deduce that the quotient manifolds are homeomorphic to handlebodies.Comment: 23 pages, 2 figure

Generalizations of Schottky groups

Schottky groups are classical examples of free groups acting properly discontinuously on the complex projective line. We discuss two different applications of similar constructions. The first gives examples of 3-dimensional Lorentzian Kleinian groups which act properly discontinuously on an open dense subset of the Einstein universe. The second gives a large class of examples of free subgroups of automorphisms groups of partially cyclically ordered spaces. We show that for a certain cyclic order on the Shilov boundary of a Hermitian symmetric space, this construction corresponds exactly to representations of fundamental groups of surfaces with boundary which have maximal Toledo invariant