Non-ergodicity of inviscid two-dimensional flow on a beta-plane and on the surface of a rotating sphere

It is shown that, for a sufficiently large value of β, two-dimensional flow on a doubly-periodic beta-plane cannot be ergodic (phase-space filling) on the phase-space surface of constant energy and enstrophy. A corresponding result holds for flow on the surface of a rotating sphere, for a sufficiently rapid rotation rate Ω. This implies that the higher-order, non-quadratic invariants are exerting a significant influence on the statistical evolution of the flow. The proof relies on the existence of a finite-amplitude Liapunov stability theorem for zonally symmetric basic states with a non-vanishing absolute-vorticity gradient. When the domain size is much larger than the size of a typical eddy, then a sufficient condition for non-ergodicity is that the wave steepness ε < 1, where ε = 2[surd radical]2Z/βU in the planar case and $\epsilon = 2^{\frac{1}{4}} a^{\frac{5}{2}}Z^{\frac{7}{4}}/\Omega U^{\frac{5}{2}}$ in the spherical case, and where Z is the enstrophy, U the r.m.s. velocity, and a the radius of the sphere. This result may help to explain why numerical simulations of unforced beta-plane turbulence (in which ε decreases in time) seem to evolve into a non-ergodic regime at large scales.

The growth rate and dimension theory of beta-expansions

In a recent paper of Feng and Sidorov they show that for β∈(1,(1+5√)/2) the set of β-expansions grows exponentially for every x∈(0,1/(β−1)). In this paper we study this growth rate further. We also consider the set of β-expansions from a dimension theory perspective.

Differential crosstalk between estrogen receptor (ER) alpha and ER beta and the thyroid hormone receptor isoforms results in flexible regulation of the consensus ERE

Crosstalk between nuclear receptors is important for conversion of external and internal stimuli to a physiologically meaningful response by cells. Previous studies from this laboratory have demonstrated crosstalk between the estrogen (ER) and thyroid hormone receptors (TR) on two estrogen responsive physiological promoters, the preproenkephalin and oxytocin receptor gene promoter. Since ERa and ERb are isoforms possessing overlapping and distinct transactivation properties, we hypothesized that the interaction of ERa and b with the various TR isoforms would not be equivalent. To explore this hypothesis, the consensus estrogen response element (ERE)derived from the Xenopus vitellogenin gene is used to investigate the differences in interaction between ERa and b isoforms and the different TR isoforms in fibroblast cells. Both the ER isoforms transactivate from the consensus ERE, though ERa transactivates to a greater extent than ERb. Although neither of the TRb isoforms have an effect on ERa transactivation from the consensus ERE, the liganded TRa1 inhibits the ERa transactivation from the consensus ERE. In contrast, the liganded TRa1 facilitates ERb-mediated transactivation. The crosstalk between the TRb isoforms with the ERa isoform, on the consensus ERE, is different from that with the ERb isoform. The use of a TRa1 mutant, which is unable to bind DNA, abolishes the ability of the TRa1 isoform to interact with either of the ER isoforms. These differences in nuclear receptor crosstalk reveal an important functional difference between isoforms, which provides a novel mechanism for neuroendocrine integration.

Distinct behavioral phenotypes in male mice lacking the thyroid hormone receptor alpha1 or beta isoforms

Thyroid hormones influence both neuronal development and anxiety via the thyroid hormone receptors (TRs). The TRs are encoded by two different genes, TRalpha and TRbeta. The loss of TRalpha1 is implicated in increased anxiety in males, possibly via a hippocampal increase in GABAergic activity. We compared both social behaviors and two underlying and related non-social behaviors, state anxiety and responses to acoustic and tactile startle in the gonadally intact TRalpha1 knockout (alpha1KO) and TRbeta (betaKO) male mice to their wild-type counterparts. For the first time, we show an opposing effect of the two TR isoforms, TRalpha1 and TRbeta, in the regulation of state anxiety, with alpha1 knockout animals (alpha1KO) showing higher levels of anxiety and betaKO males showing less anxiety compared to respective wild-type mice. At odds with the increased anxiety in non-social environments, alpha1KO males also show lower levels of responsiveness to acoustic and tactile startle stimuli. Consistent with the data that T4 is inhibitory to lordosis in female mice, we show subtly increased sex behavior in alpha1KO male mice. These behaviors support the idea that TRalpha1 could be inhibitory to ERalpha driven transcription that ultimately impacts ERalpha driven behaviors such as lordosis. The behavioral phenotypes point to novel roles for the TRs, particularly in non-social behaviors such as state anxiety and startle.