Quantum expanders and growth of group representations

Let $\pi$ be a finite dimensional unitary representation of a group $G$ with a generating symmetric $n$-element set $S\subset G$. Fix \vp>0. Assume that the spectrum of $|S|^{-1}\sum_{s\in S} \pi(s) \otimes \overline{\pi(s)}$ is included in [-1, 1-\vp] (so there is a spectral gap \ge \vp). Let $r'_N(\pi)$ be the number of distinct irreducible representations of dimension $\le N$ that appear in $\pi$. Then let R_{n,\vp}'(N)=\sup r'_N(\pi) where the supremum runs over all $\pi$ with {n,\vp} fixed. We prove that there are positive constants \delta_\vp and c_\vp such that, for all sufficiently large integer $n$ (i.e. $n\ge n_0$ with $n_0$ depending on \vp) and for all $N\ge 1$, we have \exp{\delta_\vp nN^2} \le R'_{n,\vp}(N)\le \exp{c_\vp nN^2}. The same bounds hold if, in $r'_N(\pi)$, we count only the number of distinct irreducible representations of dimension exactly $= N$.Comment: Main addition: A remark due to Martin Kassabov showing that the numbers R(N) grow faster than polynomial. v3: Minor clarification

Coupled dark energy: Towards a general description of the dynamics

In dark energy models of scalar-field coupled to a barotropic perfect fluid, the existence of cosmological scaling solutions restricts the Lagrangian of the field \vp to p=X g(Xe^{\lambda \vp}), where X=-g^{\mu\nu} \partial_\mu \vp \partial_\nu \vp /2, $\lambda$ is a constant and $g$ is an arbitrary function. We derive general evolution equations in an autonomous form for this Lagrangian and investigate the stability of fixed points for several different dark energy models--(i) ordinary (phantom) field, (ii) dilatonic ghost condensate, and (iii) (phantom) tachyon. We find the existence of scalar-field dominant fixed points (\Omega_\vp=1) with an accelerated expansion in all models irrespective of the presence of the coupling $Q$ between dark energy and dark matter. These fixed points are always classically stable for a phantom field, implying that the universe is eventually dominated by the energy density of a scalar field if phantom is responsible for dark energy. When the equation of state w_\vp for the field \vp is larger than -1, we find that scaling solutions are stable if the scalar-field dominant solution is unstable, and vice versa. Therefore in this case the final attractor is either a scaling solution with constant \Omega_\vp satisfying 0<\Omega_\vp<1 or a scalar-field dominant solution with \Omega_\vp=1.Comment: 21 pages, 5 figures; minor clarifications added, typos corrected and references updated; final version to appear in JCA

Spectral stability for subsonic traveling pulses of the Boussinesq `abc' system

We consider the spectral stability of certain traveling wave solutions of the Boussinesq `abc' system. More precisely, we consider the explicit $sech^2(x)$ like solutions of the form (\vp(x-w t), \psi(x- w t)=(\vp, const. \vp), exhibited by M. Chen (1998) and we provide a complete rigorous characterization of the spectral stability in all cases for which $a=c0$

Regulation of the formation and water permeability of endosomes from toad bladder granular cells.

Osmotic water permeability (Pf) in toad bladder is regulated by the vasopressin (VP)-dependent movement of vesicles containing water channels between the cytoplasm and apical membrane of granular cells. Apical endosomes formed in the presence of serosal VP have the highest Pf of any biological or artificial membrane (Shi and Verkman. 1989. J. Gen. Physiol. 94:1101-1115). We examine here: (a) the influence of protein kinase A and C effectors on transepithelial Pf (Pfte) in intact bladders and on the number and Pf of labeled endosomes, and (b) whether endosome Pf can be modified physically or biochemically. In paired hemibladder studies, Pfte induced by maximal serosal VP (50 mU/ml, 0.03 cm/s) was not different than that induced by 8-Br-cAMP (1 mM), forskolin (50 microM), VP + 8-Br-cAMP, or VP + forskolin. Pf was measured in endosomes labeled in intact bladders with carboxyfluorescein by a stopped-flow, fluorescence-quenching assay using an isolated microsomal suspension; the number and Pf (0.08-0.11 cm/s, 18 degrees C) of labeled endosomes was not different in bladders treated with VP, forskolin, and 8-Br-cAMP. Protein kinase C activation by 1 microM mucosal phorbol myristate acetate (PMA) induced submaximal bladder Pfte (0.015 cm/s) and endosome Pf (0.022 cm/s) in the absence of VP, but had little effect on maximal Pfte and endosome Pf induced by VP. However, PMA increased by threefold the number of apical endosomes with high Pf formed in response to serosal VP. Pf of endosomes containing the VP-sensitive water channel decreased fourfold by increasing membrane fluidity with hexanol or chloroform (0-75 mM); Pf of phosphatidylcholine liposomes (0.002 cm/s) increased 2.5-fold under the same conditions. Endosome Pf was mildly pH dependent, strongly inhibited by HgCl2, but not significantly altered by GTP gamma S, Ca, ATP + protein kinase A, and phosphatase action. We conclude that: (a) water channels cycled in endocytic vesicles are functional and not subject to physiological regulation, (b) VP and forskolin do not have cAMP-independent cellular actions, (c) activation of protein kinase C stimulates trafficking of water channels, but does not increase the number of apical membrane water channels induced by maximal VP, and (d) water channel function is sensitive to membrane fluidity. By using VP and PMA together, large quantities of endosomes containing the VP-sensitive water channel are labeled with fluid-phase endocytic markers

Study of electromagnetic decay of J/ψJ/\psi and ψ′\psi^\prime to vector and pseudoscalar

The electromagnetic decay contributions to $J/\psi(\psi^\prime)\to VP$, where $V$ and $P$ stand for vector and pseudoscalar meson, respectively, are investigated in a vector meson dominance (VMD) model. We show that $J/\psi (\psi^\prime)\to \gamma^*\to VP$ can be constrained well with the available experimental information. We find that this process has significant contributions in $\psi^\prime\to VP$ and may play a key role in understanding the deviations from the so-called "12% rule" for the branching ratio fractions between $\psi^\prime\to VP$ and $J/\psi\to VP$. We also address that the "12% rule" becomes very empirical in exclusive hadronic decay channels.Comment: Revtex, 10 pages, 1 eps figure; further discussions about the helicity conservation rule and exp. constraints are included; version to appear in Phys. Lett.

Vestibular Perception following Acute Unilateral Vestibular Lesions.

Little is known about the vestibulo-perceptual (VP) system, particularly after a unilateral vestibular lesion. We investigated vestibulo-ocular (VO) and VP function in 25 patients with vestibular neuritis (VN) acutely (2 days after onset) and after compensation (recovery phase, 10 weeks). Since the effect of VN on reflex and perceptual function may differ at threshold and supra-threshold acceleration levels, we used two stimulus intensities, acceleration steps of 0.5°/s(2) and velocity steps of 90°/s (acceleration 180°/s(2)). We hypothesised that the vestibular lesion or the compensatory processes could dissociate VO and VP function, particularly if the acute vertiginous sensation interferes with the perceptual tasks. Both in acute and recovery phases, VO and VP thresholds increased, particularly during ipsilesional rotations. In signal detection theory this indicates that signals from the healthy and affected side are still fused, but result in asymmetric thresholds due to a lesion-induced bias. The normal pattern whereby VP thresholds are higher than VO thresholds was preserved, indicating that any 'perceptual noise' added by the vertigo does not disrupt the cognitive decision-making processes inherent to the perceptual task. Overall, the parallel findings in VO and VP thresholds imply little or no additional cortical processing and suggest that vestibular thresholds essentially reflect the sensitivity of the fused peripheral receptors. In contrast, a significant VO-VP dissociation for supra-threshold stimuli was found. Acutely, time constants and duration of the VO and VP responses were reduced - asymmetrically for VO, as expected, but surprisingly symmetrical for perception. At recovery, VP responses normalised but VO responses remained shortened and asymmetric. Thus, unlike threshold data, supra-threshold responses show considerable VO-VP dissociation indicative of additional, higher-order processing of vestibular signals. We provide evidence of perceptual processes (ultimately cortical) participating in vestibular compensation, suppressing asymmetry acutely in unilateral vestibular lesions