Three results on representations of Mackey Lie algebras

I. Penkov and V. Serganova have recently introduced, for any non-degenerate pairing $W\otimes V\to\mathbb C$ of vector spaces, the Lie algebra $\mathfrak{gl}^M=\mathfrak{gl}^M(V,W)$ consisting of endomorphisms of $V$ whose duals preserve $W\subseteq V^*$. In their work, the category $\mathbb{T}_{\mathfrak{gl}^M}$ of $\mathfrak{gl}^M$-modules which are finite length subquotients of the tensor algebra $T(W\otimes V)$ is singled out and studied. In this note we solve three problems posed by these authors concerning the categories $\mathbb{T}_{\mathfrak{gl}^M}$. Denoting by $\mathbb{T}_{V\otimes W}$ the category with the same objects as $\mathbb{T}_{\mathfrak{gl}^M}$ but regarded as $V\otimes W$-modules, we first show that when $W$ and $V$ are paired by dual bases, the functor $\mathbb{T}_{\mathfrak{gl}^M}\to \mathbb{T}_{V\otimes W}$ taking a module to its largest weight submodule with respect to a sufficiently nice Cartan subalgebra of $V\otimes W$ is a tensor equivalence. Secondly, we prove that when $W$ and $V$ are countable-dimensional, the objects of $\mathbb{T}_{\mathrm{End}(V)}$ have finite length as $\mathfrak{gl}^M$-modules. Finally, under the same hypotheses, we compute the socle filtration of a simple object in $\mathbb{T}_{\mathrm{End}(V)}$ as a $\mathfrak{gl}^M$-module.Comment: 9 page

Recurrent bursts via linear processes in turbulent environments

Large-scale instabilities occurring in the presence of small-scale turbulent fluctuations are frequently observed in geophysical or astrophysical contexts but are difficult to reproduce in the laboratory. Using extensive numerical simulations, we report here on intense recurrent bursts of turbulence in plane Poiseuille flow rotating about a spanwise axis. A simple model based on the linear instability of the mean flow can predict the structure and time scale of the nearly-periodic and self-sustained burst cycles. Rotating Poiseuille flow is suggested as a prototype for future studies of low-dimensional dynamics embedded in strongly turbulent environments

Local Asymmetry and the Inner Radius of Nodal Domains

Let M be a closed Riemannian manifold of dimension n. Let f be an eigenfunction of the Laplace-Beltrami operator corresponding to an eigenvalue \lambda. We show that the volume of {f>0} inside any ball B whose center lies on {f=0} is > C|B|/\lambda^n. We apply this result to prove that each nodal domain contains a ball of radius > C/\lambda^n.Comment: 12 pages, 1 figure; minor corrections; to appear in Comm. PDE

Linking the evolution of terrestrial interiors and an early outgassed atmosphere to astrophysical observations

A terrestrial planet is molten during formation and may remain so if subject to intense insolation or tidal forces. Observations continue to favour the detection and characterisation of hot planets, potentially with large outgassed atmospheres. We aim to determine the radius of hot Earth-like planets with large outgassed atmospheres and explore differences between molten and solid silicate planets and their influence on the mass-radius relationship and transmission and emission spectra. An interior-atmosphere model, combined with static structure calculations, tracks the evolving radius of a rocky mantle that is outgassing CO$_2$ and H$_2$O. Synthetic emission and transmission spectra are generated for CO$_2$ and H$_2$O dominated atmospheres. Atmospheres dominated by CO$_2$ suppress the outgassing of H$_2$O to a greater extent than previously realised, as previous studies have applied an erroneous relationship between volatile mass and partial pressure. We therefore predict more H$_2$O can be retained by the interior during the later stages of magma ocean crystallisation. Furthermore, formation of a lid at the surface can tie outgassing of H$_2$O to the efficiency of heat transport through the lid, rather than the atmosphere's radiative timescale. Contraction of the mantle as it solidifies gives $\sim5\%$ radius decrease, which can partly be offset by addition of a relatively light species to the atmosphere. We conclude that a molten silicate mantle can increase the radius of a terrestrial planet by around $5\%$ compared to its solid counterpart, or equivalently account for a $13\%$ decrease in bulk density. An outgassing atmosphere can perturb the total radius according to its speciation. Atmospheres of terrestrial planets around M-stars that are dominated by CO$_2$ or H$_2$O can be distinguished by observing facilities with extended wavelength coverage (e.g., JWST).Comment: 19 pages, published in A&A, abstract shortene

Chemical turbulence equivalent to Nikolavskii turbulence

We find evidence that a certain class of reaction-diffusion systems can exhibit chemical turbulence equivalent to Nikolaevskii turbulence. The distinctive characteristic of this type of turbulence is that it results from the interaction of weakly stable long-wavelength modes and unstable short-wavelength modes. We indirectly study this class of reaction-diffusion systems by considering an extended complex Ginzburg-Landau (CGL) equation that was previously derived from this class of reaction-diffusion systems. First, we show numerically that the power spectrum of this CGL equation in a particular regime is qualitatively quite similar to that of the Nikolaevskii equation. Then, we demonstrate that the Nikolaevskii equation can in fact be obtained from this CGL equation through a phase reduction procedure applied in the neighborhood of a codimension-two Turing--Benjamin-Feir point.Comment: 10 pages, 3 figure

Maintaining a Wormhole with a Scalar Field

It is well known that it takes matter that violates the averaged weak energy condition to hold the throat of a wormhole open. The production of such ``exotic'' matter is usually discussed within the context of quantum field theory. In this paper I show that it is possible to produce the exotic matter required to hold a wormhole open classically. This is accomplished by coupling a scalar field to matter that satisfies the weak energy condition. The energy-momentum tensor of the scalar field and the matter separately satisfy the weak energy condition, but there exists an interaction energy-momentum tensor that does not. It is this interaction energy-momentum tensor that allows the wormhole to be maintained.Comment: 12 pages, LaTe