An absorbing boundary formulation for the stratified, linearized, ideal MHD equations based on an unsplit, convolutional perfectly matched layer

Perfectly matched layers are a very efficient and accurate way to absorb waves in media. We present a stable convolutional unsplit perfectly matched formulation designed for the linearized stratified Euler equations. However, the technique as applied to the Magneto-hydrodynamic (MHD) equations requires the use of a sponge, which, despite placing the perfectly matched status in question, is still highly efficient at absorbing outgoing waves. We study solutions of the equations in the backdrop of models of linearized wave propagation in the Sun. We test the numerical stability of the schemes by integrating the equations over a large number of wave periods.Comment: 8 pages, 7 figures, accepted, A &

Phase-field approach to heterogeneous nucleation

We consider the problem of heterogeneous nucleation and growth. The system is described by a phase field model in which the temperature is included through thermal noise. We show that this phase field approach is suitable to describe homogeneous as well as heterogeneous nucleation starting from several general hypotheses. Thus we can investigate the influence of grain boundaries, localized impurities, or any general kind of imperfections in a systematic way. We also put forward the applicability of our model to study other physical situations such as island formation, amorphous crystallization, or recrystallization.Comment: 8 pages including 7 figures. Accepted for publication in Physical Review

Recalibrating R\mathbb{R}-order trees and \mbox{Homeo}_+(S^1)-representations of link groups

In this paper we study the left-orderability of $3$-manifold groups using an enhancement, called recalibration, of Calegari and Dunfield's "flipping" construction, used for modifying \mbox{Homeo}_+(S^1)-representations of the fundamental groups of closed $3$-manifolds. The added flexibility accorded by recalibration allows us to produce \mbox{Homeo}_+(S^1)-representations of hyperbolic link exteriors so that a chosen element in the peripheral subgroup is sent to any given rational rotation. We apply these representations to show that the branched covers of families of links associated to arbitrary epimorphisms of the link group onto a finite cyclic group are left-orderable. This applies, for instance, to fibered hyperbolic strongly quasipositive links. Our result on the orderability of branched covers implies that the degeneracy locus of any pseudo-Anosov flow on an alternating knot complement must be meridional, which generalizes the known result that the fractional Dehn twist coefficient of any hyperbolic fibered alternating knot is zero. Applications of these representations to order-detection of slopes are also discussed in the paper.Comment: 43 pages, 12 figure

JSJ decompositions of knot exteriors, Dehn surgery and the LL-space conjecture

In this article, we apply slope detection techniques to study properties of toroidal $3$-manifolds obtained by performing Dehn surgeries on satellite knots in the context of the $L$-space conjecture. We show that if $K$ is an $L$-space knot or admits an irreducible rational surgery with non-left-orderable fundamental group, then the JSJ graph of its exterior is a rooted interval. Consequently, any rational surgery on a composite knot has a left-orderable fundamental group. This is the left-orderable counterpart of Krcatovich's result on the primeness of $L$-space knots, which we reprove using our methods. Analogous results on the existence of co-orientable taut foliations are proved when the knot has a fibred companion. Our results suggest a new approach to establishing the counterpart of Krcatovich's result for surgeries with co-orientable taut foliations, on which partial results have been achieved by Delman and Roberts. Finally, we prove results on left-orderable $p/q$-surgeries on knots with $|p|$ small.Comment: 25 pages, 1 appendi

Slope detection and toroidal 3-manifolds

We investigate the $L$-space conjecture for toroidal $3$-manifolds using various notions of slope detection. This leads to a proof that toroidal $3$-manifolds with small order first homology have left-orderable fundamental groups and, under certain fibring conditions, admit co-oriented taut foliations. It also allows us to show that cyclic branched covers of prime satellite knots are not $L$-spaces, have left-orderable fundamental groups and, when they have fibred companion knots, admit co-oriented taut foliations. A partial extension to prime toroidal links leads to a proof that prime quasi-alternating links are either hyperbolic or $(2, m)$-torus links. Our main technical result gives sufficient conditions for certain slopes on the boundaries of rational homology solid tori to be detected by left-orders, foliations and Heegaard Floer homology.Comment: 60 pages, 26 figure