Involutivity of integrals for sine-Gordon, modified KdV and potential KdV maps

Closed form expressions in terms of multi-sums of products have been given in \cite{Tranclosedform, KRQ} of integrals of sine-Gordon, modified Korteweg-de Vries and potential Korteweg-de Vries maps obtained as so-called $(p,-1)$-traveling wave reductions of the corresponding partial difference equations. We prove the involutivity of these integrals with respect to recently found symplectic structures for those maps. The proof is based on explicit formulae for the Poisson brackets between multi-sums of products.Comment: 24 page

The staircase method: integrals for periodic reductions of integrable lattice equations

We show, in full generality, that the staircase method provides integrals for mappings, and correspondences, obtained as traveling wave reductions of (systems of) integrable partial difference equations. We apply the staircase method to a variety of equations, including the Korteweg-De Vries equation, the five-point Bruschi-Calogero-Droghei equation, the QD-algorithm, and the Boussinesq system. We show that, in all these cases, if the staircase method provides r integrals for an n-dimensional mapping, with 2r<n, then one can introduce q<= 2r variables, which reduce the dimension of the mapping from n to q. These dimension-reducing variables are obtained as joint invariants of k-symmetries of the mappings. Our results support the idea that often the staircase method provides sufficiently many integrals for the periodic reductions of integrable lattice equations to be completely integrable. We also study reductions on other quad-graphs than the regular 2D lattice, and we prove linear growth of the multi-valuedness of iterates of high-dimensional correspondences obtained as reductions of the QD-algorithm.Comment: 40 pages, 23 Figure

Introducing a Calculus of Effects and Handlers for Natural Language Semantics

In compositional model-theoretic semantics, researchers assemble truth-conditions or other kinds of denotations using the lambda calculus. It was previously observed that the lambda terms and/or the denotations studied tend to follow the same pattern: they are instances of a monad. In this paper, we present an extension of the simply-typed lambda calculus that exploits this uniformity using the recently discovered technique of effect handlers. We prove that our calculus exhibits some of the key formal properties of the lambda calculus and we use it to construct a modular semantics for a small fragment that involves multiple distinct semantic phenomena

The Okubo-Weiss Criteria in Two-Dimensional Hydrodynamic and Magnetohydrodynamic Flows

The Okubo [2]-Weiss [3] criterion is recast by using the 2D hydrodynamic Beltrami condition (Shivamoggi et al.[13]) that approximates the slow flow-variation ansatz imposed in its derivation. This turns out to provide an interesting interpretation of the Okubo-Weiss criterion very logically in terms of the topological properties of the underlying vorticity manifold. These developments are then extended to 2D quasi-geostrophic flows (via the potential divorticity framework), magnetohydrodynamic flows and electron magnetohydrodynamic flows (via the generalized magnetic flux framework) and the Okubo-Weiss criteria for these cases are considered.Comment: 13 page

Dynamics and structure of decaying shallow dipolar vortices

The current work reports on a numerical and experimental study of the evolution of decaying dipolar vortices in a shallow fluid layer. The dynamics and the structure of such vortices are investigated as a function of both their Reynolds number Re and the aspect ratio of vertical and horizontal length scales δ. By quantifying the strength of the secondary motions (vertical motions and nonzero horizontal divergence) with respect to the swirling motions of the primary vortex cores, it was found that the three-dimensionality of a shallow (δ << 1) dipolar vortex only depends on a single parameter: δ²Re. Depending on the value of this parameter, three flow regimes are observed for shallow dipolar vortices: (1) a quasi-two-dimensional regime where the structure of the dipolar vortex remains almost unchanged throughout its lifetime, (2) a transitional regime where the structure presents some three-dimensional characteristics but remains coherent, and (3) a three-dimensional regime where the structure of the dipolar vortex acquires a complicated three-dimensional shape with a persistent spanwise vortex at its front

Higher analogues of the discrete-time Toda equation and the quotient-difference algorithm

The discrete-time Toda equation arises as a universal equation for the relevant Hankel determinants associated with one-variable orthogonal polynomials through the mechanism of adjacency, which amounts to the inclusion of shifted weight functions in the orthogonality condition. In this paper we extend this mechanism to a new class of two-variable orthogonal polynomials where the variables are related via an elliptic curve. This leads to a `Higher order Analogue of the Discrete-time Toda' (HADT) equation for the associated Hankel determinants, together with its Lax pair, which is derived from the relevant recurrence relations for the orthogonal polynomials. In a similar way as the quotient-difference (QD) algorithm is related to the discrete-time Toda equation, a novel quotient-quotient-difference (QQD) scheme is presented for the HADT equation. We show that for both the HADT equation and the QQD scheme, there exists well-posed $s$-periodic initial value problems, for almost all \s\in\Z^2. From the Lax-pairs we furthermore derive invariants for corresponding reductions to dynamical mappings for some explicit examples.Comment: 38 page

Line emission from circumstellar disks around A stars

The nature of the tenuous disks around A stars has raised considerable controversy in the literature during the past decade. The debate whether or not the disk around beta Pictoris contains gaseous molecular hydrogen is only the most recent example. Since CO is in general a poor tracer for the gas content of these low mass disks, we discuss here detailed emission line calculations for alternative tracers like C and C+, based on recent optically thin disk models by Kamp & van Zadelhoff (2001). The [CII] 157.7 mum line was searched toward Vega and beta Pictoris -- the most prominent A stars with disks -- using ISO LWS data, and a tentative detection is reported toward the latter object. From a comparison with emission line observations as well as absorption line studies of both stars, the gas-to-dust ratio is constrained to lie between 0.5 and 9 for beta Pictoris. For Vega the [CII] observations indicate an upper limit of 0.2 M_Earth for the disk gas mass. Predicted line intensities of C+ and C are presented for a range of models and appear promising species to trace the gas content in the disks around A stars with future instrumental capabilities (SOFIA, Herschel, APEX and ALMA). Searches for CO emission should focus on the J=3-2 line.Comment: 14 pages, to appear in A&A (accepted

Inertial oscillations in a confined monopolar vortex subjected to background rotation

We study the axisymmetric inertial oscillations in a confined monopolar vortex under the influence of background rotation. By first focusing on the inviscid linear dynamics, and later studying the effects of viscosity and of a no-slip bottom, we characterize the effects of rotation and confinement. It was found that background rotation allows for oscillations outside the vortex core even with frequencies larger than 2O, with O the background rotation rate. However, confinement is necessary for the system to sustain oscillations with frequencies smaller than 2O. Through the analytical solution for a small perturbation of a Rankine vortex, we obtain five regimes where the oscillations are qualitatively different, depending on their frequency. Numerical results for the linear inviscid waves sustained by a Lamb–Oseen vortex show a similar behavior. The effects of viscosity are twofold: the oscillations are damped and the vortex sustaining the oscillations is modified. When a no-slip bottom is considered, a boundary layer drives a secondary motion superimposed on the inertial oscillations. In this case, the vortex is quickly damped, but the oscillations persist due to the background rotation

Inertial oscillations in a confined vortex subjected to background rotation

No abstract