Poisson-Lie group of pseudodifferential symbols

We introduce a Lie bialgebra structure on the central extension of the Lie algebra of differential operators on the line and the circle (with scalar or matrix coefficients). This defines a Poisson--Lie structure on the dual group of pseudodifferential symbols of an arbitrary real (or complex) order. We show that the usual (second) Benney, KdV (or GL_n--Adler--Gelfand--Dickey) and KP Poisson structures are naturally realized as restrictions of this Poisson structure to submanifolds of this ``universal'' Poisson--Lie group. Moreover, the reduced (=SL_n) versions of these manifolds (W_n-algebras in physical terminology) can be viewed as subspaces of the quotient (or Poisson reduction) of this Poisson--Lie group by the dressing action of the group of functions. Finally, we define an infinite set of functions in involution on the Poisson--Lie group that give the standard families of Hamiltonians when restricted to the submanifolds mentioned above. The Poisson structure and Hamiltonians on the whole group interpolate between the Poisson structures and Hamiltonians of Benney, KP and KdV flows. We also discuss the geometrical meaning of W_\infty as a limit of Poisson algebras W_\epsilon as \epsilon goes to 0.Comment: 64 pages, no figure

Non-Local Matrix Generalizations of W-Algebras

There is a standard way to define two symplectic (hamiltonian) structures, the first and second Gelfand-Dikii brackets, on the space of ordinary linear differential operators of order $m$, $L = -d^m + U_1 d^{m-1} + U_2 d^{m-2} + \ldots + U_m$. In this paper, I consider in detail the case where the $U_k$ are $n\times n$-matrix-valued functions, with particular emphasis on the (more interesting) second Gelfand-Dikii bracket. Of particular interest is the reduction to the symplectic submanifold $U_1=0$. This reduction gives rise to matrix generalizations of (the classical version of) the {\it non-linear} $W_m$-algebras, called $V_{m,n}$-algebras. The non-commutativity of the matrices leads to {\it non-local} terms in these $V_{m,n}$-algebras. I show that these algebras contain a conformal Virasoro subalgebra and that combinations $W_k$ of the $U_k$ can be formed that are $n\times n$-matrices of conformally primary fields of spin $k$, in analogy with the scalar case $n=1$. In general however, the $V_{m,n}$-algebras have a much richer structure than the $W_m$-algebras as can be seen on the examples of the {\it non-linear} and {\it non-local} Poisson brackets of any two matrix elements of $U_2$ or $W_3$ which I work out explicitly for all $m$ and $n$. A matrix Miura transformation is derived, mapping these complicated second Gelfand-Dikii brackets of the $U_k$ to a set of much simpler Poisson brackets, providing the analogue of the free-field realization of the $W_m$-algebras.Comment: 43 pages, a reference and a remark on the conformal properties for $U_1\ne 0$ adde

Exotic clouds in the local interstellar medium

The neutral interstellar medium (ISM) inside the Local Bubble (LB) has been known to have properties typical of the warm neutral medium (WNM). However, several recent neutral hydrogen (HI) absorption experiments show evidence for the existence of at least several cold diffuse clouds inside or at the boundary of the LB, with properties highly unusual relative to the traditional cold neutral medium. These cold clouds have a low HI column density, and AU-scale sizes. As the kinematics of cold and warm gas inside the LB are similar, this suggests a possibility of all these different flavors of the local ISM belonging to the same interstellar flow. The co-existence of warm and cold phases inside the LB is exciting as it can be used to probe the thermal pressure inside the LB. In addition to cold clouds, several discrete screens of ionized scattering material are clearly located inside the LB. The cold exotic clouds inside the LB are most likely long-lived, and we expect many more clouds with similar properties to be discovered in the future with more sensitive radio observations. While physical mechanisms responsible for the production of such clouds are still poorly understood, dynamical triggering of phase conversion and/or interstellar turbulence are likely to play an important role.Comment: 10 pages, refereed, accepted for publication in the proceedings of the "From the Outer Heliosphere to the Local Bubble: Comparisons of New Observations with Theory" conference, Space Science Review

Reconciling Bayesian and frequentist evidence in the point null testing problem

Posterior probability, p-value, point null hypothesis, 62F15, 62A15,