Factorization of the current algebra and integrable top-like systems

A hierarchy of integrable hamiltonian nonlinear ODEs is associated with any decomposition of the Lie algebra of Laurent series with coefficients being elements of a semi-simple Lie algebra into a sum of the subalgebra consisting of the Taylor series and some complementary subalgebra. In the case of the Lie algebra $so(3)$ our scheme covers all classical integrable cases in the Kirchhoff problem of the motion of a rigid body in an ideal fluid. Moreover, the construction allows us to generate integrable deformations for known integrable models.Comment: 21 page

Solar-Terrestrial Simulations of CMEs with a Realistic Initiation Mechanism: Case Study for Active Region 10069

Most simulations of coronal mass ejections (CMEs) to date either focus on the interplanetary propagation of a giant plasma "blob" without paying too much attention to its origin and to the formation process or they focus on the complex evolution of the coronal magnetic field due to (sub-)photospheric motions which result in an eruption. Here, we present global simulations of CMEs where coronal motions are used to produce a realistic evolution of the coronal magnetic field and cause an eruption. We focus on active region 10069, which produced a number of eruptions in late August 2002, including the August 24, 2002 CME - a fast (~2000 km/s) eruption originating from W81-, as well as a slower eruption on August 22, 2002 (originating from W62). Using a three-dimensional magneto-hydrodynamic (MHD) simulation of these ejections with the Space Weather Modeling Framework (SWMF), we show how a realistic initiation mechanism enables us to study the deflection of the CME in the corona and in the heliosphere. Reconnection of the erupting magnetic field with that of neighboring streamers and active regions modify the solar connectivity of the field lines connecting to Earth and change the expected solar energetic particle fluxes. Comparing the results at 1 AU of our simulations with in situ observations by the ACE spacecraft, we propose an alternate solar origin for the shock wave observed at L1 on August 26.Comment: 4 pages, 2 figures, refereed proceedings for Solar Wind 1

The Spectra of Large Toeplitz Band Matrices with a Randomly Perturbed Entry

This report is concerned with the union $sp_{\Omega}^{(j,k)}T_{n}(a)$ of all possible spectra that may emerge when perturbing a large $n \times n$ Toeplitz band matrix $T_{n}(a)$ in the $(j,k)$ site by a number randomly chosen from some set $\Omega$. The main results give descriptive bounds and, in several interesting situations, even provide complete identifications of the limit of $sp_{\Omega}^{(j,k)}T_{n}(a)$ as $n \to \infty$. Also discussed are the cases of small and large sets $\Omega$ as well as the "discontinuity of the infinite volume case", which means that in general $sp_{\Omega}^{(j,k)}T_{n}(a)$ does not converge to something close to $sp_{\Omega}^{(j,k)}T_{n}(a)$ as $n \to \infty$, where $T(a)$ is the corresponding infinite Toeplitz matrix. Illustrations are provided for tridiagonal Toeplitz matrices, a notable special case. \ud \ud The second author was supported by UK Enginering and Physical Sciences Research Council Grant GR/M1241