HELIN Task Force on Electronic Archiving Report

Report of the HELIN Electronic Archiving Task Force, appointed from the HELIN Serials Committee and the HELIN Collection Development Committee

A note on q-Bernoulli numbers and polynomials

By using p-adic q-integrals, we study the q-Bernoulli numbers and polynomials of higher order.Comment: 8 page

Integrable flows and Backlund transformations on extended Stiefel varieties with application to the Euler top on the Lie group SO(3)

We show that the $m$-dimensional Euler--Manakov top on $so^*(m)$ can be represented as a Poisson reduction of an integrable Hamiltonian system on a symplectic extended Stiefel variety $\bar{\cal V}(k,m)$, and present its Lax representation with a rational parameter. We also describe an integrable two-valued symplectic map $\cal B$ on the 4-dimensional variety ${\cal V}(2,3)$. The map admits two different reductions, namely, to the Lie group SO(3) and to the coalgebra $so^*(3)$. The first reduction provides a discretization of the motion of the classical Euler top in space and has a transparent geometric interpretation, which can be regarded as a discrete version of the celebrated Poinsot model of motion and which inherits some properties of another discrete system, the elliptic billiard. The reduction of $\cal B$ to $so^*(3)$ gives a new explicit discretization of the Euler top in the angular momentum space, which preserves first integrals of the continuous system.Comment: 18 pages, 1 Figur

Mode-coupling and nonlinear Landau damping effects in auroral Farley-Buneman turbulence

The fundamental problem of Farley-Buneman turbulence in the auroral $E$-region has been discussed and debated extensively in the past two decades. In the present paper we intend to clarify the different steps that the auroral $E$-region plasma has to undergo before reaching a steady state. The mode-coupling calculation, for Farley-Buneman turbulence, is developed in order to place it in perspective and to estimate its magnitude relative to the anomalous effects which arise through the nonlinear wave-particle interaction. This nonlinear effect, known as nonlinear ``Landau damping'' is due to the coupling of waves which produces other waves which in turn lose energy to the bulk of the particles by Landau damping. This leads to a decay of the wave energy and consequently a heating of the plasma. An equation governing the evolution of the field spectrum is derived and a physical interpration for each of its terms is provided

Symmetry of the Schr\"odinger equation with variable potential

We study symmetry properties of the Schr\"odinger equation with the potential as a new dependent variable, i.e., the transformations which do not change the form of the class of equations. We also consider systems of the Schr\"odinger equations with certain conditions on the potential. In addition we investigate symmetry properties of the equation with convection term. The contact transformations of the Schr\"odinger equation with potential are obtained

A geometric interpretation of the spectral parameter for surfaces of constant mean curvature

Considering the kinematics of the moving frame associated with a constant mean curvature surface immersed in S^3 we derive a linear problem with the spectral parameter corresponding to elliptic sinh-Gordon equation. The spectral parameter is related to the radius R of the sphere S^3. The application of the Sym formula to this linear problem yields constant mean curvature surfaces in E^3. Independently, we show that the Sym formula itself can be derived by an appropriate limiting process R -> infinity.Comment: 12 page

On the structure of the B\"acklund transformations for the relativistic lattices

The B\"acklund transformations for the relativistic lattices of the Toda type and their discrete analogues can be obtained as the composition of two duality transformations. The condition of invariance under this composition allows to distinguish effectively the integrable cases. Iterations of the B\"acklund transformations can be described in the terms of nonrelativistic lattices of the Toda type. Several multifield generalizations are presented

On a class of linearizable Monge-Amp\`ere equations

Monge-Amp\`ere equations of the form, $u_{xx}u_{yy}-u_{xy}^2=F(u,u_x,u_y)$ arise in many areas of fluid and solid mechanics. Here it is shown that in the special case $F=u_y^4f(u, u_x/u_y)$, where $f$ denotes an arbitrary function, the Monge-Amp\`ere equation can be linearized by using a sequence of Amp\`ere, point, Legendre and rotation transformations. This linearization is a generalization of three examples from finite elasticity, involving plane strain and plane stress deformations of the incompressible perfectly elastic Varga material and also relates to a previous linearization of this equation due to Khabirov [7]