Symmetries of the Continuous and Discrete Krichever-Novikov Equation

A symmetry classification is performed for a class of differential-difference equations depending on 9 parameters. A 6-parameter subclass of these equations is an integrable discretization of the Krichever-Novikov equation. The dimension n of the Lie point symmetry algebra satisfies 1≤n≤5. The highest dimensions, namely n=5 and n=4 occur only in the integrable cases

Die Zweite Deutsche Spacelab-Mission D-2

Wie die erste deutsche Spacelab-Mission D-1, aber mit wesentlich erweitertem Umfang, dient das Weltraumlabor auch bei seinem Einsatz bei D-2 multidisziplinaeren Aufgaben. Sie betreffen u.a. Stroemungsmechanik, Materialwissenschaften, Gravitationsbiologie, Auswirkungen der Weltraumstrahlungen auf biologische Systeme, Einfluesse der Weltraumbedingungen auf den Menschen, Einsatz einer Ultraweitwinkelkamera fuer Untersuchungen der Milchstrasse und der Erprobung technologischer Entwicklungen im Weltraum. Berichtet wird ferner ueber den Missionsablauf, Flugmannschaft, Nutzlastbetrieb und Kommunikationssysteme. Ausfuehrlicher informiert wird dann ueber die vorstehend genannten Forschungsprogramme der Mission D-2, ueber die Raumfahrzeug- Bordmannschaft und ueber die Vorbereitung der deutschen Wissenschaftsastronauten. Zu den 93 Experimenten von D-2 steuern die Japaner 5 Experimente bei. (orig.)As with D-1, the first German spacelab mission, except that the scope of the second mission is considerably greater, the spacelab is also performing tasks in a variety of disciplines in the D-2 mission. Its spectrum of tasks concerns fluid mechanics, material sciences, gravitational biology, the effects of cosmic rays on biological systems, the influences of space environment on man, the deployment of an ultra wide-angled camera for investigating the Milky Way and trying out technological developments in space. The text also reports on the way in which the mission is intended to progress, the crew on the mission, the useful-load operation and the communication systems. The report then goes into detail on the abovementioned research programmes entailed within mission D-2, on the crew manning the spacecraft and on the preparation which the German research astronauts are undergoing. The Japanese are contributing 5 of the 93 experiments being performed on D-2. (orig.)SIGLEAvailable from TIB Hannover: RO 6006(1993,01) / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekDEGerman

Discrete Moving Frames and Discrete Integrable Systems

Group based moving frames have a wide range of applications, from the classical equivalence problems in differential geometry to more modern applications such as computer vision. Here we describe what we call a discrete group based moving frame, which is essentially a sequence of moving frames with overlapping domains. We demonstrate a small set of generators of the algebra of invariants, which we call the discrete Maurer--Cartan invariants, for which there are recursion formulae. We show that this offers significant computational advantages over a single moving frame for our study of discrete integrable systems. We demonstrate that the discrete analogues of some curvature flows lead naturally to Hamiltonian pairs, which generate integrable differential-difference systems. In particular, we show that in the centro-affine plane and the projective space, the Hamiltonian pairs obtained can be transformed into the known Hamiltonian pairs for the Toda and modified Volterra lattices respectively under Miura transformations. We also show that a specified invariant map of polygons in the centro-affine plane can be transformed to the integrable discretization of the Toda Lattice. Moreover, we describe in detail the case of discrete flows in the homogeneous 2-sphere and we obtain realizations of equations of Volterra type as evolutions of polygons on the sphere

Integrability of Difference Equations Through Algebraic Entropy and Generalized Symmetries

Given an equation arising from some application or theoretical consideration one of the first questions one might ask is: What is its behavior? It is integrable? In these lectures we will introduce two different ways for establishing (and in some sense also defining) integrability for difference equations: Algebraic Entropy and Generalized Symmetries. Algebraic Entropy deals with the degrees of growth of the solution of any kind of discrete equation (ordinary, partial or even differential-difference) and usually provides a quick test to establish if an equation is or not integrable. The approach based on Generalized Symmetries also provides tools for investigating integrable equations and to find particular solutions by symmetry reductions. The main focus of the lectures will be on the computational tools that allow us to calculate Generalized Symmetries and extract the value of the Algebraic Entropy from a finite number of iterations of the map