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Orbits of charged particles under the effect of a magnetic field are mathematically described by magnetic geodesics. They appear as solutions to a system of (nonlinear) ordinary differential equations of second order. But we are only interested in periodic solutions. To this end, we study the corresponding system of (nonlinear) parabolic equations for closed magnetic geodesics and, as a main result, eventually prove the existence of long time solutions. As generalization one can consider a system of elliptic nonlinear partial differential equations whose solutions describe the orbits of closed p-branes under the effect of a "generalized physical force". For the corresponding evolution equation, which is a system of parabolic nonlinear partial differential equations associated to the elliptic PDE, we can establish existence of short time solutions.
This paper presents shape-memory foams that can be temporarily fixed in their compressed state and be expanded on demand. Highly porous, nanocomposite foams were prepared from a solution of polyetherurethane with suspended nanoparticles (mean aggregate size 90 nm) which have an iron(III) oxide core with a silica shell. The polymer solution with suspended nanoparticles was cooled down to -20 degrees C in a two-stage process, which was followed by freeze-drying. The average pore size increases with decreasing concentration of nanoparticles from 158 mu m to 230 mu m while the foam porosity remained constant. After fixation of a temporary form of the nanocomposite foams, shape recovery can be triggered either by heat or by exposure to an alternating magnetic field. Compressed foams showed a recovery rate of up to 76 +/- 4% in a thermochamber at 80 degrees C, and a slightly lower recovery rate of up to 65 +/- 4% in a magnetic field.