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Lie systems: theory, generalisations, and applications
Lie systems form a class of systems of first-order ordinary differential
equations whose general solutions can be described in terms of certain finite
families of particular solutions and a set of constants, by means of a
particular type of mapping: the so-called superposition rule. Apart from this
fundamental property, Lie systems enjoy many other geometrical features and
they appear in multiple branches of Mathematics and Physics, which strongly
motivates their study. These facts, together with the authors' recent findings
in the theory of Lie systems, led to the redaction of this essay, which aims to
describe such new achievements within a self-contained guide to the whole
theory of Lie systems, their generalisations, and applications.Comment: 161 pages, 2 figure
Momentum transfer to small particles by aloof electron beams
The force exerted on nanoparticles and atomic clusters by fast passing
electrons like those employed in transmission electron microscopes are
calculated and integrated over time to yield the momentum transferred from the
electrons to the particles. Numerical results are offered for metallic and
dielectric particles of different sizes (0-500 nm in diameter) as well as for
carbon nanoclusters. Results for both linear and angular momentum transfers are
presented. For the electron beam currents commonly employed in electron
microscopes, the time-averaged forces are shown to be comparable in magnitude
to laser-induced forces in optical tweezers. This opens up the possibility to
study optically-trapped particles inside transmission electron microscopes.Comment: 6 pages, 5 figure
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