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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
Experimental Verification of the Number Relation at Room and Elevated Temperatures
The accuracy of the Neuber equation for predicting notch root stress-strain behavior at room temperature and at 650 C was experimentally investigated. Strains on notched specimens were measured with a non-contacting, interferometric technique and stresses were simulated with smooth specimens. Predictions of notch root stress-strain response were made from the Neuber Equation and smooth specimen behavior. Neuber predictions gave very accurate results at room temperature. However, the predicted interaction of creep and stress relaxation differed from experimentally measured behavior at 650 C
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