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In recent years using symmetry has proven to be a very useful tool to simplify computations in semidefinite programming. This dissertation examines the possibilities of exploiting discrete symmetries in three contexts: In SDP-based relaxations for polynomial optimization, in testing positivity of symmetric polynomials, and combinatorial optimization. In these contexts the thesis provides new ways for exploiting symmetries and thus deeper insight in the paradigms behind the techniques and studies a concrete combinatorial optimization question.
Der Zusammenhang fundamentaler Symmetriestrukturen mit dem Zugang moderner Naturwissenschaften zur abstrakten Beschreibung komplexer Systeme wird dargestellt. Beginnend mit der geometrischen Symmetrie der Schneeflocke und dem symmetrietheoretischen Ansatz in Platons Timaios wird der symmetrietheoretische Ansatz moderner Naturwissenschaften exemplifiziert und analysiert. Die mathematische Abstraktion wird als reduktionistische Methode verstanden.
The N/Z ratio of free nucleons from collisions of neutron-rich nuclei as a function of their momentum is studied by means of Isospin dependent Quantum Molecular Dynamics. We find that this ratio is not only sensitive to the form of the density dependence of the symmetry potential energy but also its strength determined by the symmetry energy coe cient. The uncertainties about the symmetry energy coe cient influence the accuracy of probing the density dependence of the symmetry energy by means of the N/Z ratio of free nucleons of neutron-rich nuclei.