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Neutron stars (NSs) and black holes (BHs) are born when the final collapse ofthe stellar core terminates the lives of stars more massive than about 9 Msun.This can trigger the powerful ejection of a large fraction of the star'smaterial in a core-collapse supernova (CCSN), whose extreme luminosity isenergized by the decay of radioactive isotopes such as 56Ni and 56Co. Whenevolving in close binary systems, the compact relics of such infernalcatastrophes spiral towards each other on orbits gradually decaying bygravitational-wave emission. Ultimately, the violent collision of the twocomponents forms a more massive, rapidly spinning remnant, again accompanied bythe ejection of considerable amounts of matter. These merger events can beobserved by high-energy bursts of gamma rays with afterglows andelectromagnetic transients called kilonovae, which radiate the energy releasedin radioactive decays of freshly assembled rapid neutron-capture elements. Bymeans of their mass ejection and the nuclear and neutrino reactions takingplace in the ejecta, both CCSNe and compact object mergers (COMs) are prominentsites of heavy-element nucleosynthesis and play a central role in the cosmiccycle of matter and the chemical enrichment history of galaxies. The nuclearequation of state (EoS) of NS matter, from neutron-rich to proton-dominatedconditions and with temperatures ranging from about zero to ~100 MeV, is acrucial ingredient in these astrophysical phenomena. It determines theirdynamical processes, their remnant properties even at the level of decidingbetween NS or BH, and the properties of the associated emission of neutrinos,whose interactions govern the thermodynamic conditions and theneutron-to-proton ratio for nucleosynthesis reactions in the innermost ejecta.This chapter discusses corresponding EoS dependent effects of relevance inCCSNe as well as COMs. (slightly abridged)

Dynamics and Equation of State Dependencies of Relevance for Nucleosynthesis in Supernovae and Neutron Star Mergers