In this thesis we present a study of quantum Markov semigroups. In particular, we mainly consider quantum Markov semigroups with detailed balance that are defined on finite-dimensional C*-algebras. They have an invariant density matrix ρ. Carlen and Maas showed that the evolution on the set of invertible density matrices that is given by such a semigroup is gradient flow for the relative entropy with respect to ρ for some Riemannian metric. This result is a non-commutative analog of certain diffusion equations that are gradient flow in the second order Wasserstein space. We provide a self-contained and accessible account to these issues. Moreover, we give a complete introduction to Tomita-Takesaki theory which has a close relation with quantum Markov semigroups satisfying detailed balance. Finally, we present some examples of these semigroups that arise from quantum theory.

In this thesis the stability type of y=0 is being considered for the delay differential equation y''(t) + ay(t) + by(t-1) = 0 with a and b real numbers. It is already known that y=0 is stable when b=0 and a>0 and unstable when b=0 and a<0. The aim of this project is to determine the stability of y=0 for all values of a and b. First, the general stability theory for delay differential equations was highlighted before giving an in-depth stability analysis of the equation y''(t) + ay(t) + by(t-1) = 0. It turns out that a theorem of Pontryagin (1908 -1988) is really helpful for answering these stability questions. Due to this theorem all values for a and b are determined such that y=0 is asymptotically stable for y''(t) + ay(t) + by(t-1) = 0. However, this does not cover the stability type of y=0 for all values of a and b. So more analysis was done in order to give a full answer of the stability problem. The full answer was not achieved as there are still values for a and b where the stability is unknown. Finally, numerical solutions of y''(t) + ay(t) + by(t-1) = 0 are shown to confirm the results that are obtained.