Universitätsbibliothek

We are interested in studying a system coupling the compressible Navier–Stokes equations with an elastic structure located at the boundary of the fluid domain. Initially the fluid domain is rectangular and the beam is located on the upper side of the rectangle. The elastic structure is modeled by an Euler–Bernoulli damped beam equation. We prove the local in time existence of strong solutions for that coupled system.

In this paper we apply an augmented Lagrange method to a class of semilinear ellip-tic optimal control problems with pointwise state constraints. We show strong con-vergence of subsequences of the primal variables to a local solution of the original problem as well as weak convergence of the adjoint states and weak-* convergence of the multipliers associated to the state constraint. Moreover, we show existence of stationary points in arbitrary small neighborhoods of local solutions of the original problem. Additionally, various numerical results are presented.

In this paper we introduce a theoretical framework concerned with fostering functional thinking in Grade 8 students by utilizing digital technologies. This framework is meant to be used to guide the systematic variation of tasks for implementation in the classroom while using digital technologies. Examples of problems and tasks illustrate this process. Additionally, results of an empirical investigation with Grade 8 students, which focusses on the students’ skills with digital technologies, how they utilize these tools when engaging with the developed tasks, and how they influence their functional thinking, are presented. The research aim is to investigate in which way tasks designed according to the theoretical framework could promote functional thinking while using digital technologies in the sense of the operative principle. The results show that the developed framework — Function-Operation-Matrix — is a sound basis for initiating students’ actions in the sense of the operative principle, to foster the development of functional thinking in its three aspects, namely, assignment, co-variation and object, and that digital technologies can support this process in a meaningful way.

In the present thesis we investigate algebraic and arithmetic properties of graph spectra. In particular, we study the algebraic degree of a graph, that is the dimension of the splitting field of the characteristic polynomial of the associated adjacency matrix over the rationals, and examine the question whether there is a relation between the algebraic degree of a graph and its structural properties. This generalizes the yet open question ``Which graphs have integral spectra?'' stated by Harary and Schwenk in 1974. We provide an overview of graph products since they are useful to study graph spectra and, in particular, to construct families of integral graphs. Moreover, we present a relation between the diameter, the maximum vertex degree and the algebraic degree of a graph, and construct a potential family of graphs of maximum algebraic degree. Furthermore, we determine precisely the algebraic degree of circulant graphs and find new criteria for isospectrality of circulant graphs. Moreover, we solve the inverse Galois problem for circulant graphs showing that every finite abelian extension of the rationals is the splitting field of some circulant graph. Those results generalize a theorem of So who characterized all integral circulant graphs. For our proofs we exploit the theory of Schur rings which was already used in order to solve the isomorphism problem for circulant graphs. Besides that, we study spectra of zero-divisor graphs over finite commutative rings. Given a ring \(R\), the zero-divisor graph over \(R\) is defined as the graph with vertex set being the set of non-zero zero-divisors of \(R\) where two vertices \(x,y\) are adjacent if and only if \(xy=0\). We investigate relations between the eigenvalues of a zero-divisor graph, its structural properties and the algebraic properties of the respective ring.

Stellar structure can -- in good approximation -- be described as a hydrostatic state, which which arises due to a balance between gravitational force and pressure gradient. Hydrostatic states are static solutions of the full compressible Euler system with gravitational source term, which can be used to model the stellar interior. In order to carry out simulations of dynamical processes occurring in stars, it is vital for the numerical method to accurately maintain the hydrostatic state over a long time period. In this thesis we present different methods to modify astrophysical finite volume codes in order to make them \emph{well-balanced}, preventing them from introducing significant discretization errors close to hydrostatic states. Our well-balanced modifications are constructed so that they can meet the requirements for methods applied in the astrophysical context: They can well-balance arbitrary hydrostatic states with any equation of state that is applied to model thermodynamical relations and they are simple to implement in existing astrophysical finite volume codes. One of our well-balanced modifications follows given solutions exactly and can be applied on any grid geometry. The other methods we introduce, which do no require any a priori knowledge, balance local high order approximations of arbitrary hydrostatic states on a Cartesian grid. All of our modifications allow for high order accuracy of the method. The improved accuracy close to hydrostatic states is verified in various numerical experiments.