@phdthesis{Biersack2024,
  author    = {Biersack, Florian},
  title     = {Topological Properties of Quasiconformal Automorphism Groups},
  doi       = {10.25972/OPUS-35917},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-359177},
  school      = {Universit{\"a}t W{\"u}rzburg},
  year      = {2024},
  abstract  = {The goal of this thesis is to study the topological and algebraic properties of the quasiconformal automorphism groups of simply and multiply connected domains in the complex plain, in which the quasiconformal automorphism groups are endowed with the supremum metric on the underlying domain. More precisely, questions concerning central topological properties such as (local) compactness, (path)-connectedness and separability and their dependence on the boundary of the corresponding domains are studied, as well as completeness with respect to the supremum metric. Moreover, special subsets of the quasiconformal automorphism group of the unit disk are investigated, and concrete quasiconformal automorphisms are constructed. Finally, a possible application of quasiconformal unit disk automorphisms to symmetric cryptography is presented, in which a quasiconformal cryptosystem is defined and studied.},
  subject      = {Quasikonforme Abbildung},
  language  = {en}
}
@phdthesis{Scherz2024,
  author    = {Scherz, Jan},
  title     = {Weak Solutions to Mathematical Models of the Interaction between Fluids, Solids and Electromagnetic Fields},
  doi       = {10.25972/OPUS-34920},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-349205},
  school      = {Universit{\"a}t W{\"u}rzburg},
  year      = {2024},
  abstract  = {We analyze the mathematical models of two classes of physical phenomena. The first class of phenomena we consider is the interaction between one or more insulating rigid bodies and an electrically conducting fluid, inside of which the bodies are contained, as well as the electromagnetic fields trespassing both of the materials. We take into account both the cases of incompressible and compressible fluids. In both cases our main result yields the existence of weak solutions to the associated system of partial differential equations, respectively. The proofs of these results are built upon hybrid discrete-continuous approximation schemes: Parts of the systems are discretized with respect to time in order to deal with the solution-dependent test functions in the induction equation. The remaining parts are treated as continuous equations on the small intervals between consecutive discrete time points, allowing us to employ techniques which do not transfer to the discretized setting. Moreover, the solution-dependent test functions in the momentum equation are handled via the use of classical penalization methods. The second class of phenomena we consider is the evolution of a magnetoelastic material. Here too, our main result proves the existence of weak solutions to the corresponding system of partial differential equations. Its proof is based on De Giorgi's minimizing movements method, in which the system is discretized in time and, at each discrete time point, a minimization problem is solved, the associated Euler-Lagrange equations of which constitute a suitable approximation of the original equation of motion and magnetic force balance. The construction of such a minimization problem is made possible by the realization that, already on the continuous level, both of these equations can be written in terms of the same energy and dissipation potentials. The functional for the discrete minimization problem can then be constructed on the basis of these potentials.},
  subject      = {Fluid-Struktur-Wechselwirkung},
  language  = {en}
}
@phdthesis{Jia2023,
  author    = {Jia, Xiaoxi},
  title     = {Augmented Lagrangian Methods invoking (Proximal) Gradient-type Methods for (Composite) Structured Optimization Problems},
  doi       = {10.25972/OPUS-32374},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-323745},
  school      = {Universit{\"a}t W{\"u}rzburg},
  year      = {2023},
  abstract  = {This thesis, first, is devoted to the theoretical and numerical investigation of an augmented Lagrangian method for the solution of optimization problems with geometric constraints, subsequently, as well as constrained structured optimization problems featuring a composite objective function and set-membership constraints. It is then concerned to convergence and rate-of-convergence analysis of proximal gradient methods for the composite optimization problems in the presence of the Kurdyka--{\L}ojasiewicz property without global Lipschitz assumption.},
  language  = {en}
}
@phdthesis{Dippell2023,
  author    = {Dippell, Marvin},
  title     = {Constraint Reduction in Algebra, Geometry and Deformation Theory},
  doi       = {10.25972/OPUS-30167},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-301670},
  school      = {Universit{\"a}t W{\"u}rzburg},
  year      = {2023},
  abstract  = {To study coisotropic reduction in the context of deformation quantization we introduce constraint manifolds and constraint algebras as the basic objects encoding the additional information needed to define a reduction. General properties of various categories of constraint objects and their compatiblity with reduction are examined. A constraint Serre-Swan theorem, identifying constraint vector bundles with certain finitely generated projective constraint modules, as well as a constraint symbol calculus are proved. After developing the general deformation theory of constraint algebras, including constraint Hochschild cohomology and constraint differential graded Lie algebras, the second constraint Hochschild cohomology for the constraint algebra of functions on a constraint flat space is computed.},
  subject      = {Differentialgeometrie},
  language  = {en}
}
@phdthesis{Stumpf2022,
  author    = {Stumpf, Pascal},
  title     = {On coverings and reduced residues in combinatorial number theory},
  doi       = {10.25972/OPUS-29350},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-293504},
  school      = {Universit{\"a}t W{\"u}rzburg},
  year      = {2022},
  abstract  = {Our starting point is the Jacobsthal function \(j(m)\), defined for each positive integer \(m\) as the smallest number such that every \(j(m)\) consecutive integers contain at least one integer relatively prime to \(m\). It has turned out that improving on upper bounds for \(j(m)\) would also lead to advances in understanding the distribution of prime numbers among arithmetic progressions. If \(P_r\) denotes the product of the first \(r\) prime numbers, then a conjecture of Montgomery states that \(j(P_r)\) can be bounded from above by \(r (\log r)^2\) up to some constant factor. However, the until now very promising sieve methods seem to have reached a limit here, and the main goal of this work is to develop other combinatorial methods in hope of coming a bit closer to prove the conjecture of Montgomery. Alongside, we solve a problem of Recam{\´a}n about the maximum possible length among arithmetic progressions in the least (positive) reduced residue system modulo \(m\). Lastly, we turn towards three additive representation functions as introduced by Erdős, S{\´a}rk{\"o}zy and S{\´o}s who studied their surprising different monotonicity behavior. By an alternative approach, we answer a question of S{\´a}rk{\"o}zy and demostrate that another conjecture does not hold.},
  subject      = {Kombinatorische Zahlentheorie},
  language  = {en}
}
@phdthesis{Lechner2022,
  author    = {Lechner, Theresa},
  title     = {Proximal Methods for Nonconvex Composite Optimization Problems},
  doi       = {10.25972/OPUS-28907},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-289073},
  school      = {Universit{\"a}t W{\"u}rzburg},
  year      = {2022},
  abstract  = {Optimization problems with composite functions deal with the minimization of the sum of a smooth function and a convex nonsmooth function. In this thesis several numerical methods for solving such problems in finite-dimensional spaces are discussed, which are based on proximity operators. After some basic results from convex and nonsmooth analysis are summarized, a first-order method, the proximal gradient method, is presented and its convergence properties are discussed in detail. Known results from the literature are summarized and supplemented by additional ones. Subsequently, the main part of the thesis is the derivation of two methods which, in addition, make use of second-order information and are based on proximal Newton and proximal quasi-Newton methods, respectively. The difference between the two methods is that the first one uses a classical line search, while the second one uses a regularization parameter instead. Both techniques lead to the advantage that, in contrast to many similar methods, in the respective detailed convergence analysis global convergence to stationary points can be proved without any restricting precondition. Furthermore, comprehensive results show the local convergence properties as well as convergence rates of these algorithms, which are based on rather weak assumptions. Also a method for the solution of the arising proximal subproblems is investigated. In addition, the thesis contains an extensive collection of application examples and a detailed discussion of the related numerical results.},
  subject      = {Optimierung},
  language  = {en}
}
@phdthesis{Kortum2022,
  author    = {Kortum, Joshua},
  title     = {Global Existence and Uniqueness Results for Nematic Liquid Crystal and Magnetoviscoelastic Flows},
  doi       = {10.25972/OPUS-27827},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-278271},
  school      = {Universit{\"a}t W{\"u}rzburg},
  year      = {2022},
  abstract  = {Liquid crystals and polymeric fluids are found in many technical applications with liquid crystal displays probably being the most prominent one. Ferromagnetic materials are well established in industrial and everyday use, e.g. as magnets in generators, transformers and hard drive disks. Among ferromagnetic materials, we find a subclass which undergoes deformations if an external magnetic field is applied. This effect is exploited in actuators, magnetoelastic sensors, and new fluid materials have been produced which retain their induced magnetization during the flow. A central issue consists of a proper modelling for those materials. Several models exist regarding liquid crystals and liquid crystal flows, but up to now, none of them has provided a full insight into all observed effects. On materials encompassing magnetic, elastic and perhaps even fluid dynamic effects, the mathematical literature seems sparse in terms of models. To some extent, one can unify the modeling of nematic liquid crystals and magnetoviscoelastic materials employing a so-called energetic variational approach. Using the least action principle from theoretical physics, the actual task reduces to finding appropriate energies describing the observed behavior. The procedure leads to systems of evolutionary partial differential equations, which are analyzed in this work. From the mathematical point of view, fundamental questions on existence, uniqueness and stability of solutions remain unsolved. Concerning the Ericksen-Leslie system modelling nematic liquid crystal flows, an approximation to this model is given by the so-called Ginzburg-Landau approximation. Solutions to the latter are intended to approximately represent solutions to the Ericksen-Leslie system. Indeed, we verify this presumption in two spatial dimensions. More precisely, it is shown that weak solutions of the Ginzburg-Landau approximation converge to solutions of the Ericksen-Leslie system in the energy space for all positive times of evolution. In order to do so, theory for the Euler equations invented by DiPerna and Majda on weak compactness and concentration measures is used. The second part of the work deals with a system of partial differential equations modelling magnetoviscoelastic fluids. We provide a well-posedness result in two spatial dimensions for large energies and large times. Along the verification of that conclusion, existing theory on the Ericksen-Leslie system and the harmonic map flow is deployed and suitably extended.},
  subject      = {Magnetoelastizit{\"a}t},
  language  = {en}
}
@phdthesis{Barth2022,
  author    = {Barth, Dominik},
  title     = {Computation of multi-branch-point covers and applications in Galois theory},
  doi       = {10.25972/OPUS-27702},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-277025},
  school      = {Universit{\"a}t W{\"u}rzburg},
  year      = {2022},
  abstract  = {We present a technique for computing multi-branch-point covers with prescribed ramification and demonstrate the applicability of our method in relatively large degrees by computing several families of polynomials with symplectic and linear Galois groups. As a first application, we present polynomials over \(\mathbb{Q}(\alpha,t)\) for the primitive rank-3 groups \(PSp_4(3)\) and \(PSp_4(3).C_2\) of degree 27 and for the 2-transitive group \(PSp_6(2)\) in its actions on 28 and 36 points, respectively. Moreover, the degree-28 polynomial for \(PSp_6(2)\) admits infinitely many totally real specializations. Next, we present the first (to the best of our knowledge) explicit polynomials for the 2-transitive linear groups \(PSL_4(3)\) and \(PGL_4(3)\) of degree 40, and the imprimitive group \(Aut(PGL_4(3))\) of degree 80. Additionally, we negatively answer a question by K{\"o}nig whether there exists a degree-63 rational function with rational coefficients and monodromy group \(PSL_6(2)\) ramified over at least four points. This is achieved due to the explicit computation of the corresponding hyperelliptic genus-3 Hurwitz curve parameterizing this family, followed by a search for rational points on it. As a byproduct of our calculations we obtain the first explicit \(Aut(PSL_6(2))\)-realizations over \(\mathbb{Q}(t)\). At last, we present a technique by Elkies for bounding the transitivity degree of Galois groups. This provides an alternative way to verify the Galois groups from the previous chapters and also yields a proof that the monodromy group of a degree-276 cover computed by Monien is isomorphic to the sporadic 2-transitive Conway group \(Co_3\).},
  subject      = {Galois-Theorie},
  language  = {en}
}
@phdthesis{Schmeller2022,
  author    = {Schmeller, Christof},
  title     = {Uniform distribution of zero ordinates of Epstein zeta-functions},
  doi       = {10.25972/OPUS-25199},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-251999},
  school      = {Universit{\"a}t W{\"u}rzburg},
  year      = {2022},
  abstract  = {The dissertation investigates the wide class of Epstein zeta-functions in terms of uniform distribution modulo one of the ordinates of their nontrivial zeros. Main results are a proof of a Landau type theorem for all Epstein zeta-functions as well as uniform distribution modulo one for the zero ordinates of all Epstein zeta-functions asscoiated with binary quadratic forms.},
  subject      = {Zetafunktion},
  language  = {en}
}
@phdthesis{Bartsch2021,
  author    = {Bartsch, Jan},
  title     = {Theoretical and numerical investigation of optimal control problems governed by kinetic models},
  doi       = {10.25972/OPUS-24906},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-249066},
  school      = {Universit{\"a}t W{\"u}rzburg},
  year      = {2021},
  abstract  = {This thesis is devoted to the numerical and theoretical analysis of ensemble optimal control problems governed by kinetic models. The formulation and study of these problems have been put forward in recent years by R.W. Brockett with the motivation that ensemble control may provide a more general and robust control framework for dynamical systems. Following this formulation, a Liouville (or continuity) equation with an unbounded drift function is considered together with a class of cost functionals that include tracking of ensembles of trajectories of dynamical systems and different control costs. Specifically, \$L^2\$, \$H^1\$ and \$L^1\$ control costs are taken into account which leads to non--smooth optimization problems. For the theoretical investigation of the resulting optimal control problems, a well--posedness theory in weighted Sobolev spaces is presented for Liouville and related transport equations. Specifically, existence and uniqueness results for these equations and energy estimates in suitable norms are provided; in particular norms in weighted Sobolev spaces. Then, non--smooth optimal control problems governed by the Liouville equation are formulated with a control mechanism in the drift function. Further, box--constraints on the control are imposed. The control--to--state map is introduced, that associates to any control the unique solution of the corresponding Liouville equation. Important properties of this map are investigated, specifically, that it is well--defined, continuous and Frechet differentiable. Using the first two properties, the existence of solutions to the optimal control problems is shown. While proving the differentiability, a loss of regularity is encountered, that is natural to hyperbolic equations. This leads to the need of the investigation of the control--to--state map in the topology of weighted Sobolev spaces. Exploiting the Frechet differentiability, it is possible to characterize solutions to the optimal control problem as solutions to an optimality system. This system consists of the Liouville equation, its optimization adjoint in the form of a transport equation, and a gradient inequality. Numerical methodologies for solving Liouville and transport equations are presented that are based on a non--smooth Lagrange optimization framework. For this purpose, approximation and solution schemes for such equations are developed and analyzed. For the approximation of the Liouville model and its optimization adjoint, a combination of a Kurganov--Tadmor method, a Runge--Kutta scheme, and a Strang splitting method are discussed. Stability and second--order accuracy of these resulting schemes are proven in the discrete \$L^1\$ norm. In addition, conservation of mass and positivity preservation are confirmed for the solution method of the Liouville model. As numerical optimization strategy, an adapted Krylow--Newton method is applied. Since the control is considered to be an element of \$H^1\$ and to obey certain box--constraints, a method for calculating a \$H^1\$ projection is presented. Since the optimal control problem is non-smooth, a semi-smooth adaption of Newton's method is taken into account. Results of numerical experiments are presented that successfully validate the proposed deterministic framework. After the discussion of deterministic schemes, the linear space--homogeneous Keilson--Storer master equation is investigated. This equation was originally developed for the modelling of Brownian motion of particles immersed in a fluid and is a representative model of the class of linear Boltzmann equations. The well--posedness of the Keilson--Storer master equation is investigated and energy estimates in different topologies are derived. To solve this equation numerically, Monte Carlo methods are considered. Such methods take advantage of the kinetic formulation of the Liouville equation and directly implement the behaviour of the system of particles under consideration. This includes the probabilistic behaviour of the collisions between particles. Optimal control problems are formulated with an objective that is constituted of certain expected values in velocity space and the \$L^2\$ and \$H^1\$ costs of the control. The problems are governed by the Keilson--Storer master equation and the control mechanism is considered to be within the collision kernel. The objective of the optimal control of this model is to drive an ensemble of particles to acquire a desired mean velocity and to achieve a desired final velocity configuration. Existence of solutions of the optimal control problem is proven and a Keilson--Storer optimality system characterizing the solution of the proposed optimal control problem is obtained. The optimality system is used to construct a gradient--based optimization strategy in the framework of Monte--Carlo methods. This task requires to accommodate the resulting adjoint Keilson--Storer model in a form that is consistent with the kinetic formulation. For this reason, we derive an adjoint Keilson--Storer collision kernel and an additional source term. A similar approach is presented in the case of a linear space--inhomogeneous kinetic model with external forces and with Keilson--Storer collision term. In this framework, a control mechanism in the form of an external space--dependent force is investigated. The purpose of this control is to steer the multi--particle system to follow a desired mean velocity and position and to reach a desired final configuration in phase space. An optimal control problem using the formulation of ensemble controls is stated with an objective that is constituted of expected values in phase space and \$H^1\$ costs of the control. For solving the optimal control problems, a gradient--based computational strategy in the framework of Monte Carlo methods is developed. Part of this is the denoising of the distribution functions calculated by Monte Carlo algorithms using methods of the realm of partial differential equations. A standalone C++ code is presented that implements the developed non--linear conjugated gradient strategy. Results of numerical experiments confirm the ability of the designed probabilistic control framework to operate as desired. An outlook section about optimal control problems governed by non--linear space--inhomogeneous kinetic models completes this thesis.},
  subject      = {Optimale Kontrolle},
  language  = {en}
}