@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{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{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} } @article{Weishaeupl2023, author = {Weish{\"a}upl, Sebastian}, title = {The weak Gram law for Hecke \(L\)-functions}, series = {The Ramanujan Journal}, volume = {60}, journal = {The Ramanujan Journal}, number = {4}, issn = {1382-4090}, doi = {10.1007/s11139-022-00638-5}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-324404}, pages = {981-997}, year = {2023}, abstract = {We generalize a theorem by Titchmarsh about the mean value of Hardy's \(Z\)-function at the Gram points to the Hecke \(L\)-functions, which in turn implies the weak Gram law for them. Instead of proceeding analogously to Titchmarsh with an approximate functional equation we employ a different method using contour integration.}, language = {en} } @article{LuMoenius2023, author = {Lu, Lu and M{\"o}nius, Katja}, title = {Algebraic degree of Cayley graphs over abelian groups and dihedral groups}, series = {Journal of Algebraic Combinatorics}, volume = {57}, journal = {Journal of Algebraic Combinatorics}, number = {3}, issn = {0925-9899}, doi = {10.1007/s10801-022-01190-7}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-324380}, pages = {753-761}, year = {2023}, abstract = {For a graph \(\Gamma\) , let K be the smallest field containing all eigenvalues of the adjacency matrix of \(\Gamma\) . The algebraic degree \(\deg (\Gamma )\) is the extension degree \([K:\mathbb {Q}]\). In this paper, we completely determine the algebraic degrees of Cayley graphs over abelian groups and dihedral groups.}, language = {en} } @article{GreefrathOldenburgSilleretal.2023, author = {Greefrath, Gilbert and Oldenburg, Reinhard and Siller, Hans-Stefan and Ulm, Volker and Weigand, Hans-Georg}, title = {Mathematics students' characteristics of basic mental models of the derivative}, series = {Journal f{\"u}r Mathematik-Didaktik}, volume = {44}, journal = {Journal f{\"u}r Mathematik-Didaktik}, number = {1}, issn = {0173-5322}, doi = {10.1007/s13138-022-00207-9}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-324317}, pages = {143-169}, year = {2023}, abstract = {The concept of derivative is characterised with reference to four basic mental models. These are described as theoretical constructs based on theoretical considerations. The four basic mental models—local rate of change, tangent slope, local linearity and amplification factor—are not only quantified empirically but are also validated. To this end, a test instrument for measuring students' characteristics of basic mental models is presented and analysed regarding quality criteria. Mathematics students (n = 266) were tested with this instrument. The test results show that the four basic mental models of the derivative can be reconstructed among the students with different characteristics. The tangent slope has the highest agreement values across all tasks. The agreement on explanations based on the basic mental model of rate of change is not as strongly established among students as one would expect due to framework settings in the school system by means of curricula and educational standards. The basic mental model of local linearity plays a rather subordinate role. The amplification factor achieves the lowest agreement values. In addition, cluster analysis was conducted to identify different subgroups of the student population. Moreover, the test results can be attributed to characteristics of the task types as well as to the students' previous experiences from mathematics classes by means of qualitative interpretation. These and other results of students' basic mental models of the derivative are presented and discussed in detail.}, language = {en} } @article{SillerElschenbroichGreefrathetal.2023, author = {Siller, Hans-Stefan and Elschenbroich, Hans-J{\"u}rgen and Greefrath, Gilbert and Vorh{\"o}lter, Katrin}, title = {Mathematical modelling of exponential growth as a rich learning environment for mathematics classrooms}, series = {ZDM Mathematics Education}, volume = {55}, journal = {ZDM Mathematics Education}, number = {1}, issn = {1863-9690}, doi = {10.1007/s11858-022-01433-8}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-324393}, pages = {17-33}, year = {2023}, abstract = {Mathematical concepts are regularly used in media reports concerning the Covid-19 pandemic. These include growth models, which attempt to explain or predict the effectiveness of interventions and developments, as well as the reproductive factor. Our contribution has the aim of showing that basic mental models about exponential growth are important for understanding media reports of Covid-19. Furthermore, we highlight how the coronavirus pandemic can be used as a context in mathematics classrooms to help students understand that they can and should question media reports on their own, using their mathematical knowledge. Therefore, we first present the role of mathematical modelling in achieving these goals in general. The same relevance applies to the necessary basic mental models of exponential growth. Following this description, based on three topics, namely, investigating the type of growth, questioning given course models, and determining exponential factors at different times, we show how the presented theoretical aspects manifest themselves in teaching examples when students are given the task of reflecting critically on existing media reports. Finally, the value of the three topics regarding the intended goals is discussed and conclusions concerning the possibilities and limits of their use in schools are drawn.}, language = {en} } @article{SteudingTongsomporn2023, author = {Steuding, J{\"o}rn and Tongsomporn, Janyarak}, title = {On the order of growth of Lerch zeta functions}, series = {Mathematics}, volume = {11}, journal = {Mathematics}, number = {3}, issn = {2227-7390}, doi = {10.3390/math11030723}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-303981}, year = {2023}, abstract = {We extend Bourgain's bound for the order of growth of the Riemann zeta function on the critical line to Lerch zeta functions. More precisely, we prove L(λ, α, 1/2 + it) ≪ t\(^{13/84+ϵ}\) as t → ∞. For both, the Riemann zeta function as well as for the more general Lerch zeta function, it is conjectured that the right-hand side can be replaced by t\(^ϵ\) (which is the so-called Lindel{\"o}f hypothesis). The growth of an analytic function is closely related to the distribution of its zeros.}, language = {en} } @article{HeinsRothWaldmann2023, author = {Heins, Michael and Roth, Oliver and Waldmann, Stefan}, title = {Convergent star products on cotangent bundles of Lie groups}, series = {Mathematische Annalen}, volume = {386}, journal = {Mathematische Annalen}, number = {1-2}, issn = {0025-5831}, doi = {10.1007/s00208-022-02384-x}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-324324}, pages = {151-206}, year = {2023}, abstract = {For a connected real Lie group G we consider the canonical standard-ordered star product arising from the canonical global symbol calculus based on the half-commutator connection of G. This star product trivially converges on polynomial functions on T\(^*\)G thanks to its homogeneity. We define a nuclear Fr{\´e}chet algebra of certain analytic functions on T\(^*\)G, for which the standard-ordered star product is shown to be a well-defined continuous multiplication, depending holomorphically on the deformation parameter \(\hbar\). This nuclear Fr{\´e}chet algebra is realized as the completed (projective) tensor product of a nuclear Fr{\´e}chet algebra of entire functions on G with an appropriate nuclear Fr{\´e}chet algebra of functions on \({\mathfrak {g}}^*\). The passage to the Weyl-ordered star product, i.e. the Gutt star product on T\(^*\)G, is shown to preserve this function space, yielding the continuity of the Gutt star product with holomorphic dependence on \(\hbar\).}, language = {en} } @article{JotzMehtaPapantonis2023, author = {Jotz, M. and Mehta, R. A. and Papantonis, T.}, title = {Modules and representations up to homotopy of Lie n-algebroids}, series = {Journal of Homotopy and Related Structures}, volume = {18}, journal = {Journal of Homotopy and Related Structures}, number = {1}, issn = {2193-8407}, doi = {10.1007/s40062-022-00322-x}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-324333}, pages = {23-70}, year = {2023}, abstract = {This paper studies differential graded modules and representations up to homotopy of Lie n-algebroids, for general \(n\in {\mathbb {N}}\). The adjoint and coadjoint modules are described, and the corresponding split versions of the adjoint and coadjoint representations up to homotopy are explained. In particular, the case of Lie 2-algebroids is analysed in detail. The compatibility of a Poisson bracket with the homological vector field of a Lie n-algebroid is shown to be equivalent to a morphism from the coadjoint module to the adjoint module, leading to an alternative characterisation of non-degeneracy of higher Poisson structures. Moreover, the Weil algebra of a Lie n-algebroid is computed explicitly in terms of splittings, and representations up to homotopy of Lie n-algebroids are used to encode decomposed VB-Lie n-algebroid structures on double vector bundles.}, language = {en} }