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Solving an inverse elliptic coefficient problem by convex non-linear semidefinite programming
(2021)
Several applications in medical imaging and non-destructive material testing lead to inverse elliptic coefficient problems, where an unknown coefficient function in an elliptic PDE is to be determined from partial knowledge of its solutions. This is usually a highly non-linear ill-posed inverse problem, for which unique reconstructability results, stability estimates and global convergence of numerical methods are very hard to achieve. The aim of this note is to point out a new connection between inverse coefficient problems and semidefinite programming that may help addressing these challenges. We show that an inverse elliptic Robin transmission problem with finitely many measurements can be equivalently rewritten as a uniquely solvable convex non-linear semidefinite optimization problem. This allows to explicitly estimate the number of measurements that is required to achieve a desired resolution, to derive an error estimate for noisy data, and to overcome the problem of local minima that usually appears in optimization-based approaches for inverse coefficient problems.
Mathematical modelling emphasizes the connection between mathematics and reality — still, tasks are often exclusively introduced inside the classroom. The paper examines the potential of different task settings for mathematical modelling with real objects: outdoors at the real object itself, with photographs and with a 3D model representation. It is the aim of the study to analyze how far the mathematical modelling steps of students solving the tasks differ in comparison to the settings and representations. In a qualitative study, 19 lower secondary school students worked on tasks of all three settings in a Latin square design. Their working processes in the settings are compared with a special focus on the modelling steps Simplifying and Structuring, as well as Mathematizing. The analysis by means of activity diagrams and a qualitative content analysis shows that both steps are particularly relevant when students work with real objects — independent from the three settings. Still, differences in the actual activities could be observed in the students’ discussion on the appropriateness of a model and in dealing with inaccuracies at the real object. In addition, the process of data collection shows different procedures depending on the setting which presents each of them as an enrichment for the acquisition of modelling skills.
In this article, the role of digital feedback that was provided in an outdoor mathematics education setting is taken into consideration. Using the app MathCityMap (2020) in the context of a mathematics trail, the influence of positive and/or negative feedback is examined in relation to how it influences the processes of verification and elaboration. In this context, special emphasis is placed on the students’ verification and elaboration and their relation to reasoning. In this qualitative study, 19 secondary students were filmed while solving mathematics tasks outdoors without digital support, as well as in indoor settings to enable a comparison. The results show that negative feedback in particular leads to a verification of the result. Still, an elaboration and explanation of why a result was incorrect was not often explicitly formulated by the students. Therefore, the potential of feedback is mainly seen in giving students a clear idea about the correctness of the result and searching for an alternative strategy to solve the task when in an outdoor setting.
We show that the Masur–Veech volumes and area Siegel–Veech constants can be obtained using intersection theory on strata of Abelian differentials with prescribed orders of zeros. As applications, we evaluate their large genus limits and compute the saddle connection Siegel–Veech constants for all strata. We also show that the same results hold for the spin and hyperelliptic components of the strata.
We prove that the moduli space of double covers ramified at two points Rg,2 is of general type for g≥16. Furthermore, we consider the Prym-Weierstrass divisor PW¯¯¯¯¯¯¯¯¯g in the universal curve CR¯¯¯¯¯¯¯g over the moduli space of Prym curves R¯¯¯¯g and we compute its class in PicQ(CR¯¯¯¯¯¯¯g). The pushforward to M¯¯¯¯¯¯g,1 of this class was not previously known to be in the effective cone.
We prove that the moduli space of double covers ramified at two points Rg,2 is uniruled for 3≤g≤6 and of general type for g≥16. Furthermore, we consider Prym-canonical divisorial strata in the moduli space CnR¯¯¯¯¯¯¯¯¯¯g parametrizing n-pointed Prym curves, and we compute their classes in PicQ(CnR¯¯¯¯¯¯¯¯¯¯g).
We prove that the moduli space of double covers ramified at two points Rg,2 is of general type for g≥16. Furthermore, we consider the Prym-Weierstrass divisor PW¯¯¯¯¯¯¯¯¯g in the universal curve CR¯¯¯¯¯¯¯g over the moduli space of Prym curves R¯¯¯¯g and we compute its class in PicQ(CR¯¯¯¯¯¯¯g). The pushforward to M¯¯¯¯¯¯g,1 of this class was not previously known to be in the effective cone.
We show that in the theory of Daniell integration iterated integrals may always be formed, and the order of integration may always be interchanged. By this means, we discuss product integrals and show that the related Fubini theorem holds in full generality. The results build on a density theorem on Riesz tensor products due to Fremlin, and on the Fubini–Stone theorem.
We show that in the theory of Daniell integration iterated integrals may always be formed, and the order of integration may always be interchanged. By this means, we discuss product integrals and show that the related Fubini theorem holds in full generality. The results build on a density theorem on Riesz tensor products due to Fremlin, and on the Fubini-Stone Theorem.