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In this article we provide a stack-theoretic framework to study the universal tropical Jacobian over the moduli space of tropical curves. We develop two approaches to the process of tropicalization of the universal compactified Jacobian over the moduli space of curves -- one from a logarithmic and the other from a non-Archimedean analytic point of view. The central result from both points of view is that the tropicalization of the universal compactified Jacobian is the universal tropical Jacobian and that the tropicalization maps in each of the two contexts are compatible with the tautological morphisms. In a sequel we will use the techniques developed here to provide explicit polyhedral models for the logarithmic Picard variety.
Foundations of geometry
(2020)
In this thesis, the focus is on the actions of primary school children using digital and analogue materials in comparable mathematical situations. To emphasise actions on different materials in the mathematical learning process, a semiotic perspective according to C. S. Peirce (CP 1931-35) on mathematics learning is adopted. This theoretical research perspective highlights the activity itself on diagrams as a mathematical activity and brings actions to the forefront of interest. The actions on comparable digital and analogue diagrams are the basis for the reconstruction of mathematical interpretations of learners in 3rd and 4th grade.
The research questions investigate to what extent possible differences between the reconstructed interpretations of the learners can be attributed to the different materials and what influence the material has on the mathematical relationships that the learners take into account in their actions to manipulate the diagram.
For the reconstruction of the diagram interpretations based on the learners' actions on the material, a semiotic specification of Vogel's (2017) adaptation of Mayring's (2014) context analysis is used. This specification is based on Peirce's triadic theory of signs (Billion, 2023). The reconstructed interpretations of the analogue and digital diagrams are compared in a second step to identify possible differences and similarities.
The results of the qualitative analyses show, among other things, that despite the different actions of the learners on the digital and analogue diagrams, it is possible to reconstruct the same diagram interpretations if the learners establish the same mathematical relationships between the parts of the diagrams in their actions. There are also passages in the analyses where the same diagram interpretations cannot be reconstructed based on the actions on the digital and analogue materials. If the digital material acts as a tool and automatically creates several relationships between the parts of the diagram triggered by an action, then the reconstruction of the learners' diagram interpretations based on the analysis of their actions is partially possible. If the tool automatically establishes relationships, these must then be interpreted by the learners using gestures and phonetic utterances to understand the newly created diagram. Thus, a tool changes how mathematical relationships are expressed, because learners no longer have to interpret the relationships before their actions to manipulate the diagram itself, but afterwards through gestures and phonetic utterances. Regarding diagrammatic reasoning according to Peirce (NEM IV), this means that with analogue material the focus is on the construction and manipulation of diagrams through rule-guided actions, whereas with digital material, which functions as a tool, there is more emphasis on observing the results of the manipulations on the diagram.
At the end of the thesis, a recommendation for teachers on how to design mathematics lessons for primary school children using digital and analogue materials will be derived from the results.
The literature cited in this summary can be found in the references of the presented thesis.
In 1999, Merino and Welsh conjectured that evaluations of the Tutte polynomial of a graph satisfy an inequality. In this short article, we show that the conjecture generalized to matroids holds for the large class of all split matroids by exploiting the structure of their lattice of cyclic flats. This class of matroids strictly contains all paving and copaving matroids.
We present a massively parallel framework for computing tropicalizations of algebraic varieties which can make use of symmetries using the workflow management system GPI-Space and the computer algebra system Singular. We determine the tropical Grassmannian TGr0(3,8). Our implementation works efficiently on up to 840 cores, computing the 14763 orbits of maximal cones under the canonical S8-action in about 20 minutes. Relying on our result, we show that the Gröbner structure of TGr0(3,8) refines the 16-dimensional skeleton of the coarsest fan structure of the Dressian Dr(3,8), except for 23 orbits of special cones, for which we construct explicit obstructions to the realizability of their tropical linear spaces. Moreover, we propose algorithms for identifying maximal-dimensional cones which belong to positive tropicalizations of algebraic varieties. We compute the positive Grassmannian TGr+(3,8) and compare it to the cluster complex of the classical Grassmannian Gr(3,8).
Thought structures of modelling task solutions and their connection to the level of difficulty
(2015)
Although efforts have been made to integrate the concept of mathematical modelling in school, among others PISA and TIMSS revealed weaknesses of not only German students in the field of mathematical modelling. There may be various reasons starting from educational policy via curricular issues to practical instructional concerns. Studies show that mathematical modelling has not been arrived yet in everyday school class (Blum &BorromeoFerri, 2009, p. 47). Thus, the proportion of mathematical modelling in everyday school classes is low (Jordan et al., 2006). When focusing on the teachers’ point of view there are difficulties which may contribute to avoid modelling tasks in class. The development of reasonable modelling tasks, estimating the task space, valuating the task difficulty and assessing the student solutions are difficulties which occur to an increasing degree compared to ordinary mathematics tasks.The project MokiMaS (transl.: modeling competency in math classes of secondary education) aims at providing inter-year modelling tasks, whose task space and level of difficulty is known, together with an evaluation scheme. In particular a theory based method has been developed to determine the level of difficulty of modelling tasks on the basis of thought structures, representing the cognitive load of solution approaches. The current question is whether this method leads to a realistic rating. To go further into that question an evaluation scheme has been developed which is guided by the daily assessment work of teachers, to investigate the relation of task difficulty and student performance.
Mathematical arguments are central components of mathematics and play a role in certain types of modelling of potential mathematical giftedness. However, particular characteristics of arguments are interpreted differently in the context of mathematical giftedness. Some models of giftedness see no connection, whereas other models consider the formulation of complete and plausible arguments as a partial aspect of giftedness. Furthermore, longitudinal changes in argumentation characteristics remain open. This leads to the research focus of this article, which is to identify and describe the changes of argumentation products in potentially mathematically gifted children over a longer period. For this purpose, the argumentation products of children from third to sixth grade are collected throughout a longitudinal study and examined with respect to the use of examples and generalizations. The analysis of all products results in six different types of changes in the characteristics of the argumentation products identified over the survey period and case studies are used to illustrate student use of examples and generalizations of these types. This not only reveals the general importance of the use of examples in arguments. For one type, an increase in generalized arguments can be observed over the survey period. The article will conclude with a discussion of the role of argument characteristics in describing potential mathematical giftedness.
Interactional niche in the development of geometrical and spatial thinking in the familial context
(2016)
In the analysis of mathematics education in early childhood it is necessary to consider the familial context, which has a significant influence on development in early childhood. Many reputable international research studies emphasize that the more children experience mathematical situations in their families, the more different emerging forms of participation occur for the children that enable them to learn mathematics in the early years. In this sense mathematical activities in the familial context are cornerstones of children’s mathematical development, which is also affected by the ethnic, cultural, educational and linguistic features of their families. Germany has a population of approximately 82 million, about 7.2 million of whom are immigrants (Statisches Bundesamt 2009, pp.28-32). Children in immigrant families grow up with multiculturalism and multilingualism, therefore these children are categorized as a risk group in Germany. “Early Steps in Mathematics Learning – Family Study” (erStMaL-FaSt) is the one of the first familial studies in Germany to deal with the impact of familial socialization on mathematics learning. The study enables us to observe children from different ethnic groups with their family members in different mathematical play situations. The family study (erStMaL-FaSt) is empirically performed within the framework of the erStMaL (Early Steps in Mathematics Learning) project, which relates to the investigation of longitudinal mathematical cognitive development in preschool and early primary-school ages from a socio-constructivist perspective. This study uses two selected mathematical domains, Geometry and Measurement, and four play situations within these two mathematical domains.
My PhD study is situated in erStMaL-FaSt. Therefore, in the beginning of this first chapter, I briefly touch upon IDeA Centre and the erStMaL project and then elaborate on erStMaL-FaSt. As parts of my research concepts, I specify two themes of erStMaL-FaSt: family and play. Thereafter I elaborate upon my research interest. The aim of my study is the research and development of theoretical insights in the functioning of familial interactions for the formation of geometrical (spatial) thinking and learning of children of Turkish ethnic background. Therefore, still in Chapter 1, I present some background on the Turkish people who live in Germany and the spatial development of the children.
This study is designed as a longitudinal study and constructed from interactionist and socio-constructivist perspectives. From a socio-constructivist perspective the cognitive development of an individual is constitutively bound to the participation of this individual in a variety of social interactions. In this regard the presence of each family member provides the child with some “learning opportunities” that are embedded in the interactive process of negotiation of meaning about mathematical play. During the interaction of such various mathematical learning situations, there occur different emerging forms of participation and support. For the purpose of analysing the spatial development of a child in interaction processes in play situations with family members, various statuses of participation are constructed and theoretically described in terms of the concept of the “interactional niche in the development of mathematical thinking in the familial context” (NMT-Family) (Acar & Krummheuer, 2011), which is adapted to the special needs of familial interaction processes. The concept of the “interactional niche in the development of mathematical thinking” (NMT) consists of the “learning offerings” provided by a group or society, which are specific to their culture and are categorized as aspects of “allocation”, and of the situationally emerging performance occurring in the process of meaning negotiation, both of which are subsumed under the aspect of the “situation”, and of the individual contribution of the particular child, which constitutes the aspect of “child’s contribution” (Krummheuer 2011a, 2011b, 2012, 2014; Krummheuer & Schütte 2014). Thereby NMT-Family is constructed as a subconcept of NMT, which offers the advantage of closer analyses and comparisons between familial mathematical learning occasions in early childhood and primary school ages.
Within the scope of NMT-Family, a “mathematics learning support system” (MLSS) is an interactional system which may emerge between the child and the family members in the course of the interaction process of concrete situations in play (Krummheuer & Acar Bayraktar, 2011). All these topics are addressed in Chapter 2 as theoretical approaches and in Chapter 3 as the research method of this study. In Chapter 4 the data collection and analysis is clarified in respect of these approaches...
In an earlier paper we proposed a recursive model for epidemics; in the present paper we generalize this model to include the asymptomatic or unrecorded symptomatic people, which we call dark people (dark sector). We call this the SEPARd-model. A delay differential equation version of the model is added; it allows a better comparison to other models. We carry this out by a comparison with the classical SIR model and indicate why we believe that the SEPARd model may work better for Covid-19 than other approaches.
In the second part of the paper we explain how to deal with the data provided by the JHU, in particular we explain how to derive central model parameters from the data. Other parameters, like the size of the dark sector, are less accessible and have to be estimated more roughly, at best by results of representative serological studies which are accessible, however, only for a few countries. We start our country studies with Switzerland where such data are available. Then we apply the model to a collection of other countries, three European ones (Germany, France, Sweden), the three most stricken countries from three other continents (USA, Brazil, India). Finally we show that even the aggregated world data can be well represented by our approach.
At the end of the paper we discuss the use of the model. Perhaps the most striking application is that it allows a quantitative analysis of the influence of the time until people are sent to quarantine or hospital. This suggests that imposing means to shorten this time is a powerful tool to flatten the curves.
We deal with the shape reconstruction of inclusions in elastic bodies. For solving this inverse problem in practice, data fitting functionals are used. Those work better than the rigorous monotonicity methods from Eberle and Harrach (Inverse Probl 37(4):045006, 2021), but have no rigorously proven convergence theory. Therefore we show how the monotonicity methods can be converted into a regularization method for a data-fitting functional without losing the convergence properties of the monotonicity methods. This is a great advantage and a significant improvement over standard regularization techniques. In more detail, we introduce constraints on the minimization problem of the residual based on the monotonicity methods and prove the existence and uniqueness of a minimizer as well as the convergence of the method for noisy data. In addition, we compare numerical reconstructions of inclusions based on the monotonicity-based regularization with a standard approach (one-step linearization with Tikhonov-like regularization), which also shows the robustness of our method regarding noise in practice.
We deal with the reconstruction of inclusions in elastic bodies based on monotonicity methods and construct conditions under which a resolution for a given partition can be achieved. These conditions take into account the background error as well as the measurement noise. As a main result, this shows us that the resolution guarantees depend heavily on the Lamé parameter μ and only marginally on λ.
The Calderón problem with finitely many unknowns is equivalent to convex semidefinite optimization
(2023)
We consider the inverse boundary value problem of determining a coefficient function in an elliptic partial differential equation from knowledge of the associated Neumann-Dirichlet-operator. The unknown coefficient function is assumed to be piecewise constant with respect to a given pixel partition, and upper and lower bounds are assumed to be known a-priori.
We will show that this Calderón problem with finitely many unknowns can be equivalently formulated as a minimization problem for a linear cost functional with a convex non-linear semidefinite constraint. We also prove error estimates for noisy data, and extend the result to the practically relevant case of finitely many measurements, where the coefficient is to be reconstructed from a finite-dimensional Galerkin projection of the Neumann-Dirichlet-operator.
Our result is based on previous works on Loewner monotonicity and convexity of the Neumann-Dirichlet-operator, and the technique of localized potentials. It connects the emerging fields of inverse coefficient problems and semidefinite optimization.
Uniqueness and Lipschitz stability in electrical impedance tomography with finitely many electrodes
(2019)
For the linearized reconstruction problem in electrical impedance tomography with the complete electrode model, Lechleiter and Rieder (2008 Inverse Problems 24 065009) have shown that a piecewise polynomial conductivity on a fixed partition is uniquely determined if enough electrodes are being used. We extend their result to the full non-linear case and show that measurements on a sufficiently high number of electrodes uniquely determine a conductivity in any finite-dimensional subset of piecewise-analytic functions. We also prove Lipschitz stability, and derive analogue results for the continuum model, where finitely many measurements determine a finite-dimensional Galerkin projection of the Neumann-to-Dirichlet operator on a boundary part.
In this short note, we investigate simultaneous recovery inverse problems for semilinear elliptic equations with partial data. The main technique is based on higher order linearization and monotonicity approaches. With these methods at hand, we can determine the diffusion, cavity and coefficients simultaneously by knowing the corresponding localized Dirichlet-Neumann operators.
The purpose of the paper is to initiate the development of the theory of Newton Okounkov bodies of curve classes. Our denition is based on making a fundamental property of NewtonOkounkov bodies hold also in the curve case: the volume of the NewtonOkounkov body of a curve is a volume-type function of the original curve. This construction allows us to conjecture a new relation between NewtonOkounkov bodies, we prove it in certain cases.
Although everyone is familiar with using algorithms on a daily basis, formulating, understanding and analysing them rigorously has been (and will remain) a challenging task for decades. Therefore, one way of making steps towards their understanding is the formulation of models that are portraying reality, but also remain easy to analyse. In this thesis we take a step towards this way by analyzing one particular problem, the so-called group testing problem. R. Dorfman introduced the problem in 1943. We assume a large population and in this population we find a infected group of individuals. Instead of testing everybody individually, we can test group (for instance by mixing blood samples). In this thesis we look for the minimum number of tests needed such that we can say something meaningful about the infection status. Furthermore we assume various versions of this problem to analyze at what point and why this problem is hard, easy or impossible to solve.
We derive a simple criterion that ensures uniqueness, Lipschitz stability and global convergence of Newton’s method for the finite dimensional zero-finding problem of a continuously differentiable, pointwise convex and monotonic function. Our criterion merely requires to evaluate the directional derivative of the forward function at finitely many evaluation points and for finitely many directions. We then demonstrate that this result can be used to prove uniqueness, stability and global convergence for an inverse coefficient problem with finitely many measurements. We consider the problem of determining an unknown inverse Robin transmission coefficient in an elliptic PDE. Using a relation to monotonicity and localized potentials techniques, we show that a piecewise-constant coefficient on an a-priori known partition with a-priori known bounds is uniquely determined by finitely many boundary measurements and that it can be uniquely and stably reconstructed by a globally convergent Newton iteration. We derive a constructive method to identify these boundary measurements, calculate the stability constant and give a numerical example.
Several novel imaging and non-destructive testing technologies are based on reconstructing the spatially dependent coefficient in an elliptic partial differential equation from measurements of its solution(s). In practical applications, the unknown coefficient is often assumed to be piecewise constant on a given pixel partition (corresponding to the desired resolution), and only finitely many measurement can be made. This leads to the problem of inverting a finite-dimensional non-linear forward operator F: D(F)⊆Rn→Rm , where evaluating ℱ requires one or several PDE solutions.
Numerical inversion methods require the implementation of this forward operator and its Jacobian. We show how to efficiently implement both using a standard FEM package and prove convergence of the FEM approximations against their true-solution counterparts. We present simple example codes for Comsol with the Matlab Livelink package, and numerically demonstrate the challenges that arise from non-uniqueness, non-linearity and instability issues. We also discuss monotonicity and convexity properties of the forward operator that arise for symmetric measurement settings.
This text assumes the reader to have a basic knowledge on Finite Element Methods, including the variational formulation of elliptic PDEs, the Lax-Milgram-theorem, and the Céa-Lemma. Section 3 also assumes that the reader is familiar with the concept of Fréchet differentiability.
We show that the metrisability of an oriented projective surface is equivalent to the existence of pseudo-holomorphic curves. A projective structure p and a volume form σ on an oriented surface M equip the total space of a certain disk bundle Z→M with a pair (Jp,Jp,σ) of almost complex structures. A conformal structure on M corresponds to a section of Z→M and p is metrisable by the metric g if and only if [g]:M→Z is a pseudo-holomorphic curve with respect to Jp and Jp,dAg.
In this article we use techniques from tropical and logarithmic geometry to construct a non-Archimedean analogue of Teichmüller space T¯g whose points are pairs consisting of a stable projective curve over a non-Archimedean field and a Teichmüller marking of the topological fundamental group of its Berkovich analytification. This construction is closely related to and inspired by the classical construction of a non-Archimedean Schottky space for Mumford curves by Gerritzen and Herrlich. We argue that the skeleton of non-Archimedean Teichmüller space is precisely the tropical Teichmüller space introduced by Chan–Melo–Viviani as a simplicial completion of Culler–Vogtmann Outer space. As a consequence, Outer space turns out to be a strong deformation retract of the locus of smooth Mumford curves in T¯g.