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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.
We study the asymptotics of Dirichlet eigenvalues and eigenfunctions of the fractional Laplacian (−Δ)s in bounded open Lipschitz sets in the small order limit s→0+. While it is easy to see that all eigenvalues converge to 1 as s→0+, we show that the first order correction in these asymptotics is given by the eigenvalues of the logarithmic Laplacian operator, i.e., the singular integral operator with Fourier symbol 2log|ξ|. By this we generalize a result of Chen and the third author which was restricted to the principal eigenvalue. Moreover, we show that L2-normalized Dirichlet eigenfunctions of (−Δ)s corresponding to the k-th eigenvalue are uniformly bounded and converge to the set of L2-normalized eigenfunctions of the logarithmic Laplacian. In order to derive these spectral asymptotics, we establish new uniform regularity and boundary decay estimates for Dirichlet eigenfunctions for the fractional Laplacian. As a byproduct, we also obtain corresponding regularity properties of eigenfunctions of the logarithmic Laplacian.
In the model of randomly perturbed graphs we consider the union of a deterministic graph G with minimum degree αn and the binomial random graph G(n, p). This model was introduced by Bohman, Frieze, and Martin and for Hamilton cycles their result bridges the gap between Dirac’s theorem and the results by Pósa and Korshunov on the threshold in G(n, p). In this note we extend this result in G ∪G(n, p) to sparser graphs with α = o(1). More precisely, for any ε > 0 and α: N ↦→ (0, 1) we show that a.a.s. G ∪ G(n, β/n) is Hamiltonian, where β = −(6 + ε) log(α). If α > 0 is a fixed constant this gives the aforementioned result by Bohman, Frieze, and Martin and if α = O(1/n) the random part G(n, p) is sufficient for a Hamilton cycle. We also discuss embeddings of bounded degree trees and other spanning structures in this model, which lead to interesting questions on almost spanning embeddings into G(n, p).
For a class of Cannings models we prove Haldane’s formula, π(sN)∼2sNρ2, for the fixation probability of a single beneficial mutant in the limit of large population size N and in the regime of moderately strong selection, i.e. for sN∼N−b and 0<b<1/2. Here, sN is the selective advantage of an individual carrying the beneficial type, and ρ2 is the (asymptotic) offspring variance. Our assumptions on the reproduction mechanism allow for a coupling of the beneficial allele’s frequency process with slightly supercritical Galton–Watson processes in the early phase of fixation.
The problem of unconstrained or constrained optimization occurs in many branches of mathematics and various fields of application. It is, however, an NP-hard problem in general. In this thesis, we examine an approximation approach based on the class of SAGE exponentials, which are nonnegative exponential sums. We examine this SAGE-cone, its geometry, and generalizations. The thesis consists of three main parts:
1. In the first part, we focus purely on the cone of sums of globally nonnegative exponential sums with at most one negative term, the SAGE-cone. We ex- amine the duality theory, extreme rays of the cone, and provide two efficient optimization approaches over the SAGE-cone and its dual.
2. In the second part, we introduce and study the so-called S-cone, which pro- vides a uniform framework for SAGE exponentials and SONC polynomials. In particular, we focus on second-order representations of the S-cone and its dual using extremality results from the first part.
3. In the third and last part of this thesis, we turn towards examining the con- ditional SAGE-cone. We develop a notion of sublinear circuits leading to new duality results and a partial characterization of extremality. In the case of poly- hedral constraint sets, this examination is simplified and allows us to classify sublinear circuits and extremality for some cases completely. For constraint sets with certain conditions such as sets with symmetries, conic, or polyhedral sets, various optimization and representation results from the unconstrained setting can be applied to the constrained case.
The aim of this bachelor thesis is to compare and empirically test the use of classification to improve the topic models Latent Dirichlet Allocation (LDA) and Author Topic Modeling
(ATM) in the context of the social media platform Twitter. For this purpose, a corpus was classified with the Dewey Decimal Classification (DDC) and then used to train the topic models. A second dataset, the unclassified corpus, was used for comparison. The assumption that the use of classification could improve the topic models did not prove true for the LDA topic model. Here, a sufficiently good improvement of the models could not be achieved. The ATM model, on the other hand, could be improved by using the classification. In general, the ATM model performed significantly better than the LDA model. In the context of the social media platform Twitter, it can thus be seen that the ATM model is superior to the LDA model and can additionally be improved by classifying the data.
We provide extensions of the dual variational method for the nonlinear Helmholtz equation from Evéquoz and Weth. In particular we prove the existence of dual ground state solutions in the Sobolev critical case, extend the dual method beyond the standard Stein Tomas and Kenig Ruiz Sogge range and generalize the method for sign changing nonlinearities.
We study continuous dually epi-translation invariant valuations on certain cones of convex functions containing the space of finite-valued convex functions. Using the homogeneous decomposition of this space, we associate a certain distribution to any homogeneous valuation similar to the Goodey-Weil embedding for translation invariant valuations on convex bodies. The support of these distributions induces a corresponding notion of support for the underlying valuations, which imposes certain restrictions on these functionals, and we study the relation between the support of a valuation and its domain. This gives a partial answer to the question which dually epi-translation invariant valuations on finite-valued convex functions can be extended to larger cones of convex functions.
We also study topological properties of spaces of valuations with support contained in a fixed compact set. As an application of these results, we introduce the class of smooth valuations on convex functions and show that the subspace of smooth dually epi-translation invariant valuations is dense in the space of continuous dually epi-translation invariant valuation on finite-valued convex functions. These smooth valuations are given by integrating certain smooth differential forms over the graph of the differential of a convex function. We use this construction to give a characterization of a dense subspace of all continuous valuations on finite-valued convex functions that are rotation invariant as well as dually epi-translation invariant.
Using results from Alesker's theory of smooth valuations on convex bodies, we also show that any smooth valuation can be written as a convergent sum of mixed Hessian valuations. In particular, mixed Hessian valuations span a dense subspace, which is a version of McMullen’s conjecture for valuations on convex functions.
In this paper we deal with an implementation as well as numerical experiments for the coupling of interior and exterior problems of the elastodynamic wave equation with transparent boundary conditions in 3D as described in a previous paper by this author. In more detail, the FEM‐BEM‐coupling as well as the time discretization by using leapfrog and convolution quadrature is considered. Our aim is to provide an insight into the necessary steps of the implementation. Based on this, we present numerical experiments for a non‐convex domain and analyze the errors.
We contribute to the foundations of tropical geometry with a view toward formulating tropical moduli problems, and with the moduli space of curves as our main example. We propose a moduli functor for the moduli space of curves and show that it is representable by a geometric stack over the category of rational polyhedral cones. In this framework, the natural forgetful morphisms between moduli spaces of curves with marked points function as universal curves.
Our approach to tropical geometry permits tropical moduli problems—moduli of curves or otherwise—to be extended to logarithmic schemes. We use this to construct a smooth tropicalization morphism from the moduli space of algebraic curves to the moduli space of tropical curves, and we show that this morphism commutes with all of the tautological morphisms.
The specific temporal evolution of bacterial and phage population sizes, in particular bacterial depletion and the emergence of a resistant bacterial population, can be seen as a kinetic fingerprint that depends on the manifold interactions of the specific phage–host pair during the course of infection. We have elaborated such a kinetic fingerprint for a human urinary tract Klebsiella pneumoniae isolate and its phage vB_KpnP_Lessing by a modeling approach based on data from in vitro co-culture. We found a faster depletion of the initially sensitive bacterial population than expected from simple mass action kinetics. A possible explanation for the rapid decline of the bacterial population is a synergistic interaction of phages which can be a favorable feature for phage therapies. In addition to this interaction characteristic, analysis of the kinetic fingerprint of this bacteria and phage combination revealed several relevant aspects of their population dynamics: A reduction of the bacterial concentration can be achieved only at high multiplicity of infection whereas bacterial extinction is hardly accomplished. Furthermore the binding affinity of the phage to bacteria is identified as one of the most crucial parameters for the reduction of the bacterial population size. Thus, kinetic fingerprinting can be used to infer phage–host interactions and to explore emergent dynamics which facilitates a rational design of phage therapies.
In vivo functional diversity of midbrain dopamine neurons within identified axonal projections
(2019)
Functional diversity of midbrain dopamine (DA) neurons ranges across multiple scales, from differences in intrinsic properties and connectivity to selective task engagement in behaving animals. Distinct in vitro biophysical features of DA neurons have been associated with different axonal projection targets. However, it is unknown how this translates to different firing patterns of projection-defined DA subpopulations in the intact brain. We combined retrograde tracing with single-unit recording and labelling in mouse brain to create an in vivo functional topography of the midbrain DA system. We identified differences in burst firing among DA neurons projecting to dorsolateral striatum. Bursting also differentiated DA neurons in the medial substantia nigra (SN) projecting either to dorsal or ventral striatum. We found differences in mean firing rates and pause durations among ventral tegmental area (VTA) DA neurons projecting to lateral or medial shell of nucleus accumbens. Our data establishes a high-resolution functional in vivo landscape of midbrain DA neurons.
We study empirically and analytically growth and fluctuation of firm size distribution. An empirical analysis is carried out on a US data set on firm size, with emphasis on one-time distribution as well as growth-rate probability distribution. Both Pareto's law and Gibrat's law are often used to study firm size distribution. Their theoretical relationship is discussed, and it is shown how they are complementable with a bimodal distribution of firm size. We introduce economic mechanisms that suggest a bimodal distribution of firm size in the long run. The mechanisms we study have been known in the economic literature since long. Yet, they have not been studied in the context of a dynamic decision problem of the firm. Allowing for these mechanism thus will give rise to heterogeneity of firms with respect to certain characteristics. We then present different types of tests on US data on firm size which indicate a bimodal distribution of firm size.
Strong convergence rates for numerical approximations of stochastic partial differential equations
(2018)
In this thesis and in the research articles which this thesis consists of, respectively, we focus on strong convergence rates for numerical approximations of stochastic partial differential equations (SPDEs). In Part I of this thesis, i.e., Chapter 2 and Chapter 3, we study higher order numerical schemes for SPDEs with multiplicative trace class noise based on suitable Taylor expansions of the Lipschitz continuous coefficients of the SPDEs under consideration. More precisely, Chapter 2 proves strong convergence rates for a linear implicit Euler-Milstein scheme for SPDEs and is based on an unpublished manuscript written by the author of this thesis. This chapter extends an earlier result1 by slightly lowering the assumptions posed on the diffusion coefficient and a different approximation of the semigroup. In Chapter 3 we introduce an exponential Wagner-Platen type numerical scheme for SPDEs and prove that this numerical approximation method converges in the strong sense with oder up to 3/2−. Moreover, we illustrate how the (mixed) iterated stochastic-deterministic integrals, that are part of our numerical scheme, can be simulated exactly under suitable assumptions.
The second part of this thesis, i.e. Chapter 4 and Chapter 5, is devoted to strong convergence rates for numerical approximations of SPDEs with superlinearly growing nonlinearities driven by additive space-time white noise. More specifically, in Chapter 4, we prove strong convergence with rate in the time variable for a class of nonlinearity-truncated numerical approximation schemes for SPDEs and provide examples that fit into our abstract setting like stochastic Allen-Cahn equations. Finally, in Chapter 5, we extend this result with spatial approximations and establish strong convergence rates for a class of full-discrete nonlinearity truncated numerical approximation schemes for SPDEs. Moreover, we apply our strong convergence result to stochastic Allen-Cahn equations and provide lower and upper bounds which show that our strong convergence result can, in general, not essentially be improved.
To crack the neural code and read out the information neural spikes convey, it is essential to understand how the information is coded and how much of it is available for decoding. To this end, it is indispensable to derive from first principles a minimal set of spike features containing the complete information content of a neuron. Here we present such a complete set of coding features. We show that temporal pairwise spike correlations fully determine the information conveyed by a single spiking neuron with finite temporal memory and stationary spike statistics. We reveal that interspike interval temporal correlations, which are often neglected, can significantly change the total information. Our findings provide a conceptual link between numerous disparate observations and recommend shifting the focus of future studies from addressing firing rates to addressing pairwise spike correlation functions as the primary determinants of neural information.
In 1957, Craig Mooney published a set of human face stimuli to study perceptual closure: the formation of a coherent percept on the basis of minimal visual information. Images of this type, now known as “Mooney faces”, are widely used in cognitive psychology and neuroscience because they offer a means of inducing variable perception with constant visuo-spatial characteristics (they are often not perceived as faces if viewed upside down). Mooney’s original set of 40 stimuli has been employed in several studies. However, it is often necessary to use a much larger stimulus set. We created a new set of over 500 Mooney faces and tested them on a cohort of human observers. We present the results of our tests here, and make the stimuli freely available via the internet. Our test results can be used to select subsets of the stimuli that are most suited for a given experimental purpose.
Motivation: The topic of this paper is the estimation of alignments and mutation rates based on stochastic sequence-evolution models that allow insertions and deletions of subsequences ("fragments") and not just single bases. The model we propose is a variant of a model introduced by Thorne, Kishino, and Felsenstein (1992). The computational tractability of the model depends on certain restrictions in the insertion/deletion process; possible effects we discuss.
Results: The process of fragment insertion and deletion in the sequence-evolution model induces a hidden Markov structure at the level of alignments and thus makes possible efficient statistical alignment algorithms. As an example we apply a sampling procedure to assess the variability in alignment and mutation parameter estimates for HVR1 sequences of human and orangutan, improving results of previous work. Simulation studies give evidence that estimation methods based on the proposed model also give satisfactory results when applied to data for which the restrictions in the insertion/deletion process do not hold.
Availability: The source code of the software for sampling alignments and mutation rates for a pair of DNA sequences according to the fragment insertion and deletion model is freely available from www.math.uni-frankfurt.de/~stoch/software/mcmcsalut under the terms of the GNU public license (GPL, 2000).
Viewing of ambiguous stimuli can lead to bistable perception alternating between the possible percepts. During continuous presentation of ambiguous stimuli, percept changes occur as single events, whereas during intermittent presentation of ambiguous stimuli, percept changes occur at more or less regular intervals either as single events or bursts. Response patterns can be highly variable and have been reported to show systematic differences between patients with schizophrenia and healthy controls. Existing models of bistable perception often use detailed assumptions and large parameter sets which make parameter estimation challenging. Here we propose a parsimonious stochastic model that provides a link between empirical data analysis of the observed response patterns and detailed models of underlying neuronal processes. Firstly, we use a Hidden Markov Model (HMM) for the times between percept changes, which assumes one single state in continuous presentation and a stable and an unstable state in intermittent presentation. The HMM captures the observed differences between patients with schizophrenia and healthy controls, but remains descriptive. Therefore, we secondly propose a hierarchical Brownian model (HBM), which produces similar response patterns but also provides a relation to potential underlying mechanisms. The main idea is that neuronal activity is described as an activity difference between two competing neuronal populations reflected in Brownian motions with drift. This differential activity generates switching between the two conflicting percepts and between stable and unstable states with similar mechanisms on different neuronal levels. With only a small number of parameters, the HBM can be fitted closely to a high variety of response patterns and captures group differences between healthy controls and patients with schizophrenia. At the same time, it provides a link to mechanistic models of bistable perception, linking the group differences to potential underlying mechanisms.
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.
Bipartite graphs occur in many parts of mathematics, and their embeddings into orientable compact surfaces are an old subject. A new interest comes from the fact that these embeddings give dessins d’enfants providing the surface with a unique structure as a Riemann surface and algebraic curve. In this paper, we study the (surprisingly many different) dessins coming from the graphs of finite cyclic projective planes. It turns out that all reasonable questions about these dessins — uniformity, regularity, automorphism groups, cartographic groups, defining equations of the algebraic curves, their fields of definition, Galois actions — depend on cyclic orderings of difference sets for the projective planes. We explain the interplay between number theoretic problems concerning these cyclic ordered difference sets and topological properties of the dessin like e.g. the Wada property that every vertex lies on the border of every cell.
The purpose of the present paper is to explain the fake projective plane constructed by J. H. Keum from the point of view of arithmetic ball quotients. Beside the ball quotient associated with the fake projective plane, we also analize two further naturally related ball quotients whose minimal desingularizations lead to two elliptic surfaces, one already considered by J. H. Keum as well as the one constructed by M. N. Ishida in terms of p-adic uniformization.
2000 Mathematics Subject Classification: 11F23,14J25,14J27
Can variances of latent variables be scaled in such a way that they correspond to eigenvalues?
(2017)
The paper reports an investigation of whether sums of squared factor loadings obtained in confirmatory factor analysis correspond to eigenvalues of exploratory factor analysis. The sum of squared factor loadings reflects the variance of the corresponding latent variable if the variance parameter of the confirmatory factor model is set equal to one. Hence, the computation of the sum implies a specific type of scaling of the variance. While the investigation of the theoretical foundations suggested the expected correspondence between sums of squared factor loadings and eigenvalues, the necessity of procedural specifications in the application, as for example the estimation method, revealed external influences on the outcome. A simulation study was conducted that demonstrated the possibility of exact correspondence if the same estimation method was applied. However, in the majority of realized specifications the estimates showed similar sizes but no correspondence.
In this paper, a translation of the visual description technique HyCharts to Hybrid Data-Flow Graphs (HDFG) is given. While HyCharts combine a data-flow and a control-flow oriented formalism for the specification of the architecture and the behavior of hybrid systems, HDFG allow the efficient and homogeneous internal representation of hybrid systems in computers and their automatic manipulation. HDFG represent a system as a data-flow network built from a set of fundamental functions.
The translation permits to combine the advantages of the different description techniques: The use of HyCharts for specification supports the abstract and formal interactive specification of hybrid systems, while HDFG permit the tool based optimization of hybrid systems and the synthesis of mixed-signal prototypes.
We study exchangeable coalescent trees and the evolving genealogical trees in models for neutral haploid populations.
We show that every exchangeable infinite coalescent tree can be obtained as the genealogical tree of iid samples from a random marked metric measure space when the marks are added to the metric distances. We apply this representation to generalize the tree-valued Fleming-Viot process to include the case with dust in which the genealogical trees have isolated leaves.
Using the Donnelly-Kurtz lookdown approach, we describe all individuals ever alive in the population model by a random complete and separable metric space, the lookdown space, which we endow with a family of sampling measures. This yields a pathwise construction of tree-valued Fleming-Viot processes. In the case of coming down from infinity, we also read off a process whose state space is endowed with the Gromov-Hausdorff-Prohorov topology. This process has additional jumps at the extinction times of parts of the population.
In the case with only binary reproduction events, we construct the lookdown space also from the Aldous continuum random tree by removing the root and the highest leaf, and by deforming the metric in a way that corresponds to the time change that relates the Fleming-Viot process with a Dawson-Watanabe process. The sampling measures on the lookdown space are then image measures of the normalized local time measures.
We also show invariance principles for Markov chains that describe the evolving genealogy in Cannings models. For such Markov chains with values in the space of distance matrix distributions, we show convergence to tree-valued Fleming-Viot processes under the conditions of Möhle and Sagitov for the convergence of the genealogy at a fixed time to a coalescent with simultaneous multiple mergers. For the convergence of Markov chains with values in the space of marked metric measure spaces, an additional assumption is needed in the case with dust.
Random constraint satisfaction problems have been on the agenda of various sciences such as discrete mathematics, computer science, statistical physics and a whole series of additional areas of application since the 1990s at least. The objective is to find a state of a system, for instance an assignment of a set of variables, satisfying a bunch of constraints. To understand the computational hardness as well as the underlying random discrete structures of these problems analytically and to develop efficient algorithms that find optimal solutions has triggered a huge amount of work on random constraint satisfaction problems up to this day. Referring to this context in this thesis we present three results for two random constraint satisfaction problems. ...
Based on a non-rigorous formalism called the “cavity method”, physicists have made intriguing predictions on phase transitions in discrete structures. One of the most remarkable ones is that in problems such as random k-SAT or random graph k-coloring, very shortly before the threshold for the existence of solutions there occurs another phase transition called condensation [Krzakala et al., PNAS 2007]. The existence of this phase transition seems to be intimately related to the difficulty of proving precise results on, e. g., the k-colorability threshold as well as to the performance of message passing algorithms. In random graph k-coloring, there is a precise conjecture as to the location of the condensation phase transition in terms of a distributional fixed point problem. In this paper we prove this conjecture, provided that k exceeds a certain constant k0.
The condensation phase transition and the number of solutions in random graph and hypergraph models
(2016)
This PhD thesis deals with two different types of questions on random graph and random hypergraph structures.
One part is about the proof of the existence and the determination of the location of the condensation phase transition. This transition will be investigated for large values of $k$ in the problem of $k$-colouring random graphs and in the problem of 2-colouring random $k$-uniform hypergraphs, where in the latter case we investigate a more general model with finite inverse temperature.
The other part deals with establishing the limiting distribution of the number of solutions in these structures in density regimes below the condensation threshold.
Algorithms for the Maximum Cardinality Matching Problem which greedily add edges to the solution enjoy great popularity. We systematically study strengths and limitations of such algorithms, in particular of those which consider node degree information to select the next edge. Concentrating on nodes of small degree is a promising approach: it was shown, experimentally and analytically, that very good approximate solutions are obtained for restricted classes of random graphs. Results achieved under these idealized conditions, however, remained unsupported by statements which depend on less optimistic assumptions.
The KarpSipser algorithm and 1-2-Greedy, which is a simplified variant of the well-known MinGreedy algorithm, proceed as follows. In each step, if a node of degree one (resp. at most two) exists, then an edge incident with a minimum degree node is picked, otherwise an arbitrary edge is added to the solution.
We analyze the approximation ratio of both algorithms on graphs of degree at most D. Families of graphs are known for which the expected approximation ratio converges to 1/2 as D grows to infinity, even if randomization against the worst case is used. If randomization is not allowed, then we show the following convergence to 1/2: the 1-2-Greedy algorithm achieves approximation ratio (D-1)/(2D-3); if the graph is bipartite, then the more restricted KarpSipser algorithm achieves the even stronger factor D/(2D-2). These guarantees set both algorithms apart from other famous matching heuristics like e.g. Greedy or MRG: these algorithms depend on randomization to break the 1/2-barrier even for paths with D=2. Moreover, for any D our guarantees are strictly larger than the best known bounds on the expected performance of the randomized variants of Greedy and MRG.
To investigate whether KarpSipser or 1-2-Greedy can be refined to achieve better performance, or be simplified without loss of approximation quality, we systematically study entire classes of deterministic greedy-like algorithms for matching. Therefore we employ the adaptive priority algorithm framework by Borodin, Nielsen, and Rackoff: in each round, an adaptive priority algorithm requests one or more edges by formulating their properties---like e.g. "is incident with a node of minimum degree"---and adds the received edges to the solution. No constraints on time and space usage are imposed, hence an adaptive priority algorithm is restricted only by its nature of picking edges in a greedy-like fashion. If an adaptive priority algorithm requests edges by processing degree information, then we show that it does not surpass the performance of KarpSipser: our D/(2D-2)-guarantee for bipartite graphs is tight and KarpSipser is optimal among all such "degree-sensitive" algorithms even though it uses degree information merely to detect degree-1 nodes. Moreover, we show that if degrees of both nodes of an edge may be processed, like e.g. the Double-MinGreedy algorithm does, then the performance of KarpSipser can only be increased marginally, if at all. Of special interest is the capability of requesting edges not only by specifying the degree of a node but additionally its set of neighbors. This enables an adaptive priority algorithm to "traverse" the input graph. We show that on general degree-bounded graphs no such algorithm can beat factor (D-1)/(2D-3). Hence our bound for 1-2-Greedy is tight and this algorithm performs optimally even though it ignores neighbor information. Furthermore, we show that an adaptive priority algorithm deteriorates to approximation ratio exactly 1/2 if it does not request small degree nodes. This tremendous decline of approximation quality happens for graphs on which 1-2-Greedy and KarpSipser perform optimally, namely paths with D=2. Consequently, requesting small degree nodes is vital to beat factor 1/2.
Summarizing, our results show that 1-2-Greedy and KarpSipser stand out from known and hypothetical algorithms as an intriguing combination of both approximation quality and conceptual simplicity.
Given an Abelian semi-group (A, +), an A-valued curvature measure is a valuation with values in A-valued measures. If A = R, complete classifications of Hausdorff-continuous translation-invariant SO(n)-invariant valuations and curvature measures were obtained by Hadwiger and Schneider, respectively. More recently, characterisation results have been achieved for curvature measures with values in A = Sym^p R^n and A = Sym^2 Λ^q R^n for p, q ≥ 1 with varying assumptions as for their invariance properties.
In the present work, we classify all smooth translation-invariant SO(n)-covariant curvature measures with values in any SO(n)-representation in terms of certain differential forms on the sphere bundle S R^n and describe their behaviour under the globalisation map. The latter result also yields a similar classification of all continuous SO(n)-module-valued SO(n)-covariant valuations. Furthermore, a decomposition of the space of smooth translation-
invariant scalar-valued curvature measures as an SO(n)-module is obtained. As a corollary, we construct explicit bases of continuous translation-invariant scalar-valued valuations and smooth translation-invariant scalar-valued curvature measures.
Die Populationsgenetik beschäftigt sich mit dem Einfluss von zufälliger Reproduktion, Rekombination, Migration, Mutation und Selektion auf die genetische Struktur einer Population.
In dieser Arbeit mit dem englischen Titel "Ancestral lines under mutation and selection" wird das Zusammenspiel von zufälliger Reproduktion, gerichteter Selektion und Zweiwegmutation untersucht.
Dazu betrachten wir eine haploide Population in der jedes Individuum zu jedem Zeitpunkt genau einen von zwei Typen aus S:={0,1} trägt. Dabei sei 1 der neutrale und 0 der selektiv bevorzugte Typ. Im Diffusionslimes sehr großer Populationen modellieren wir den Prozess der Frequenz der Typ-0-Individuen durch eine Wright-Fisher-Diffusion X:=(X_t) mit Mutation und gerichteter Selektion.
Zu jedem Zeitpunkt s gibt es genau ein Individuum, dessen Nachkommen ab einem bestimmten zukünftigen Zeitpunkt t>s die gesamte Population ausmachen werden. Wir nennen dieses Individuum den gemeinsamen Vorfahren zum Zeitpunkt s, da alle Individuen zu allen Zeitpunkten r>t von ihm abstammen. Sei R_{s} dessen Typ zum Zeitpunkt s. Wir nehmen an, dass der Prozess X zum Zeitpunkt 0 im Gleichgewicht ist und definieren die Wahrscheinlichkeit, dass der gemeinsame Vorfahre zum Zeitpunkt 0 Typ 0 hat, durch h(x):= P(R_{0}=0|X_{0}=x). Eine Darstellung von h(x) wurde bereits von Fearnhead (2002) und Taylor (2007) gefunden und dort mit vorwiegend analytischen Methoden bewiesen. In dieser Arbeit entwickeln wir in Kapitel 3 ein neues Teilchenbild, den pruned lookdown ancestral selection graph (pruned LD-ASG), der für sich selbst genommen interessant ist und eine neue probabilistische Interpretation der Darstellung von h(x) liefert.
Durch Erweiterung des Teilchenbildes auf Nachkommenverteilungen mit schweren Tails und mit Hilfe einer Siegmund Dualität gelingt es uns in Kapitel 4 das Resultat für h(x) von klassischen Wright-Fisher-Diffusionen auf Lambda-Wright-Fisher-Diffuison zu erweitern.
Eine Verbindung zwischen Ideen von Taylor (2007), der den gemeinsamen Prozess (X,R) untersucht hat, und einem von Fearnhead (2002) betrachteten Prozess (R,V), der die Entwicklung des Typs R des gemeinsamen Vorfahren in einer Umgebung von V sogenannten virtuellen Linien beschreibt, stellen wir in Kapitel 6 her. Wir bestimmen die gemeinsame Dynamik des Tripels (X,R,V). In Kapitel 7 betrachten wir ein diskretes Bild mit endlicher Populationsgröße N und schlagen dort eine Brücke zu Resultaten von Kluth, Hustedt und Baake (2013).
Des Weiteren entwickeln wir in Kapitel 5 dieser Arbeit einen Algorithmus zur Simulation der Typen einer Stichprobe von m Individuen, die aus einer Wright-Fisher-Population mit Mutation und Selektion im Gleichgewicht gezogen wird. Mittels dieses Algorithmus illustrieren wir die Typenverteilung für verschiedene Parameterwerte und Stichprobengrößen.
European Music Portfolio (EMP) – Maths: 'Sounding ways into mathematics' : teacher’s handbook
(2016)
Music and mathematics share an odd character: many people believe that they are not good at one or the other (or both). However, ‘I cannot sing’ or ‘I never understood mathematics’ will probably not keep them from having successful careers, and nor will it change the opinions others have about them.
The project ‘European Music Portfolio – Sounding Ways into Mathematics’ (EMP-Maths) aims towards a different understanding with regards to this character. Everyone can sing and make music, and everyone can do mathematics. Both topics are integral parts of our life and society. What needs to be improved is our ability to give students opportunities to like them.
This teacher’s handbook presents activities with different mathematical and musical content in order to offer teachers resources, ideas and examples. These activities are designed to be expandable, adaptable to different contexts, and adjustable to the needs of each teacher and their students. Furthermore, these activities are not just planned to be carried out individually; a teaching unit could be used to make sense of them, or they could even be developed in connection with each other.
Apart from this teacher’s handbook, the project provides a continuing professional development (CPD) course, a webpage (http://maths.emportfolio.eu) from which all materials can be downloaded, and an online collaboration platform. A general overview of related literature and research is available in separate documents. Additional teacher booklets provide related materials and a brief overview of the theoretical background, and are the basis for the CPD courses. The project ‘Sounding Ways into Mathematics’ is related to the EMP-Languages project ‘A Creative Way into Languages’ (http://emportfolio.eu/emp/).