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n this paper we study invasion probabilities and invasion times of cooperative parasites spreading in spatially structured host populations. The spatial structure of the host population is given by a random geometric graph on [0,1]n, n∈N, with a Poisson(N)-distributed number of vertices and in which vertices are connected over an edge when they have a distance of at most rN∈Θ(Nβ−1n) for some 0<β<1 and N→∞. At a host infection many parasites are generated and parasites move along edges to neighbouring hosts. We assume that parasites have to cooperate to infect hosts, in the sense that at least two parasites need to attack a host simultaneously. We find lower and upper bounds on the invasion probability of the parasites in terms of survival probabilities of branching processes with cooperation. Furthermore, we characterize the asymptotic invasion time.
An important ingredient of the proofs is a comparison with infection dynamics of cooperative parasites in host populations structured according to a complete graph, i.e. in well-mixed host populations. For these infection processes we can show that invasion probabilities are asymptotically equal to survival probabilities of branching processes with cooperation.
Furthermore, we build in the proofs on techniques developed in [BP22], where an analogous invasion process has been studied for host populations structured according to a configuration model.
We substantiate our results with simulations.
Muller's ratchet, in its prototype version, models a haploid, asexual population whose size~N is constant over the generations. Slightly deleterious mutations are acquired along the lineages at a constant rate, and individuals carrying less mutations have a selective advantage. The classical variant considers {\it fitness proportional} selection, but other fitness schemes are conceivable as well. Inspired by the work of Etheridge et al. ([EPW09]) we propose a parameter scaling which fits well to the ``near-critical'' regime that was in the focus of [EPW09] (and in which the mutation-selection ratio diverges logarithmically as N→∞). Using a Moran model, we investigate the``rule of thumb'' given in [EPW09] for the click rate of the ``classical ratchet'' by putting it into the context of new results on the long-time evolution of the size of the best class of the ratchet with (binary) tournament selection, which (other than that of the classical ratchet) follows an autonomous dynamics up to the time of its extinction. In [GSW23] it was discovered that the tournament ratchet has a hierarchy of dual processes which can be constructed on top of an Ancestral Selection graph with a Poisson decoration. For a regime in which the mutation/selection-ratio remains bounded away from 1, this was used in [GSW23] to reveal the asymptotics of the click rates as well as that of the type frequency profile between clicks. We will describe how these ideas can be extended to the near-critical regime in which the mutation-selection ratio of the tournament ratchet converges to 1 as N→∞.
The category of abelian varieties over Fq is shown to be anti-equivalent to a category of Z-lattices that are modules for a non-commutative pro-ring of endomorphisms of a suitably chosen direct system of abelian varieties over Fq. On full subcategories cut out by a finite set w of conjugacy classes of Weil q-numbers, the anti-equivalence is represented by what we call w-locally projective abelian varieties.
We consider ground state solutions u ∈ H2(RN) of biharmonic (fourth-order) nonlinear Schrodinger equations of the form ¨2u + 2au + bu − |u| p−2u = 0 in RN with positive constants a, b > 0 and exponents 2 < p < 2∗, where 2∗ = 2N N−4 if N > 4 and 2∗ = ∞ if N ≤ 4. By exploiting a connection to the adjoint Stein–Tomas inequality on the unit sphere and by using trial functions due to Knapp, we prove a general symmetry breaking result by showing that all ground states u ∈ H2(RN) in dimension N ≥ 2 fail to be radially symmetric for all exponents 2 < p < 2N+2 N−1 in a suitable regime of a, b > 0. As applications of our main result, we also prove symmetry breaking for a minimization problem with constrained L2-mass and for a related problem on the unit ball in RN subject to Dirichlet boundary conditions.
Using limit linear series on chains of curves, we show that closures of certain Brill-Noether loci contain a product of pointed Brill-Noether loci of small codimension. As a result, we obtain new non-containments of Brill-Noether loci, in particular that dimensionally expected non-containments hold for expected maximal Brill-Noether loci. Using these degenerations, we also give a new proof that Brill-Noether loci with expected codimension −ρ≤⌈g/2⌉ have a component of the expected dimension. Additionally, we obtain new non-containments of Brill-Noether loci by considering the locus of the source curves of unramified double covers.
We prove that the projectivized strata of differentials are not contained in pointed Brill-Noether divisors, with only a few exceptions. For a generic element in a stratum of differentials, we show that many of the associated pointed Brill-Noether loci are of expected dimension. We use our results to study the Auel-Haburcak Conjecture: We obtain new non-containments between maximal Brill-Noether loci in Mg. Our results regarding quadratic differentials imply that the quadratic strata in genus 6 are uniruled.
Geometry is part of the core of mathematics. It has been relevant ever since people have interacted with nature and its phenomena. Geometry’s relevance to the teaching and learning of mathematics can be emphasized, too. Nevertheless, a current potential shift in the topics of mathematics education to the detriment of geometry might be emerging. That is, other topics related to mathematics are seeming to grow in importance in comparison to geometry. Despite this, or perhaps because of it, geometry is an important component of current research in mathematics education. In the literature review, we elaborate relevant foci on the basis of current conference proceedings. By means of about 50 journal articles, five main topics are elaborated in more detail: geometric thinking and practices, geometric contents and topics, teacher education in geometry, argumentation and proof in geometry, as well as the use of digital tools for the teaching and learning of geometry. Conclusions and limitations for current and future research on geometry are formulated at the end of the article. In particular, the transfer to the practices of geometric teaching is explored on the basis of the elaborated research findings in order to combine both aspects of the teaching and learning of geometry.
The free energy of TAP-solutions for the SK-model of mean field spin glasses can be expressed as a nonlinear functional of local terms: we exploit this feature in order to contrive abstract REM-like models which we then solve by a classical large deviations treatment. This allows to identify the origin of the physically unsettling quadratic (in the inverse of temperature) correction to the Parisi free energy for the SK-model, and formalizes the true cavity dynamics which acts on TAP-space, i.e. on the space of TAP-solutions. From a non-spin glass point of view, this work is the first in a series of refinements which addresses the stability of hierarchical structures in models of evolving populations.
The free energy of TAP-solutions for the SK-model of mean field spin glasses can be expressed as a nonlinear functional of local terms: we exploit this feature in order to contrive abstract REM-like models which we then solve by a classical large deviations treatment. This allows to identify the origin of the physically unsettling quadratic (in the inverse of temperature) correction to the Parisi free energy for the SK-model, and formalizes the true cavity dynamics which acts on TAP-space, i.e. on the space of TAP-solutions. From a non-spin glass point of view, this work is the first in a series of refinements which addresses the stability of hierarchical structures in models of evolving populations.
During my initial days here in Frankfurt, in October 2020 amidst the pandemic crisis, all my notes revolved around three articles by Bolthausen and Kistler, which now form the starting point of this work.
The ones introduced by Bolthausen and Kistler are abstract mean field spin glass models, reminiscent of Derrida’s Generalized Random Energy Model (GREM), which generalize the GREM while remaining rigorously solvable through large deviations methods and within a classical Boltzmann-Gibbs formalism. This allows to establish, by means of a second moment method, the associated free energy at the thermodynamic limit as an orthodox, infinite-dimensional, Boltzmann-Gibbs variational principle.
Dual Parisi formulas for the limiting free energy associated with these Hamiltonians hold, and are revealed to be the finite-dimensional (”collapsed”) versions of the classical, infinite-dimensional Boltzmann-Gibbs principles.
In the 2nd chapter of this thesis, we uncover the hidden yet essential connection between real-world spin glasses, like the Sherrington-Kirkpatrick (SK) model and the random energy models. The crucial missing element is that of TAP-free energies: integrating it with the framework introduced by Bolthausen and Kistler results in a correction to the Parisi formula for the free energy, which brings it much, much closer to the ”true” Parisi solution for the SK-model. In other words, we can identify the principles that transform the classical Boltzmann-Gibbs maximization into the unorthodox (and puzzling) Parisi minimization.
This arguably stands as the primary achievement of this work.
Komplexität und Zufälligkeit
(1978)
Muller's ratchet, in its prototype version, models a haploid, asexual population whose size~N is constant over the generations. Slightly deleterious mutations are acquired along the lineages at a constant rate, and individuals carrying less mutations have a selective advantage. The classical variant considers {\it fitness proportional} selection, but other fitness schemes are conceivable as well. Inspired by the work of Etheridge et al. ([EPW09]) we propose a parameter scaling which fits well to the ``near-critical'' regime that was in the focus of [EPW09] (and in which the mutation-selection ratio diverges logarithmically as N→∞). Using a Moran model, we investigate the``rule of thumb'' given in [EPW09] for the click rate of the ``classical ratchet'' by putting it into the context of new results on the long-time evolution of the size of the best class of the ratchet with (binary) tournament selection, which (other than that of the classical ratchet) follows an autonomous dynamics up to the time of its extinction. In [GSW23] it was discovered that the tournament ratchet has a hierarchy of dual processes which can be constructed on top of an Ancestral Selection graph with a Poisson decoration. For a regime in which the mutation/selection-ratio remains bounded away from 1, this was used in [GSW23] to reveal the asymptotics of the click rates as well as that of the type frequency profile between clicks. We will describe how these ideas can be extended to the near-critical regime in which the mutation-selection ratio of the tournament ratchet converges to 1 as N→∞.
Motivated by the question of the impact of selective advantage in populations with skewed reproduction mechanims, we study a Moran model with selection. We assume that there are two types of individuals, where the reproductive success of one type is larger than the other. The higher reproductive success may stem from either more frequent reproduction, or from larger numbers of offspring, and is encoded in a measure Λ for each of the two types. Our approach consists of constructing a Λ-asymmetric Moran model in which individuals of the two populations compete, rather than considering a Moran model for each population. Under certain conditions, that we call the "partial order of adaptation", we can couple these measures. This allows us to construct the central object of this paper, the Λ−asymmetric ancestral selection graph, leading to a pathwise duality of the forward in time Λ-asymmetric Moran model with its ancestral process. Interestingly, the construction also provides a connection to the theory of optimal transport. We apply the ancestral selection graph in order to obtain scaling limits of the forward and backward processes, and note that the frequency process converges to the solution of an SDE with discontinous paths. Finally, we derive a Griffiths representation for the generator of the SDE and use it to find a semi-explicit formula for the probability of fixation of the less beneficial of the two types.
Motivated by the question of the impact of selective advantage in populations with skewed reproduction mechanims, we study a Moran model with selection. We assume that there are two types of individuals, where the reproductive success of one type is larger than the other. The higher reproductive success may stem from either more frequent reproduction, or from larger numbers of offspring, and is encoded in a measure Λ for each of the two types. Our approach consists of constructing a Λ-asymmetric Moran model in which individuals of the two populations compete, rather than considering a Moran model for each population. Under certain conditions, that we call the ``partial order of adaptation'', we can couple these measures. This allows us to construct the central object of this paper, the Λ−asymmetric ancestral selection graph, leading to a pathwise duality of the forward in time Λ-asymmetric Moran model with its ancestral process. Interestingly, the construction also provides a connection to the theory of optimal transport. We apply the ancestral selection graph in order to obtain scaling limits of the forward and backward processes, and note that the frequency process converges to the solution of an SDE with discontinous paths. Finally, we derive a Griffiths representation for the generator of the SDE and use it to find a semi-explicit formula for the probability of fixation of the less beneficial of the two types.
Motivated by the question of the impact of selective advantage in populations with skewed reproduction mechanisms, we study a Moran model with selection. We assume that there are two types of individuals, where the reproductive success of one type is larger than the other. The higher reproductive success may stem from either more frequent reproduction, or from larger numbers of offspring, and is encoded in a measure Λ for each of the two types. Λ-reproduction here means that a whole fraction of the population is replaced at a reproductive event. Our approach consists of constructing a Λ-asymmetric Moran model in which individuals of the two populations compete, rather than considering a Moran model for each population. Provided the measure are ordered stochastically, we can couple them. This allows us to construct the central object of this paper, the Λ−asymmetric ancestral selection graph, leading to a pathwise duality of the forward in time Λ-asymmetric Moran model with its ancestral process. We apply the ancestral selection graph in order to obtain scaling limits of the forward and backward processes, and note that the frequency process converges to the solution of an SDE with discontinuous paths. Finally, we derive a Griffiths representation for the generator of the SDE and use it to find a semi-explicit formula for the probability of fixation of the less beneficial of the two types.
Therapy evasion – and subsequent disease progression – is a major challenge in current oncology. An important role in this context seems to be played by various forms of cancer cell dormancy. For example, therapy-induced dormancy, over short timescales, can create serious obstacles to aggressive treatment approaches such as chemotherapy, and long-term dormancy may lead to relapses and metastases even many years after an initially successful treatment. The underlying dormancy-related mechanisms are complex and highly diverse, so that the analysis even of basic patterns of the population-level consequences of dormancy requires abstraction and idealization, as well as the identification of the relevant specific scenarios.
In this paper, we focus on a situation in which individual cancer cells may switch into and out of a dormant state both spontaneously as well as in response to treatment, and over relatively short time-spans. We introduce a mathematical ‘toy model’, based on stochastic agent-based interactions, for the dynamics of cancer cell populations involving individual short-term dormancy, and allow for a range of (multi-drug) therapy protocols. Our analysis shows that in our idealized model, even a small initial population of dormant cells can lead to therapy failure under classical (and in the absence of dormancy successful) single-drug treatments. We further investigate the effectiveness of several multidrug regimes (manipulating dormant cancer cells in specific ways) and provide some basic rules for the design of (multi-)drug treatment protocols depending on the types and parameters of dormancy mechanisms present in the population.
Highlights
• We study dormancy in the ‘rare mutation’ regime of stochastic adaptive dynamics.
• We first derive the polymorphic evolution sequence, based on prior work.
• Our evolutionary branching criterion extends a result by Champagnat and Méléard.
• In a classical model dormancy can favour evolutionary branching.
• Dormancy also affects several more population characteristics.
Abstract
In this paper, we investigate the consequences of dormancy in the ‘rare mutation’ and ‘large population’ regime of stochastic adaptive dynamics. Starting from an individual-based micro-model, we first derive the Polymorphic Evolution Sequence of the population, based on a previous work by Baar and Bovier (2018). After passing to a second ‘small mutations’ limit, we arrive at the Canonical Equation of Adaptive Dynamics, and state a corresponding criterion for evolutionary branching, extending a previous result of Champagnat and Méléard (2011).
The criterion allows a quantitative and qualitative analysis of the effects of dormancy in the well-known model of Dieckmann and Doebeli (1999) for sympatric speciation. In fact, quite an intuitive picture emerges: Dormancy enlarges the parameter range for evolutionary branching, increases the carrying capacity and niche width of the post-branching sub-populations, and, depending on the model parameters, can either increase or decrease the ‘speed of adaptation’ of populations. Finally, dormancy increases diversity by increasing the genetic distance between subpopulations.
For genus g=r(r+1)2+1, we prove that via the forgetful map, the universal Prym-Brill-Noether locus Rrg has a unique irreducible component dominating the moduli space Rg of Prym curves.
For genus g=2i≥4 and the length g−1 partition μ=(4,2,…,2,−2,…,−2) of 0, we compute the first coefficients of the class of D¯¯¯¯(μ) in PicQ(R¯¯¯¯g), where D(μ) is the divisor consisting of pairs [C,η]∈Rg with η≅OC(2x1+x2+⋯+xi−1−xi−⋯−x2i−1) for some points x1,…,x2i−1 on C. We further provide several enumerative results that will be used for this computation.
For genus g=2i≥4 and the length g−1 partition μ=(4,2,…,2,−2,…,−2) of 0, we compute the first coefficients of the class of D¯¯¯¯(μ) in PicQ(R¯¯¯¯g), where D(μ) is the divisor consisting of pairs [C,η]∈Rg with η≅OC(2x1+x2+⋯+xi−1−xi−⋯−x2i−1) for some points x1,…,x2i−1 on C. We further provide several enumerative results that will be used for this computation.
For genus g=2i≥4 and the length g−1 partition μ=(4,2,…,2,−2,…,−2) of 0, we compute the first coefficients of the class of D¯¯¯¯(μ) in PicQ(R¯¯¯¯g), where D(μ) is the divisor consisting of pairs [C,η]∈Rg with η≅OC(2x1+x2+⋯+xi−1−xi−⋯−x2i−1) for some points x1,…,x2i−1 on C. We further provide several enumerative results that will be used for this computation.
Between his arrival in Frankfurt in 1922 and and his proof of his famous finiteness theorem for integral points in 1929, Siegel had no publications. He did, however, write a letter to Mordell in 1926 in which he explained a proof of the finiteness of integral points on hyperelliptic curves. Recognizing the importance of this argument (and Siegel's views on publication), Mordell sent the relevant extract to be published under the pseudonym "X".
The purpose of this note is to explain how to optimize Siegel's 1926 technique to obtain the following bound. Let K be a number field, S a finite set of places of K, and f∈oK,S[t] monic of degree d≥5 with discriminant Δf∈o×K,S. Then: #|{(x,y):x,y∈oK,S,y2=f(x)}|≤2rankJac(Cf)(K)⋅O(1)d3⋅([K:Q]+#|S|).
This improves bounds of Evertse-Silverman and Bombieri-Gubler from 1986 and 2006, respectively.
The main point underlying our improvement is that, informally speaking, we insist on "executing the descents in the presence of only one root (and not three) until the last possible moment".
For genus g=2i≥4 and the length g−1 partition μ=(4,2,…,2,−2,…,−2) of 0, we compute the first coefficients of the class of D¯¯¯¯(μ) in PicQ(R¯¯¯¯g), where D(μ) is the divisor consisting of pairs [C,η]∈Rg with η≅OC(2x1+x2+⋯+xi−1−xi−⋯−x2i−1) for some points x1,…,x2i−1 on C. We further provide several enumerative results that will be used for this computation.
For genus g=2i≥4 and the length g−1 partition μ=(4,2,…,2,−2,…,−2) of 0, we compute the first coefficients of the class of D¯¯¯¯(μ) in PicQ(R¯¯¯¯g), where D(μ) is the divisor consisting of pairs [C,η]∈Rg with η≅OC(2x1+x2+⋯+xi−1−xi−⋯−x2i−1) for some points x1,…,x2i−1 on C. We further provide several enumerative results that will be used for this computation.
We prove that the projectivized strata of differentials are not contained in pointed Brill-Noether divisors, with only a few exceptions. For a generic element in a stratum of differentials, we show that many of the associated pointed Brill-Noether loci are of expected dimension. We use our results to study the Auel-Haburcak Conjecture: We obtain new non-containments between maximal Brill-Noether loci in Mg. Our results regarding quadratic differentials imply that the quadratic strata in genus 6 are uniruled.
Affine Bruhat--Tits buildings are geometric spaces extracting the combinatorics of algebraic groups. The building of PGL parametrizes flags of subspaces/lattices in or, equivalently, norms on a fixed finite-dimensional vector space, up to homothety. It has first been studied by Goldman and Iwahori as a piecewise-linear analogue of symmetric spaces. The space of seminorms compactifies the space of norms and admits a natural surjective restriction map from the Berkovich analytification of projective space that factors the natural tropicalization map. Inspired by Payne's result that the analytification is the limit of all tropicalizations, we show that the space of seminorms is the limit of all tropicalized linear embeddings ι:Pr↪Pn and prove a faithful tropicalization result for compactified linear spaces. The space of seminorms is in fact the tropical linear space associated to the universal realizable valuated matroid.
We use recent results by Bainbridge–Chen–Gendron–Grushevsky–Möller on compactifications of strata of abelian differentials to give a comprehensive solution to the realizability problem for effective tropical canonical divisors in equicharacteristic zero. Given a pair (Γ,D) consisting of a stable tropical curve Γ and a divisor D in the canonical linear system on Γ, we give a purely combinatorial condition to decide whether there is a smooth curve X over a non-Archimedean field whose stable reduction has Γ as its dual tropical curve together with an effective canonical divisor KX that specializes to D.
We show that the non-Archimedean skeleton of the d-th symmetric power of a smooth projective algebraic curve X is naturally isomorphic to the d-th symmetric power of the tropical curve that arises as the non-Archimedean skeleton of X. The retraction to the skeleton is precisely the specialization map for divisors. Moreover, we show that the process of tropicalization naturally commutes with the diagonal morphisms and the Abel-Jacobi map and we exhibit a faithful tropicalization for symmetric powers of curves. Finally, we prove a version of the Bieri-Groves Theorem that allows us, under certain tropical genericity assumptions, to deduce a new tropical Riemann-Roch-Theorem for the tropicalization of linear systems.
Using the notion of a root datum of a reductive group G we propose a tropical analogue of a principal G-bundle on a metric graph. We focus on the case G=GLn, i.e. the case of vector bundles. Here we give a characterization of vector bundles in terms of multidivisors and use this description to prove analogues of the Weil--Riemann--Roch theorem and the Narasimhan--Seshadri correspondence. We proceed by studying the process of tropicalization. In particular, we show that the non-Archimedean skeleton of the moduli space of semistable vector bundles on a Tate curve is isomorphic to a certain component of the moduli space of semistable tropical vector bundles on its dual metric graph.
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.
Die Arbeit befasst sich mit einer Vereinfachung des von Devroye (1999) geprägten Begriffs der random split trees und verallgemeinert diesen im Sinne von Janson (2019) auf unbeschränkten Verzweigungsgrad. Diese Verallgemeinerung deckt auch preferential attachment trees mit linearen Gewichten ab, wofür ein Beweis von Janson (2019) aufbereitet wird. Zusätzlich bleiben die von Devroye (1999) nachgewiesenen Eigenschaften über die Tiefe der hinzugefügten Knoten erhalten.
Aus Sicht der Pädagogischen Psychologie ist Lernen ein Prozess, bei dem es zu überdauernden Änderungen im Verhaltenspotenzial als Folge von Erfahrungen kommt. Aus konstruktivistischer Perspektive lässt sich Lernen am besten als eine individuelle Konstruktion von Wissen infolge des Entdeckens, Transformierens und Interpretierens komplexer Informationen durch den Lernenden selbst beschreiben. Erkennt der Lernende den Sinn und übernimmt, erweitert oder verändert ihn für sich selbst, so ist der Grundstein für nachhaltiges Lernen gelegt.
Lernen ist ein sehr individueller Prozess. Schule muss also individuelles Lernen auch im Klassenverband ermöglichen und der Lehrende muss zum Lerncoach werden, da sonst kein individuelles und eigenaktives Lernen möglich ist. Das Unterrichtskonzept des forschend-entdeckenden Lernens bietet genau diese Möglichkeit. Es erlaubt die Erfüllung der drei Grundbedürfnisse eines Menschen nach Kompetenz, Autonomie und sozialer Eingebundenheit und ermöglicht damit Motivation, Leistung und Wohlbefinden (Ryan & Deci, 2004).
Forschend-entdeckendes Lernen im Mathematikunterricht ist schrittweise geprägt von folgenden Merkmalen:
- eine problemorientierte Organisation
- selbstständiges, eigenaktives und eigenverantwortliches Lernen der Schülerinnen und Schüler
- individuelle Lernwege und Lernprozesse
- Entwicklung eigener Fragestellungen und Vorgehensweisen der Lernenden
- eigenes Aufstellen von Hypothesen und Vermutungen; Überprüfung der Vermutungen; Dokumentation, Interpretation und Präsentation der Ergebnisse
- eine fördernde Atmosphäre, in der die Lernenden nach und nach forschende Arbeitstechniken vermitteln bekommen
- kooperative Lernformen und damit Förderung von Team- und Kommunikationsfähigkeit
- Unterrichtsinhalte mit hohem Realitäts- und Sinnbezug, gesellschaftlicher Relevanz, Möglichkeiten der Interdisziplinarität
- Stetige Angebote der Unterstützung
Das entdeckende Lernen kann als Vorstufe des forschenden Lernens gesehen werden, da hier der wissenschaftliche Fokus noch nicht so stark ausgeprägt ist. Um alle Phasen auf dem Weg zu annähernd wissenschaftlichen forschenden Lernens anzusprechen, verwenden wir den Begriff des forschend-entdeckenden Lernens.
Voraussetzung ist, dass die Lehrkräfte das forschende Lernen als aktiven, produktiven und selbstbestimmten Lernprozess selbst zuvor erlebt haben müssen. Unter anderem können die Lehrkräfte Unterrichtsprozesse danach besser planen und währenddessen unterstützen, da sie selbst forschend-entdeckendem Lernen „ausgesetzt“ waren und vergleichbare Prozesse durchlebt haben.
Hiermit wird deutlich, dass forschendes Lernen nicht bedeuten kann, dass die Schülerinnen und Schüler auf sich gestellt sind. Die gezielte Unterstützung der Lernenden beim Entdecken und Forschen durch die Lehrkraft ist für einen ertragreichen Lernerfolg unverzichtbar und muss Teil der Vorbereitung und des Prozesses sein.
Internationale Studien zeigen, dass forschend-entdeckende Unterrichtsansätze (inquiry-based learning IBL) im Mathematikunterricht bei geeigneter Umsetzung Lernen verbessern, Lernerfolg und Lernleistung steigern und Freude gegenüber Mathematikunterricht erhöhen können. Die Implementierung dieses Unterrichtsansatzes ist trotz der positiven Ergebnisse nicht alltäglich.
Um neue Unterrichtskonzepte in den Schulalltag zu bringen beziehungsweise um bestehende Unterrichtskonzepte neu in den Schulalltag zu bringen bedarf es Fortbildungen zur Professionalisierung von Lehrerinnen und Lehrern.
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.
Eine Woche lang präsentieren Wissenschaftler*innen Ergebnisse aus der mathematikdidaktischen Forschung und Lehr-Lern-Konzepte für mathematisches Lernen von Schüler*innen sowie für das mathematische und mathematikdidaktische Lernen in den verschiedenen Phasen der Lehrer*innenbildung. Der UniReport sprach mit den Organisatorinnen der Tagung – Prof.in Dr. Susanne Schnell, Prof.in Dr. Rose Vogel und Prof.in Dr. Jessica Hoth – über das Programm der Tagung und über die künftige Ausrichtung des Faches Mathematik.
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.
[Nachruf] Wolfgang Schwarz
(2013)
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.
Wir konnten unseren eigenen Weg gehen, jeder von uns hatte am Ende ein anderes Ergebnis und es war keines falsch. Das macht für mich die Qualität beim Lernen aus, dass mir genug Platz für meine Gedanken gegeben wird und ich ernst genommen werde. […] Dieses Gefühl ist bis heute nicht verloren gegangen und der Gedanke, wie es sein könnte, hilft mir, aus mir raus zukommen und andere zu motivieren, das ebenfalls zu tun, um auch um mich herum anregende Gespräche zu führen, die an die während der Akademie geführten heranreichen. (Feedback einer Teilnehmerin der HSAKA-M 2018)
Bildung durch Wissenschaft im Sinne des Forschenden Lernens ist ein zentrales Thema schulischer Bildung und findet beispielsweise im Konzept Kultur.Forscher! eine didaktische, schulische Umsetzung und wird vom Wissenschaftsrat als Leitgedanke ebenfalls für Universitäten mit dem Ziel empfohlen, Studium und Lehre deutlicher an der Forschung auszurichten.
Die Erfahrung, "…dass alles auch ganz anders sein könnte" ist die wohl wichtigste Erfahrung in Bildungsprozessen. Die Entdeckung von Möglichkeiten, Perspektivwechseln und transformatorischen Selbst-Bildungsprozessen ist zentral für eine gelungene kulturelle Bildungssituation. (Birgit Mandel, 2005).
Die Hessischen Schülerakademien zur Förderung besonders engagierter und begabter junger Menschen wurden bewusst als ein Unterfangen des Forschenden Lernens gegründet und fühlen sich diesem Leitgedanken im Kontext kultureller Bildung verpflichtet. Dieser Satz klingt zunächst einmal gut und zeitgemäß. Doch was steckt genau dahinter?
Das Zusammentreffen zu Beginn der Sommerferien von 60 wissbegierigen und experimentierfreudigen Schülerinnen und Schülern mit einem ebensolchen Team aus Hochschullehrenden und Kulturschaffenden, versprach wie immer eine intensive und aufregende Zeit zu werden. Diese positive Erwartung wurde auch voll erfüllt und gipfelte am Gästenachmittag mit Eltern, Verwandten, Freunden und interessierten Besuchern in einen feierlich-fröhlichen Abschluss mit spannenden und auch überraschenden Werkschauen der Kurse. Ein besonderes Highlight war die großformatige Gestaltung eines Modells der BURG FÜRSTENECK als interdisziplinäres Ergebnis des Hauptkurses Mathematik und des Wahlkurses Modellbau.
Als wir im Herbst 2015 auf den Homepages von BURG FÜRSTENECK und der Schülerakademie unsere Ausschreibung für die Akademie 2016 veröffentlichten, ahnten wir noch nicht, dass wir uns weitere Werbung mit dem jährlichen Flyer, den wir zum Jahreswechsel an die hessischen Gymnasien und Gesamtschulen mit gymnasialen Zweig versenden, hätten (fast) sparen können. Zu unserer Überraschung und großer Freude zählten wir bereits im Februar 2016 "58" Anmeldungen von Schülerinnen und Schülern. Die Werbung hat uns im Anschluss über 20 weitere Bewerbungen beschert und in die unangenehme Situation gebracht, (zu) vielen Schülerinnen und Schülern absagen bzw. sie auf das nächste Jahr vertrösten zu müssen.
Nur eine Institution, die sich verändern kann, kann auch bestehen – das gilt mit Sicherheit im besonderen Maße für Bildungseinrichtungen. Veränderungen können jedoch in unterschiedlichem Gewand daherkommen. Manche geschehen unerwartet und verursachen dadurch vielleicht Probleme, andere hingegen bahnen sich so langsam an, dass ihre Effekte geradezu überraschend wirken können. Die in ihrer Geschwindigkeit unerwartete Einführung des Praxissemesters in der ersten, universitären Phase der Lehrerausbildung in Hessen ist eine solche problematische Veränderung für die Hessische Schülerakademie (Oberstufe), weil sie deren bisher gültige Integration in die schulpraktischen Studienanteile der studentischen BetreuerInnen nicht mehr vorsieht – ein Umstand, der Akademieleitung und Kuratorium ebenso wie unsere Kooperationspartner an der Universität und im Kultusministerium jetzt schon seit über zwei Jahren intensiv beschäftigt.
Wer gern mitzählt, wird vielleicht festgestellt haben, dass im Sommer 2017 die zwanzigste Hessische Schülerakademie stattfand – dreizehn Oberstufenakademien waren es seit 2004, sieben für die Mittelstufe kamen seit 2011 hinzu. Zwanzig erfolgreiche Akademien bieten nicht nur Anlass zur Freude, sie bilden auch die solide Grundlage für einen selbstbewussten Blick in die Zukunft. Im nächsten Frühjahr lädt daher die Akademie Burg Fürsteneck gemeinsam mit dem Hessischen Kultusministerium zu einem interdisziplinären Symposium ein, bei dem die Hessische Schülerakademie und das Programm KulturSchule im Mittelpunkt stehen: Unter dem Titel "Kulturelle Bildung auf dem Weg" beschäftigen sich vom 2. bis zum 4. März 2018 Fachleute aus Wissenschaft und Praxis auf Burg Fürsteneck mit den "Qualitätsbedingungen in der Kulturellen Bildung am Beispiel der Schülerakademien und der Kulturschulen in Hessen".
Das Akademiejahr 2018 hatte neben den beiden Schülerakademien für die Mittelstufe und die Oberstufe noch einen weiteren Höhepunkt: das Symposium "Kulturelle Bildung auf dem Weg" (vom 2. bis 4. März 2018, ausgerichtet von Burg Fürsteneck gemeinsam mit dem Schulentwicklungsprogramm KulturSchule des Hessischen Kultusministeriums und dem Weiterbildungsmaster Kulturelle Bildung an Schulen der Uni Marburg). Es wurde von unserem Schirmherrn, Kultusminister Prof. Dr. R. Alexander Lorz, eröffnet und hatte unter anderem das Ziel, in der Begegnung von Bildungsexpert*innen und -praktiker*innen eine Fachdebatte über "Qualitätsbedingungen in der Kulturellen Bildung am Beispiel der Schülerakademien und der Kulturschulen in Hessen" anzustoßen.
Kaum ein Name ist so eng mit dem "Projekt HSAKA" verbunden wie der von Wolf Aßmus: Seit der ersten Hessischen Schülerakademie für die Oberstufe im Jahre 2004 ist er als Leiter des Physik-Kurses dabei; die Gründung der Mittelstufenakademie 2011 wurde von ihm tatkräftig unterstützt und gefördert; einen Sitz im Kuratorium hat er ebenso übernommen wie das Amt des Ersten Vorsitzenden des Trägervereins von Burg Fürsteneck – der inzwischen pensionierte Professor für Festkörperphysik verkörpert geradezu die Idee vom "Un-Ruhestand". Wer mag es ihm da verübeln, wenn Wolf beschließt, im nächsten Sommer mal mehr Zeit mit seinen Enkeln zu verbringen, statt auf die Burg zu fahren? Weil es daher 2020 zum ersten Mal eine Oberstufenakademie ohne Wolf und ohne Physik-Kurs geben wird (stattdessen Philosophie und Informatik), haben wir auf der vergangenen Akademie die Gelegenheit genutzt, Wolf für 15 Jahre Schülerakademie zu danken. Genauer gesagt: für 15 Jahre, 16 Fachkurse in Physik (15 auf der Oberstufenakademie und einer bei der Mittelstufe), 15 kursübergreifende Naturkunde-Angebote, für die Betreuung Dutzender Studierender und weit über 200 Schüler*innen, für unzählige gemeinsame Aha-Erlebnisse und humorvolle Geschichten, für unermüdliches Engagement und geduldigen Beistand – und nicht zuletzt für viele, viele Liter Speiseeis. Unsere Dankbarkeit wollen wir hier mit allen Leser*innen dieser Dokumentation teilen.
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).
Die Mathematik ist gleichermaßen eine Kulturwissenschaft mit langer Tradition als auch treibende Kraft hinter vielen modernen Technologien und damit Schlüsseldisziplin des Informationszeitalters. Zum einen zielt die Mathematik darauf ab, abstrakte Strukturen und ihre Zusammenhänge zu verstehen; zum anderen entwickelt sie kraftvolle Methoden, um Frage- und Problemstellungen in zahlreichen Wissenschaftsdisziplinen zu behandeln. Moderne Anwendungen der Mathematik liegen beispielsweise in den Bereichen der Datensicherheit und -kompression, der Verkehrssteuerung, der Bewertung und Optimierung von Finanzinstrumenten oder der medizinischen Operationsplanung.
In dieser Broschüre stellen wir Ihnen das Profil der Frankfurter Mathematik in Forschung und Lehre sowie speziell die Studiengänge
• Bachelor Mathematik
• Master Mathematik
vor. An der Goethe-Universität ist es auch möglich, Mathematik auf Lehramt (L1, L2, L3, L5) zu studieren. ...
Gleichungen mit mehreren Unbekannten zu lösen, üben Schüler schon in der Mittelstufe. Für die einen ist es eine spannende mathematische Knobelei, für die anderen eher Quälerei. Doch den wenigsten ist bewusst, wie viele Leben dadurch jeden Tag gerettet werden. Die moderne medizinische Bildgebung beruht darauf, sehr viele Gleichungen nach sehr vielen Unbekannten aufzulösen.
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.