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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...
Concentration of multivariate random recursive sequences arising in the analysis of algorithms
(2006)
Stochastic analysis of algorithms can be motivated by the analysis of randomized algorithms or by postulating on the sets of inputs of the same length some probability distributions. In both cases implied random quantities are analyzed. Here, the running time is of great concern. Characteristics like expectation, variance, limit law, rates of convergence and tail bounds are studied. For the running time, beside the expectation, upper bounds on the right tail are particularly important, since one wants to know large values of the running time not taking place with possibly high probability. In the first chapter game trees are analyzed. The worst case runnig time of Snir's randomized algorithm is specified and its expectation, asymptotic behavior of the variance, a limit law with uniquely characterized limit and tail bounds are identified. Furthermore, a limit law for the value of the game tree under Pearl's probabilistic modell is proved. In the second chapter upper and lower bounds for the Wiener Index of random binary search trees are identified. In the third chapter tail bounds for the generation size of multitype Galton-Watson processes (with immigration) are derived, depending on their offspring distribution. Therefore, the method used to prove the tail bounds in the first chapter is generalized.
Random ordinary differential equations (RODEs) are ordinary differential equations (ODEs) which have a stochastic process in their vector field functions. RODEs have been used in a wide range of applications such as biology, medicine, population dynamics and engineering and play an important role in the theory of random dynamical systems, however, they have been long overshadowed by stochastic differential equations.
Typically, the driving stochastic process has at most Hoelder continuous sample paths and the resulting vector field is, thus, at most Hoelder continuous in time, no matter how smooth the vector function is in its original variables, so the sample paths of the solution are certainly continuously differentiable, but their derivatives are at most Hoelder continuous in time. Consequently, although the classical numerical schemes for ODEs can be applied pathwise to RODEs, they do not achieve their traditional orders.
Recently, Gruene and Kloeden derived the explicit averaged Euler scheme by taking the average of the noise within the vector field. In addition, new forms of higher order Taylor-like schemes for RODEs are derived systematically by Jentzen and Kloeden.
However, it is still important to build higher order numerical schemes and computationally less expensive schemes as well as numerically stable schemes and this is the motivation of this thesis. The schemes by Gruene and Kloeden and Jentzen and Kloeden are very general, so RODEs with special structure, i.e., RODEs with Ito noise and RODEs with affine structure, are focused and numerical schemes which exploit these special structures are investigated.
The developed numerical schemes are applied to several mathematical models in biology and medicine. In order to see the performance of the numerical schemes, trajectories of solutions are illustrated. In addition, the error vs. step sizes as well as the computational costs are compared among newly developed schemes and the schemes in literature.
In der vorliegenden Arbeit untersuchen wir die Verteilung der Nullstellen Dirichletscher L-Reihen auf oder in der Nähe der kritischen Geraden. Diese Funktionen und ihre Nullstellen stehen im Mittelpunkt des Interesses bei einer Vielzahl klassischer zahlentheoretischer Fragestellungen; beispielsweise besagt die Verallgemeinerte Riemannsche Vermutung, daß sämtliche Nullstellen dieser Funktionen auf der kritischen Geraden liegen. Unsere Ergebnisse gehen unter anderem über die besten bislang bekannten Abschätzungen - für den Anteil der Nullstellen der Dirichletschen L-Reihen, die auf der kritischen Geraden liegen, - für den Anteil einfacher beziehungsweise m-facher Nullstellen sowie - über Nullstellen in der Nähe der kritischen Geraden hinaus. Wir setzen hiermit Arbeiten von A. Selberg, N. Levinson, J. B. Conrey und anderen fort und verallgemeinern Ergebnisse, die für die Riemannsche #-Funktion gültig sind, auf alle Dirichletschen LReihen beziehungsweise verbessern bisherige Resultate. Nach einer ausführlicheren Darstellung der Hintergründe zeigen wir einen Satz über Mittelwerte "geglätteter" L-Reihen, d.h. mit einem geeigneten Dirichlet-Polynom multiplizierte L-Reihen. Solche Mittelwertsätze stellen ein wesentliches Hilfsmittel zur Untersuchung der Nullstellenverteilung dar. Die in unserem Hauptsatz gegebene asymptotische Darstellung dieses Mittelwertes können wir dann nutzen, um die genannten Ergebnisse herzuleiten.
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.
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.
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.
A stochastic model for the joint evaluation of burstiness and regularity in oscillatory spike trains
(2013)
The thesis provides a stochastic model to quantify and classify neuronal firing patterns of oscillatory spike trains. A spike train is a finite sequence of time points at which a neuron has an electric discharge (spike) which is recorded over a finite time interval. In this work, these spike times are analyzed regarding special firing patterns like the presence or absence of oscillatory activity and clusters (so called bursts). These bursts do not have a clear and unique definition in the literature. They are often fired in response to behaviorally relevant stimuli, e.g., an unexpected reward or a novel stimulus, but may also appear spontaneously. Oscillatory activity has been found to be related to complex information processing such as feature binding or figure ground segregation in the visual cortex. Thus, in the context of neurophysiology, it is important to quantify and classify these firing patterns and their change under certain experimental conditions like pharmacological treatment or genetical manipulation. In neuroscientific practice, the classification is often done by visual inspection criteria without giving reproducible results. Furthermore, descriptive methods are used for the quantification of spike trains without relating the extracted measures to properties of the underlying processes.
For that reason, a doubly stochastic point process model is proposed and termed 'Gaussian Locking to a free Oscillator' - GLO. The model has been developed on the basis of empirical observations in dopaminergic neurons and in cooperation with neurophysiologists. The GLO model uses as a first stage an unobservable oscillatory background rhythm which is represented by a stationary random walk whose increments are normally distributed. Two different model types are used to describe single spike firing or clusters of spikes. For both model types, the distribution of the random number of spikes per beat has different probability distributions (Bernoulli in the single spike case or Poisson in the cluster case). In the second stage, the random spike times are placed around their birth beat according to a normal distribution. These spike times represent the observed point process which has five easily interpretable parameters to describe the regularity and the burstiness of the firing patterns.
It turns out that the point process is stationary, simple and ergodic. It can be characterized as a cluster process and for the bursty firing mode as a Cox process. Furthermore, the distribution of the waiting times between spikes can be derived for some parameter combination. The conditional intensity function of the point process is derived which is also called autocorrelation function (ACF) in the neuroscience literature. This function arises by conditioning on a spike at time zero and measures the intensity of spikes x time units later. The autocorrelation histogram (ACH) is an estimate for the ACF. The parameters of the GLO are estimated by fitting the ACF to the ACH with a nonlinear least squares algorithm. This is a common procedure in neuroscientific practice and has the advantage that the GLO ACF can be computed for all parameter combinations and that its properties are closely related to the burstiness and regularity of the process. The precision of estimation is investigated for different scenarios using Monte-Carlo simulations and bootstrap methods.
The GLO provides the neuroscientist with objective and reproducible classification rules for the firing patterns on the basis of the model ACF. These rules are inspired by visual inspection criteria often used in neuroscientific practice and thus support and complement usual analysis of empirical spike trains. When applied to a sample data set, the model is able to detect significant changes in the regularity and burst behavior of the cells and provides confidence intervals for the parameter estimates.
We consider the long-time behaviour of spatially extended random populations with locally dependent branching. We treat two classes of models: 1) Systems of continuous-time random walks on the d-dimensional grid with state dependent branching rate. While there are k particles at a given site, a branching event occurs there at rate s(k), and one of the particles is replaced by a random number of offspring (according to a fixed distribution with mean 1 and finite variance). 2) Discrete-time systems of branching random walks in random environment. Given a space-time i.i.d. field of random offspring distributions, all particles act independently, the offspring law of a given particle depending on its position and generation. The mean number of children per individual, averaged over the random environment, equals one The long-time behaviour is determined by the interplay of the motion and the branching mechanism: In the case of recurrent symmetrised individual motion, systems of the second type become locally extinct. We prove a comparison theorem for convex functionals of systems of type one which implies that these systems also become locally extinct in this case, provided that the branching rate function grows at least linearly. Furthermore, the analysis of a caricature model leads to the conjecture that local extinction prevails generically in this case. In the case of transient symmetrised individual motion the picture is more complex: Branching random walks with state dependent branching rate converge towards a non-trivial equilibrium, which preserves the initial intensity, whenever the branching rate function grows subquadratically. Systems of type 1) and systems of type 2) with quadratic branching rate function show very similar behaviour. They converge towards a non-trivial equilibrium if a conditional exponential moment of the collision time of two random walks of an order that reflects the variability in the branching mechanism is finite almost surely. The equilibrium population has finite variance of the local particle number if the corresponding unconditional exponential moment is finite. These results are proved by means of genealogical representations of the locally size-biased population. Furthermore, we compute the threshold values for existence of conditional exponential moments of the collision time of two random walks in terms of the entropy of the transition functions, using tools from large deviations theory. Our results prove in particular that - in contrast to the classical case of independent branching - there is a regime of equilibria with variance of the local number of particles.
In dieser Arbeit werden die mathematischen Grundlagen zur Konstruktion der primären Felder der minimalen Modelle der konformen Quantenfeldtheorie beschrieben. Wir untersuchen Verma und Fock-Moduln der Virasoro-Algebra und klassifizieren diese Moduln bezüglich der Struktur der (ko-) singulären Vektoren. Wir definieren die Vertex-Operatoren zwischen gewissen Fock-Moduln (die eine kanonische Hilbertraumstruktur besitzen) und beweisen verschiedene Eigenschaften dieser Operatoren: Unter bestimmten Voraussetzungen sind Vertex-Operatoren dicht definierte, nicht abschließbare Operatoren zwischen den Fock-Moduln. Radialgeordnete Produkte von Vertex-Operatoren existieren auf einem dichten Teilraum. Wir beweisen Kommutatorrelationen zwischen Vertex-Operatoren und den Generatoren der Virasoro-Algebra. Dann definieren wir die integrierten Vertex-Operatoren und zeigen, daß diese Operatoren im wesentlichen wieder die Eigenschaften der nichtintegrierten Vertex-Operatoren haben. Gewisse integrierte Vertex-Operatoren können mit konformen Felder identifiziert werden. Ein unter den Vertex-Operatoren invarianter Unterraum der Fock-Moduln kann mit dem physikalischen Zustandsraum identifiziert werden.