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Institute
- Mathematik (365) (remove)
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 der Arbeit wird ein Testverfahren zum Prüfen der Varianzhomogenität der Lebenszeiten eines Erneuerungsprozesses entwickelt. Das Verfahren basiert auf der "Filtered-Derivative"-Methode. Zur Herleitung des Annahmebereichs werden zunächst Bootstrap-Permutationen genutzt, bevor zu einer asymptotischen Methode übergangen wird. Ein entsprechender funktionaler Grenzwertsatz wird skizziert. Aufbauend auf dem Test wird ein Multiple-Filter-Algorithmus zur genauen Detektion der Varianz-Change-Points besprochen. Schließlich folgt die Inklusion von vorher detektierten Ratenänderungen in das Verfahren. Der Test und der Algorithmus werden in Simulationsstudien evaluiert. Abschließend erfolgt eine Anwendung auf EEG-Daten.
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.
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.
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".
The relation between the complexity of a time-switched dynamics and the complexity of its control sequence depends critically on the concept of a non-autonomous pullback attractor. For instance, the switched dynamics associated with scalar dissipative affine maps has a pullback attractor consisting of singleton component sets. This entails that the complexity of the control sequence and switched dynamics, as quantified by the topological entropy, coincide. In this paper we extend the previous framework to pullback attractors with nontrivial components sets in order to gain further insights in that relation. This calls, in particular, for distinguishing two distinct contributions to the complexity of the switched dynamics. One proceeds from trajectory segments connecting different component sets of the attractor; the other contribution proceeds from trajectory segments within the component sets. We call them “macroscopic” and “microscopic” complexity, respectively, because only the first one can be measured by our analytical tools. As a result of this picture, we obtain sufficient conditions for a switching system to be more complex than its unswitched subsystems, i.e., a complexity analogue of Parrondo’s paradox.
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.
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.
ranching Processes in Random Environment (BPREs) $(Z_n:n\geq0)$ are the generalization of Galton-Watson processes where \lq in each generation' the reproduction law is picked randomly in an i.i.d. manner. The associated random walk of the environment has increments distributed like the logarithmic mean of the offspring distributions. This random walk plays a key role in the asymptotic behavior. In this paper, we study the upper large deviations of the BPRE $Z$ when the reproduction law may have heavy tails. More precisely, we obtain an expression for the limit of $-\log \mathbb{P}(Z_n\geq \exp(\theta n))/n$ when $n\rightarrow \infty$. It depends on the rate function of the associated random walk of the environment, the logarithmic cost of survival $\gamma:=-\lim_{n\rightarrow\infty} \log \mathbb{P}(Z_n>0)/n$ and the polynomial rate of decay $\beta$ of the tail distribution of $Z_1$. This rate function can be interpreted as the optimal way to reach a given "large" value. We then compute the rate function when the reproduction law does not have heavy tails. Our results generalize the results of B\"oinghoff $\&$ Kersting (2009) and Bansaye $\&$ Berestycki (2008) for upper large deviations. Finally, we derive the upper large deviations for the Galton-Watson processes with heavy tails.
Based on a non-rigorous formalism called the “cavity method”, physicists have made intriguing predictions on phase transitions in discrete structures. One of the most remarkable ones is that in problems such as random k-SAT or random graph k-coloring, very shortly before the threshold for the existence of solutions there occurs another phase transition called condensation [Krzakala et al., PNAS 2007]. The existence of this phase transition seems to be intimately related to the difficulty of proving precise results on, e. g., the k-colorability threshold as well as to the performance of message passing algorithms. In random graph k-coloring, there is a precise conjecture as to the location of the condensation phase transition in terms of a distributional fixed point problem. In this paper we prove this conjecture, provided that k exceeds a certain constant k0.