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Buch
Beschreibende Statistik mit Fathom
(2010-11-11)
Das vorliegende Dokument ist in einem Gemeinschaftsprojekt der Universität Kassel und der Elisabeth-Knipping-Schule Kassel entstanden. Im Rahmen der fachbezogenen schulpraktischen Stu-dien für das Fach Mathematik ist eine Unterrichtsreihe zur Beschreibenden Statistik mit Softwareein-satz für die Fachoberschule Klasse 11 entwickelt und umgesetzt worden. Dieses Dokument fasst Ideen, Materialien und didaktische Kommentare der durchgeführten Unterrichtsreihe in aufbereiteter Form zusammen. Viele der konzeptionellen Ansätze ...
Buch
Stochastische Simulation von Zufallsexperimenten mit Fathom
(Franzbecker, Hildesheim, 2010)
In der Mathematikdidaktik gibt es die weit verbreitete Auffassung, durch die Verwendung von Simulationen Lernprozesse zu unterstützen. Dies hat mich im Rahmen meiner Dissertation dazu bewogen, erstens eine Werkzeuganalyse des Simulationspotentials der Software Fathom durchzuführen und zweitens exemplarische Analysen dazu, wie Lernende mit der Software arbeiten.
Bei der Werkzeuganalyse standen vor allem folgende zwei Fragen im Mittelpunkt: Was bietet die Software an Simulationspotential? Wie gut und leicht lassen ...
Preprint
On Solution Sets of Information Inequalities
(2011)
We investigate solution sets of a special kind of linear inequality systems. In particular, we derive characterizations of these sets in terms of minimal solution sets. The studied inequalities emerge as information inequalities in the context of Bayesian networks. This allows to deduce important properties of Bayesian networks, which is important within causal inference.
Dissertation
Automatic computation of continued fraction representations as solutions of explicit differential equations
(2019)
The main focus of this thesis is to present a variation of an algorithm first presented by Maulat and Salvy, with which it is possible to algorithmically guess as well as prove continued fraction expansions of analytical expressions with the help of ordinary differential equations.
Dissertation
Free Resolutions from Involutive Bases
(2016-11-02)
We show that the theory of involutive bases can be combined with discrete algebraic Morse Theory. For a graded k[x0 ...,xn]-module M, this yields a free resolution G, which in general is not minimal. We see that G is isomorphic to the resolution induced by an involutive basis. It is possible to identify involutive bases inside the resolution G. The shape of G is given by a concrete description. Regarding the differential dG, several rules are established for its computation, which are based on the fact that in the ...
Aufsatz
The Impact of Visualizing Nested Sets. An Empirical Study on Tree Diagrams and Unit Squares
(2017-01-06)
It is an ongoing debate, what properties of visualizations increase people’s performance when solving Bayesian reasoning tasks. In the discussion of the properties of two visualizations, i.e., the tree diagram and the unit square, we emphasize how both visualizations make relevant subset relations transparent. Actually, the unit square with natural frequencies reveals the subset relation that is essential for the Bayes’ rule in a numerical and geometrical way whereas the tree diagram with natural frequencies does it ...
Habilitation
Algorithmic Methods for Mixed Recurrence Equations, Zeros of Classical Orthogonal Polynomials and Classical Orthogonal Polynomial Solutions of Three-Term Recurrence Equations
(2019-07)
Using an algorithmic approach, we derive classes of mixed recurrence equations satisfied by classical orthogonal polynomials. Starting from certain structure relations satisfied by classical orthogonal polynomials or their connection formulae, we show that our mixed recurrence equations are structurally valid. However, they couldn't be easily obtained with classical methods and for this reason, our algorithmic approach is important. The main algorithmic tool used here is an extended version of Zeilberger's ...
Dissertation
Explicit Description Of Isogeny And Isomorphism Classes Of Drinfeld Modules Of Higher Rank Over Finite Fields
(2020)
When jumping from the number fields theory to the function fields theory, one cannot miss the deep analogy between rank 1 Drinfeld modules and the group of root of unity and the analogy between rank 2 Drinfeld modules and elliptic curves. But so far, there is no known structure in number fields theory that is analogous to the Drinfeld modules of higher rank r ≥ 3. In this thesis we investigate the classes of those Drinfeld modules of higher rank r ≥ 3 defined over a finite field L. We describe explicitly the Weil ...
Preprint
Characterization theorem for classical orthogonal polynomials on non-uniform lattices: The functional approach
(2010)
Using the functional approach, we state and prove a characterization theorem for classical orthogonal polynomials on non-uniform lattices (quadratic lattices of a discrete or a q-discrete variable) including the Askey-Wilson polynomials. This theorem proves the equivalence between seven characterization properties, namely the Pearson equation for the linear functional, the second-order divided-difference equation, the orthogonality of the derivatives, the Rodrigues formula, two types of structure relations,and the ...
Preprint
On the relationship between the Method of Least Squares and Gram-Schmidt orthogonalization
(2010)
The method of Least Squares is due to Carl Friedrich Gauss. The Gram-Schmidt
orthogonalization method is of much younger date. A method for solving Least Squares Problems is developed which automatically results in the appearance of the Gram-Schmidt orthogonalizers. Given these orthogonalizers an induction-proof is available for solving Least Squares Problems.