Institut für Mathematik
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The determination of the atmospheric aerosol size distribution is an inverse illposed problem. The shape and the material composition of the air-carried particles are two substantial model parameters. Present evaluation algorithms only used an approximation with spherical homogeneous particles. In this paper we propose a new numerically efficient recursive algorithm for inhomogeneous multilayered coated and absorbing particles. Numerical results of real existing particles show that the influence of the two parameters on the model is very important and therefore cannot be ignored.
The ill-posed problem of aerosol size distribution determination from a small number of backscatter and extinction measurements was solved successfully with a mollifier method which is advantageous since the ill-posed part is performed on exactly given quantities, the points r where n(r) is evaluated may be freely selected. A new twodimensional model for the troposphere is proposed.
In this thesis we mainly generalize two theorems from Mackaay-Picken and Picken (2002, 2004). In the first paper, Mackaay and Picken show that there is a bijective correspondence between Deligne 2-classes $\xi \in \check{H}^2(M,\mathcal{D}^2)$ and holonomy maps from the second thin-homotopy group $\pi_2^2(M)$ to $U(1)$. In the second one, a generalization of this theorem to manifolds with boundaries is given: Picken shows that there is a bijection between Deligne 2-cocycles and a certain variant of 2-dimensional topological quantum field theories. In this thesis we show that these two theorems hold in every dimension. We consider first the holonomy case, and by using simplicial methods we can prove that the group of smooth Deligne $d$-classes is isomorphic to the group of smooth holonomy maps from the $d^{th}$ thin-homotopy group $\pi_d^d(M)$ to $U(1)$, if $M$ is $(d-1)$-connected. We contrast this with a result of Gajer (1999). Gajer showed that Deligne $d$-classes can be reconstructed by a different class of holonomy maps, which not only include holonomies along spheres, but also along general $d$-manifolds in $M$. This approach does not require the manifold $M$ to be $(d-1)$-connected. We show that in the case of flat Deligne $d$-classes, our result differs from Gajers, if $M$ is not $(d-1)$-connected, but only $(d-2)$-connected. Stiefel manifolds do have this property, and if one applies our theorem to these and compare the result with that of Gajers theorem, it is revealed that our theorem reconstructs too many Deligne classes. This means, that our reconstruction theorem cannot live without the extra assumption on the manifold $M$, that is our reconstruction needs less informations about the holonomy of $d$-manifolds in $M$ at the price of assuming $M$ to be $(d-1)$-connected. We continue to show, that also the second theorem can be generalized: By introducing the concept of Picken-type topological quantum field theory in arbitrary dimensions, we can show that every Deligne $d$-cocycle induces such a $d$-dimensional field theory with two special properties, namely thin-invariance and smoothness. We show that any $d$-dimensional topological quantum field theory with these two properties gives rise to a Deligne $d$-cocycle and verify that this construction is surjective and injective, that is both groups are isomorphic.
Since 1971, the Freudenthal Institute has developed an approach to mathematics education named Realistic Mathematics Education (RME). The philosophy of RME is based on Hans Freudenthal’s concept of ‘mathematics as a human activity’. Prof. Hans Freudenthal (1905-1990), a mathematician and educator, believes that ‘ready-made mathematics’ should not be taught in school. By contrast, he urges that students should be offered ‘realistic situations’ so that they can rediscover from informal to formal mathematics. Although mathematics education in Vietnam has some achievements, it still encounters several challenges. Recently, the reform of teaching methods has become an urgent task in Vietnam. It appears that Vietnamese mathematics education lacks necessary theoretical frameworks. At first sight, the philosophy of RME is suitable for the orientation of the teaching method reform in Vietnam. However, the potential of RME for mathematics education as well as the ability of applying RME to teaching mathematics is still questionable in Vietnam. The primary aim of this dissertation is to research into abilities of applying RME to teaching and learning mathematics in Vietnam and to answer the question “how could RME enrich Vietnamese mathematics education?”. This research will emphasize teaching geometry in Vietnamese middle school. More specifically, the dissertation will implement the following research tasks: • Analyzing the characteristics of Vietnamese mathematics education in the ‘reformed’ period (from the early 1980s to the early 2000s) and at present; • Implementing a survey of 152 middle school teachers’ ideas from several Vietnamese provinces and cities about Vietnamese mathematics education; • Analyzing RME, including Freudenthal’s viewpoints for RME and the characteristics of RME; • Discussing how to design RME-based lessons and how to apply these lessons to teaching and learning in Vietnam; • Experimenting RME-based lessons in a Vietnamese middle school; • Analyzing the feedback from the students’ worksheets and the teachers’ reports, including the potentials of RME-based lessons for Vietnamese middle school and the difficulties the teachers and their students encountered with RME-based lessons; • Discussing proposals for applying RME-based lessons to teaching and learning mathematics in Vietnam, including making suggestions for teachers who will apply these lessons to their teaching and designing courses for in-service teachers and teachers-in training. This research reveals that although teachers and students may encounter some obstacles while teaching and learning with RME-based lesson, RME could become a potential approach for mathematics education and could be effectively applied to teaching and learning mathematics in Vietnamese school.
Als ich anfing, ein Thema für meine Promotion zu erarbeiten, fand ich Massentests ziemlich beeindruckend. TIMSS: über 500000 Schüler getestet. PISA: 180000 Schüler getestet. Ich wollte diese Datenbasis nutzen, um Erkenntnisse für die Gestaltung von Unterricht zu gewinnen. Leider kam ich damit nicht weit. Je tiefer ich mich mit den Tests und den dahinterstehenden Theorien befasste, desto deutlicher schälte sich heraus, dass mit diesen Tests keine neue Erkenntnis generiert werden kann. Fast alle Schlussfolgerungen, die aus den Tests gezogen werden, konnten gar nicht aus den Tests selbst gewonnen werden. Ich konzentrierte mich zunehmend auf die Testaufgaben, weil die Geltung der Aussage eines Tests an der Aufgabe erzeugt wird: In der Aufgabe gerinnt das, was die Tester als „mathematische Leistungsfähigkeit“ konstruieren. Der Schüler wiederum hat nur die Aufgabe vor sich. Es gibt nur „gelöst“ (ein Punkt) und „ungelöst“ (kein Punkt). Damit der Schüler den Punkt bekommt, muss er an der richtigen Stelle ankreuzen, oder er muss etwas hinschrei-ben, wofür der Auswerter einen Punkt gibt. In der Dissertation wird untersucht, was die Aufgaben testen, was also alles in das Konstrukt von „mathematischer Leistungsfähigkeit“ einfließt, und ob es das ist, was der Test testen soll. Es stellte sich durchaus erstaunliches heraus: - Oftmals gibt es so viele Möglichkeiten, zur gewünschten Lösung (die nicht in jedem Fall die richtige Lösung ist) zu gelangen, dass man nicht benennen kann, welche Fähigkeit die Aufgabe eigentlich misst. Das Konstrukt „mathematische Leistungsfähigkeit“ wird damit zu einem zufälligen. - Es werden Komponenten von Testfähigkeit mitgemessen: Viele Aufgaben enthalten Irritationen, welche von testerfahrenen Schülern leichter überwunden werden können als von testunerfahrenen. Es gibt Aufgaben, die gelöst werden können, ohne dass man über die Fähigkeit verfügt, die getestet werden soll. Umgekehrt gibt es Aufgaben, die man eventuell nicht lösen kann, obwohl man über diese Fähigkeit verfügt. Als Kernkompetenz von Testfähigkeit stellt sich heraus, weder das gestellte mathematische Problem noch die angeblichen realen Proble-me ernst zu nehmen, sondern sich statt dessen auf das zu konzentrieren, was die Tester angekreuzt oder hinge-schrieben sehen wollen. Prinzipiell erweist es sich als günstig, mittelmäßig zu arbeiten, auf intellektuelle Tiefe in der Auseinandersetzung mit den Aufgaben also zu verzichten. - Man kann bei Multiple-Choice-Tests raten. Die PISA-Gruppe behauptet zwar, dieses Problem technisch über-winden zu können, dies erweist sich aber als Fehleinschätzung. - Sowohl bei TIMSS als auch bei PISA stellt sich heraus, dass die vorgeblich verwendeten didaktischen und psychologischen Theorien lediglich theoretische Mäntel für eine theoriearme Testerstellung sind. Am Beispiel der Theorie der mentalen Situationsmodelle (zur Bearbeitung von realitätsnahen Aufgaben) wird dies ausführlich exemplarisch ausgearbeitet. Das Problem reproduziert sich in anderen Theoriefeldern. Die Tests werden nicht durch Operationalisierungen von Messkonstrukten erstellt, sondern durch systematisches Zusammenstückeln von Aufgaben. - Bei PISA sollte „Mathematical Literacy“ getestet werden. Verkürzt sollte das die Fähigkeit sein, „die Rolle, die Mathematik in der Welt spielt, zu erkennen und zu verstehen, begründete mathematische Urteile abzugeben und sich auf eine Weise mit der Mathematik zu befassen, die den Anforderungen des gegenwärtigen und künftigen Lebens einer Person als eines konstruktiven, engagierten und reflektierten Bürgers entspricht“ (PISA-Eigendarstellung). Von all dem kann angesichts der Aufgaben keine Rede sein. - Bei der Untersuchung des PISA-Tests drängte sich ein mathematikdidaktischer Habitus auf, der eine separate Untersuchung erzwang. Ich habe ihn unter dem Stichwort der „Abkehr von der Sache“ zusammengefasst. Er ist geprägt von Zerstörungen des Mathematischen bei gleichzeitiger Überbetonung des Fachsprachlichen und durch Verwerfungen des Mathematischen und des Realen bei realitätsnahen Aufgaben. Letzteres gründet in der Nicht-beachtung der Authentizität sowohl des Realen als auch des Mathematischen. Die Arbeit versammelt neben den Untersuchungen zu TIMSS und PISA ein ausführliches Kapitel über das Prob-lem des Testens und eine Darstellung der Methodologie und Praxis der Objektiven Hermeneutik.
Our work goes in two directions. At first we want to transfer definitions, concepts and results of the theory of hyperidentities and solid varieties from the total to the partial case. (1) We prove that the operators chi^A_RNF and chi^E_RNF are only monotone and additive and we show that the sets of all fixed points of these operators are characterized only by three instead of four equivalent conditions for the case of closure operators. (2) We prove that V is n − SF-solid iff clone^SF V is free with respect to itself, freely generated by the independent set {[fi(x_1, . . . , x_n)]Id^SF_n V | i \in I}. (3) We prove that if V is n-fluid and ~V |P(V ) =~V −iso |P(V ) then V is kunsolid for k >= n (where P(V ) is the set of all V -proper hypersubstitutions of type \tau ). (4) We prove that a strong M-hyperquasi-equational theory is characterized by four equivalent conditions. The second direction of our work is to follow ideas which are typical for the partial case. (1) We characterize all minimal partial clones which are strongly solidifyable. (2)We define the operator Chi^A_Ph where Ph is a monoid of regular partial hypersubstitutions.Using this concept, we define the concept of a Phyp_R(\tau )-solid strong regular variety of partial algebras and we prove that a PHyp_R(\tau )-solid strong regular variety satisfies four equivalent conditions.
Semiclassical asymptotics for the scattering amplitude in the presence of focal points at infinity
(2006)
We consider scattering in $\R^n$, $n\ge 2$, described by the Schr\"odinger operator $P(h)=-h^2\Delta+V$, where $V$ is a short-range potential. With the aid of Maslov theory, we give a geometrical formula for the semiclassical asymptotics as $h\to 0$ of the scattering amplitude $f(\omega_-,\omega_+;\lambda,h)$ $\omega_+\neq\omega_-$) which remains valid in the presence of focal points at infinity (caustics). Crucial for this analysis are precise estimates on the asymptotics of the classical phase trajectories and the relationship between caustics in euclidean phase space and caustics at infinity.
We analyze the notions of monotonicity and complete monotonicity for Markov Chains in continuous-time, taking values in a finite partially ordered set. Similarly to what happens in discrete-time, the two notions are not equivalent. However, we show that there are partially ordered sets for which monotonicity and complete monotonicity coincide in continuous time but not in discrete-time.
We analyze the asymptotic behavior in the limit epsilon to zero for a wide class of difference operators H_epsilon = T_epsilon + V_epsilon with underlying multi-well potential. They act on the square summable functions on the lattice (epsilon Z)^d. We start showing the validity of an harmonic approximation and construct WKB-solutions at the wells. Then we construct a Finslerian distance d induced by H and show that short integral curves are geodesics and d gives the rate for the exponential decay of Dirichlet eigenfunctions. In terms of this distance, we give sharp estimates for the interaction between the wells and construct the interaction matrix.
We consider a system of infinitely many hard balls in R<sup>d undergoing Brownian motions and submitted to a smooth pair potential. It is modelized by an infinite-dimensional stochastic differential equation with a local time term. We prove that the set of all equilibrium measures, solution of a detailed balance equation, coincides with the set of canonical Gibbs measures associated to the hard core potential added to the smooth interaction potential.