The Many Faces of the Mathematical Modeling Cycle

In literature about mathematical modeling a diversity can be seen in ways of presenting the modeling cycle. Every year, students in the Bachelor’s program Applied Mathematics of the Eindhoven University of Technology, after having completed a series of mathematical modeling projects, have been prompted with a simple three-step representation of the modeling cycle. This representation consisted out of 1) problem translation into a mathematical model, 2) the solution to mathematical problem, and 3) interpretation of the solution in the context of the original problem. The students’ task was to detail and complete this representation. Their representations also showed a great diversity. This diversity is investigated and compared with the representations of the students’ teachers. The representations with written explanations of 77 students and 20 teachers are analyzed with respect to the presence of content aspects such as problem analysis, worlds/models/knowledge other than mathematical, verification, validation, communication and reflection at the end of the modeling process. Also form aspects such as iteration and complexity are analyzed. The results show much diversity within both groups concerning the presence or absence of aspects. Validation is present most, reflection least. Only iteration (one is passing the modeling cycle) more than once is significantly more present in the teachers’ group than in the students’ group. While accepting diversity as a natural phenomenon, the authors plea for incorporating all aspects mentioned into mathematical modeling education

Words or Images: the PhD Thesis of Kees Hoogland

Op 8 november is Kees Hoogland gepromoveerd op het proefschrift Images of Numeracy, Investigating the effects of visual representations of problem situations in contextual mathematical problem solving. Het onderzoek van Kees, onder andere oud-hoofdredacteur van Euclides, gaat over de vraag of bij realistische rekenopgaven zoals die in de rekentoetsen 2F voorkomen hetrepresenteren van de probleemsituatie in de vorm van een plaatje, meestal een foto, tot betere prestatie leidt dan bij het presenteren van de probleemsituatie in woorden. In de woorden van Kees:In presenting contextual mathematical problems, what is the effect on student performance of changing a descriptive representation of the problem situation to a mainly depictive one?In deze bijdrage volt per hoofdstuk van het proefschrift een samenvatting. Daarna volgen een aantal afsluitende opmerkingen

Routing trains through a railway station based on a Node Packing model

In this paper we describe the problem of routing trains through a railway station. This routing problem is a subproblem of the automatic generation of timetables for the Dutch railway system. The problem of routing trains through a railway station is the problem of assigning each of the involved trains to a route through the railway station, given the detailed layout of the railway network within the station and given the arrival and departure times of the trains. When solving this routing problem, several aspects such as capacity, safety, and customer service have to be taken into account. In this paper we describe this routing problem in terms of a Weighted Node Packing Problem. Furthermore, we describe an algorithm for solving this routing problem to optimality. The algorithm is based on preprocessing, valid inequalities, and a branch-and-cut approach. The preprocessing techniques aim at identifying super uous nodes which can be removed from the problem instance. The characteristics of the preprocessing techniques with respect to propagation are investigated. We also present the results of a computational study in which the model, the preprocessing techniques and the algorithm are tested based on data related to the railway stations Arnhem, Hoorn and Utrecht in the Netherlands.mathematical economics and econometrics ;

On-Surface Covalent Linking of Organic Building Blocks on a Bulk Insulator

Kittelmann M, Rahe P, Nimmrich M, Hauke CM, Gourdon A, Kühnle A. On-Surface Covalent Linking of Organic Building Blocks on a Bulk Insulator. ACS Nano. 2011;5(10):8420-8425.On-surface synthesis in ultrahigh vacuum provides a promising strategy for creating thermally and chemically stable molecular structures at surfaces. The two-dimensional confinement of the educts, the possibility of working at higher (or lower) temperatures in the absence of solvent, and the templating effect of the surface bear the potential of preparing compounds that cannot be obtained in solution. Moreover, covalently linked conjugated molecules allow for efficient electron transport and are, thus, particularly interesting for future molecular electronics applications. When having these applications in mind, electrically insulating substrates are mandatory to provide sufficient decoupling of the molecular structure from the substrate surface. So far, however, on-surface synthesis has been achieved only on metallic substrates. Here we demonstrate the covalent linking of organic molecules on a bulk insulator, namely, calcite. We deliberately employ the strong electrostatic interaction between the carboxylate groups of halide-substituted benzoic adds and the surface calcium cations to prevent molecular desorption and to reach homolytic cleavage temperatures. This allows for the formation of aryl radicals and intermolecular coupling. By varying the number and position of the halide substitution, we rationally design the resulting structures, revealing straight lines, zigzag structures, and dimers, thus providing clear evidence for the covalent linking. Our results constitute an important step toward exploiting on-surface synthesis for molecular electronics and optics applications, which require electrically insulating rather than metallic supporting substrates

Micrometre-long covalent organic fibres by photoinitiated chain-growth radical polymerization on an alkali-halide surface

On-surface polymerization is a promising technique to prepare organic functional nanomaterials that are challenging to synthesize in solution, but it is typically used on metal substrates, which play a catalytic role. Previous examples on insulating surfaces have involved intermediate self-assembled structures, which face high barriers to diffusion, or annealing to higher temperatures, which generally causes rapid dewetting and desorption of the monomers. Here we report the photoinitiated radical polymerization, initiated from a two-dimensional gas phase, of a dimaleimide monomer on an insulating KCl surface. Polymer fibres up to 1 μm long are formed through chain-like rather than step-like growth. Interactions between potassium cations and the dimaleimide’s oxygen atoms facilitate the propagation of the polymer fibres along a preferred axis of the substrate over long distances. Density functional theory calculations, non-contact atomic force microscopy imaging and manipulations at room temperature were used to explore the initiation and propagation processes, as well as the structure and stability of the resulting one-dimensional polymer fibres