An application of combinatorial optimization in the printing industry

Offset printing is a common method to produce large amounts of printed matter. We consider a real-world offset printing process that is used to imprint customer-specific designs on napkin pouches. The print- ing technology used yields a number of specific constraints. The planning problem consists of allocating designs to printing-plate slots such that the given customer demand for each design is fulfilled, all technologi- cal and organizational constraints are met and the total overproduction and setup costs are minimized. We formulate this planning problem as a mixed-binary linear program, and we develop a multi-pass matching-based savings heuristic. We report computational results for a set of problem instances devised from real-world data.

Explicit mathematics and operational set theory: some ontological comparisons

We discuss several ontological properties of explicit mathematics and operational set theory: global choice, decidable classes, totality and extensionality of operations, function spaces, class and set formation via formulas that contain the definedness predicate and applications.

Bipartite graphs and quasipositive surfaces

We present a simple combinatorial model for quasipositive surfaces and positive braids, based on embedded bipartite graphs. As a first application, we extend the well-known duality on standard diagrams of torus links to twisted torus links. We then introduce a combinatorial notion of adjacency for bipartite graph links and discuss its potential relation with the adjacency problem for plane curve singularities.

Conformational diversity versus nucleic acid triplex stability, a combinatorial study

The stability of a triple helix formed between a DNA duplex and an incoming oligonucleotide strand strongly depends on the solvent conditions and on intrinsic chemical and conformational factors. Attempts to increase triple helix stability in the past included chemical modification of the backbone, sugar ring, and bases in the third strand. However, the predictive power of such modifications is still rather poor. We therefore developed a method that allows for rapid screening of conformationally diverse third strand oligonucleotides for triplex stability in the parallel pairing motif to a given DNA double helix sequence. Combinatorial libraries of oligonucleotides of the requisite (fixed) base composition and length that vary in their sugar unit (ribose or deoxyribose) at each position were generated. After affinity chromatography against their corresponding immobilized DNA target duplex, utilizing a temperature gradient as the selection criterion, the oligonucleotides forming the most stable triple helices were selected and characterized by physicochemical methods. Thus, a series of oligonucleotides were identified that allowed us to define basic rules for triple helix stability in this conformationally diverse system. It was found that ribocytidines in the third strand increase triplex stability relative to deoxyribocytidines independently of the neighboring bases and position along the strand. However, remarkable sequence-dependent differences in stability were found for (deoxy)thymidines and uridines

A new model construction by making a detour via intuitionistic theories II: Interpretability lower bound of Feferman's explicit mathematics T0

We partially solve a long-standing problem in the proof theory of explicit mathematics or the proof theory in general. Namely, we give a lower bound of Feferman’s system T0 of explicit mathematics (but only when formulated on classical logic) with a concrete interpretat ion of the subsystem Σ12-AC+ (BI) of second order arithmetic inside T0. Whereas a lower bound proof in the sense of proof-theoretic reducibility or of ordinalanalysis was already given in 80s, the lower bound in the sense of interpretability we give here is new. We apply the new interpretation method developed by the author and Zumbrunnen (2015), which can be seen as the third kind of model construction method for classical theories, after Cohen’s forcing and Krivine’s classical realizability. It gives us an interpretation between classical theories, by composing interpretations between intuitionistic theories.

How gender fair are German schoolbooks in the twenty-first century? An analysis of language and illustrations in schoolbooks for mathematics and German

Schoolbooks convey not only school-relevant knowledge; they also influence the development of stereotypes about different social groups. Particularly during the 1970s and 1980s, many studies analysed schoolbooks and criticised the overall predominance of male persons and of traditional role allocations. Since that time, women’s and men’s occupations and social functions have changed considerably. The present research investigated gender portrayals in schoolbooks for German and mathematics that were recently published in Germany. We examined the proportions of female and male persons in pictures and texts and categorized their activities, occupational and parental roles. Going beyond previous studies, we added two criteria: the use of gender-fair language and the spatial arrangements of persons in pictures. Our results show that schoolbooks for German contained almost balanced depictions of girls and boys, whereas women were less frequently shown than men. In mathematics books, males outnumbered females in general. Across both types of books, female and male persons were engaged in many different activities, not only gendertyped ones; however, male persons were more often described via their profession than females. Use of gender-fair language has found its way into schoolbooks but is not used consistently. Books for German were more gender fair in terms of linguistic forms than books for mathematics. For spatial arrangements, we found no indication for gender biases. The results are discussed with a focus on how schoolbooks can be optimized to contribute to gender equality.

How Plankton Swim: An Interdisciplinary Approach for Using Mathematics & Physics to Understand the Biology of the Natural World

The authors have developed and field-tested high school-level curricular materials that guide students to use biology, mathematics, and physics to understand plankton and how these tiny organisms move in a world where their intuition does not apply. The authors chose plankton as the focus of their materials primarily because the challenges faced by plankton are novel problems to most students, forcing adoption of new perspectives and making the study of plankton exciting. Additional reasons that they chose plankton to focus on include their ecological importance, their availability to most teachers and students, the ease with which they can be collected and observed, and the current focus of some scientific researchers on their movement and behavior. These curricular materials include a series of inquiry-based, hands-on exercises designed to be accessible to students with a range of backgrounds. Many of these materials could be adapted for use by middle-school, and/or college-level students. In this article, the authors describe sample lessons, summarize what worked well, and flag obstacles they encountered while integrating mathematics and physics into the biology classroom.

Promoting Student Collaboration in a Detracked, Heterogeneous Secondary Mathematics Classroom

Detracking and heterogeneous groupwork are two educational practices that have been shown to have promise for affording all students needed learning opportunities to develop mathematical proficiency. However, teachers face significant pedagogical challenges in organizing productive groupwork in these settings. This study offers an analysis of one teacher’s role in creating a classroom system that supported student collaboration within groups in a detracked, heterogeneous geometry classroom. The analysis focuses on four categories of the teacher’s work that created a set of affordances to support within group collaborative practices and links the teacher’s work with principles of complex systems.

Introductory Mathematics for Quantum Chemistry

An elementary discussion of some of the mathematics employed in studying Quantum Chemistry in a style appropriate for persons who have not taken advanced mathematical instruction.

Supporting Whole-Class Collaborative Inquiry in a Secondary Mathematics Classroom

Recent mathematics education reform efforts call for the instantiation of mathematics classroom environments where students have opportunities to reason and construct their understandings as part of a community of learners. Despite some successes, traditional models of instruction still dominate the educational landscape. This limited success can be attributed, in part, to an underdeveloped understanding of the roles teachers must enact to successfully organize and participate in collaborative classroom practices. Towards this end, an in-depth longitudinal case study of a collaborative high school mathematics classroom was undertaken guided by the following two questions: What roles do these collaborative practices require of teacher and students? How does the community’s capacity to engage in collaborative practices develop over time? The analyses produced two conceptual models: one of the teacher’s role, along with specific instructional strategies the teacher used to organize a collaborative learning environment, and the second of the process by which the class’s capacity to participate in collaborative inquiry practices developed over time.