Expanding context and domain: A cross-curricular activity in mathematics and physics

This article is based on my 15 years of experience as a teacher of mathematics and physics in the Danish Gymnasium (high school), and it gives an example of an interdisciplinary course between mathematics and physics. The course is centered around the concept of exponential functions. The starting point is that concepts are rooted in practice and gain their meaning through application, and the concept of a function is regarded as a tool for modelling real-world situations. It is the intention to teach a course that emphasizes factors that promote transfer of the concept and use of the various representations of the concept, to make it more practical and meaningful for the students. It is concluded that a coordinated cross-curricular activity between mathematics and physics, by offering a great variety of domain relations and context settings, has a great potential for creating a learning environment where the students, through applicational and modelling activities, are engaged actively in constructing and using knowledge.     

Perspectives of cross-curricular activity in Japanese mathematics education

Historically, we can find cross-curricular activities (“life unit study”) after World War II in Japan. When we consider the arguments of “life unit study”, the following three approaches will serve as reference: the approach orientated to the independent aspect, the approach orientated to the societal aspect, the approach orientated to the scientific aspect. By reflecting the critique of “life unit study”, it is suggested that a balanced approach should be carried out. In other words, teachers had better focus on one aspect according to the aim of classroom teaching, and balance three approaches on the whole.     

Supporting mathematical literacy: examples from a cross-curricular project

Mathematical literacy implies the capacity to apply mathematical knowledge to various and context-related problems in a functional, flexible and practical way. Improving mathematical literacy requires a learning environment that stimulates students cognitively as well as allowing them to collect practical experiences through connections with the real world. In order to achieve this, students should be confronted with many different facets of reality. They should be given the opportunity to participate in carrying out experiments, to be exposed to verbal argumentative discussions and to be involved in model-building activities. This leads to the idea of integrating science into maths education. Two sequences of lessons were developed and tried out at the University of Education Schwäbisch Gmünd integrating scientific topics and methods into maths lessons at German secondary schools. The results show that the scientific activities and their connection with reality led to well-based discussions. The connection between the phenomenon and the model remained remarkably close during the entire series of lessons. At present the sequences of lessons are integrated in the European ScienceMath project, a joint project between universities and schools in Denmark, Finland, Slovenia and Germany (see www.sciencemath.ph-gmuend.de).     

A view on multiple recurrence

In this note we present a proof of multiple recurrence for ergodic systems (and thereby of Szemerédi’s theorem) being a mixture of three known proofs. It is based on a conditional version of the Jacobs–de Leeuw–Glicksberg decomposition and properties of the Gowers–Host–Kra uniformity seminorms.     

Mathematics and cross-curricular activities: Bridges exist for crossing them

The paper presents some thoughts on the relations of mathematics teaching and learning and the cross-curricular activities. Some Spantsh examples are reviewed and some insights for future actions are given.     

A general view on computing communities

We define a community structure of a graph as a partition of the vertices into at least two sets with the property that each vertex has connections to relatively many vertices in its own set compared to any other set in the partition and refer to the sets in such a partition as communities  . We show that it is NP-hard to compute a community containing a given set of vertices. On the other hand, we show how to compute a community structure in polynomial time for any connected graph containing at least four vertices except the star graph _Sn$S_n$. Finally, we generalize our results and formally show that counterintuitive aspects are unavoidable for any definition of a community structure with a polynomial time algorithm for computing communities containing specific vertices.     

A linear set view on KM-arcs

In this paper, we study KM-arcs of type t, i.e., point sets of size \(q+t\) in \(\mathrm {PG}(2,q)\) such that every line contains 0, 2 or t of its points. We use field reduction to give a different point of view on the class of translation arcs. Starting from a particular \(\mathbb {F}_2\)-linear set, called an i -club, we reconstruct the projective triads, the translation hyperovals as well as the translation arcs constructed by Korchmáros-Mazzocca, Gács-Weiner and Limbupasiriporn. We show the KM-arcs of type \(q/4\) recently constructed by Vandendriessche are translation arcs and fit in this family. Finally, we construct a family of KM-arcs of type \(q/4\). We show that this family, apart from new examples that are not translation KM-arcs, contains all translation KM-arcs of type \(q/4\).     

A new view on some characterizations of simplices

The author wishes to thank Professor R. Schneider for helpful discussion, especially for referring his attention to the useful notionincluded double cone.     

A geometric view on learning Bayesian network structures

We recall the basic idea of an algebraic approach to learning Bayesian network (BN) structures, namely to represent every BN structure by a certain (uniquely determined) vector, called a standard imset. The main result of the paper is that the set of standard imsets is the set of vertices (=extreme points) of a certain polytope. Motivated by the geometric view, we introduce the concept of the geometric neighborhood for standard imsets, and, consequently, for BN structures. Then we show that it always includes the inclusion neighborhood, which was introduced earlier in connection with the greedy equivalence search (GES) algorithm. The third result is that the global optimum of an affine function over the polytope coincides with the local optimum relative to the geometric neighborhood.To illustrate the new concept by an example, we describe the geometric neighborhood in the case of three variables and show it differs from the inclusion neighborhood. This leads to a simple example of the failure of the GES algorithm if data are not “generated” from a perfectly Markovian distribution. The point is that one can avoid this failure if the search technique is based on the geometric neighborhood instead. We also found out what is the geometric neighborhood in the case of four and five variables.     

Criss-cross methods: A fresh view on pivot algorithms

Criss-cross methods are pivot algorithms that solve linear programming problems in one phase starting with any basic solution. The first finite criss-cross method was invented by Chang, Terlaky and Wang independently. Unlike the simplex method that follows a monotonic edge path on the feasible region, the trace of a criss-cross method is neither monotonic (with respect to the objective function) nor feasibility preserving. The main purpose of this paper is to present mathematical ideas and proof techniques behind finite criss-cross pivot methods. A recent result on the existence of a short admissible pivot path to an optimal basis is given, indicating shortest pivot paths from any basis might be indeed short for criss-cross type algorithms. The origins and the history of criss-cross methods are also touched upon.     

A tropical view on Bruhat-Tits buildings and their compactifications

We relate some features of Bruhat-Tits buildings and their compactifications to tropical geometry. If G is a semisimple group over a suitable non-Archimedean field, the stabilizers of points in the Bruhat-Tits building of G and in some of its compactifications are described by tropical linear algebra. The compactifications we consider arise from algebraic representations of G. We show that the fan which is used to compactify an apartment in this theory is given by the weight polytope of the representation and that it is related to the tropicalization of the hypersurface given by the character of the representation.     

Valuation theory: A constructive view

The theory of valuations on fields is developed in the constructive spirit of Errett Bishop. As a consequence of the general theory we are able to construct all nonarchimedean valuations on algebraic number fields and compute their ramification indices and residue class degrees. The notion of a field with a valuation for which the infimum of the values of any polynomial function can be computed plays an important role. Numerous limiting counterexamples are provided.     

A mathematical view of biological complexity

This essay is a trial on giving some mathematical ideas about the concept of biological complexity, trying to explore four different attributes considered to be essential to characterize a complex system in a biological context: decomposition, heterogeneous assembly, self-organization, and adequacy. It is a theoretical and speculative approach, opening some possibilities to further numerical and experimental work, illustrated by references to several researches that applied the concepts presented here.