A new look at school algebra—Past,present, and future

The articles in this issue arose from the keynote presentations and discussions of the Algebra Working Group of the Seventh International Congress on Mathematical Education (ICME-7) held in Québec City, Canada, in August 1992 (official title of the Working Group: The Place of Algebra in Secondary and Tertiary Education). Before presenting an overview of the articles and how they all fit together, let me first say a few words about the origins and structure of the Working Group.     

An axiomatic look at the Erdős-Trost problem

The Erd?s-Trost problem can be formulated in the following way: “If the triangle XY Z is inscribed in the triangle ABC—with X, Y, and Z on the sides BC, CA, and AB, respectively—then one of the areas of the triangles BXZ, CXY , AY Z is less than or equal to the area of the triangle XY Z.” There are many different solutions for this problem. In this note we take up a very elementary proof (due to Szekeres) and deduce that the class of ordered translation planes is the level in the hierarchy of affine planes where the Erd?s-Trost statement still holds true. We also look at the conditions an absolute plane needs to satisfy for the validity of the Erd?s-Trost statement.     

Bridging incommensurable discourses – A commognitive look at instructional design in the zone of proximal development

There are points in the mathematics curriculum where the “rules of the game” change, for example, the meaning and method of multiplication when negative numbers are introduced. At these junctions the new mathematical discourse may be in conflict with learners’ current discourse. Learners may have little intrinsic motivation to accept new rules whose productiveness they cannot yet appreciate, hence, their first steps in the emerging discourse may need to be ritualized - socially motivated by the teacher’s approval. In this article we ask how careful crafting of task situations can support teachers in leading learners into a new discourse. We propose interdiscursivity – the blending of discursive elements from different discourses – as a mechanism for designing task situations to support learners in taking their first steps in an emerging discourse. On the basis of three examples, we suggest that this mechanism may support participation that is intrinsically motivated (explorative).     

A modular look at Farkas’ identities

Recently H. Farkas introduced a new simple arithmetic function and found an identity which involves this function. It is immediate to rewrite this identity as an identity between modular forms and reprove it in this way. We discuss natural generalizations of Farkas’ identity. Surprisingly, in a certain sense, there is only one identity which is an exact analogue of that found by Farkas. At the same time, we present a way to produce infinitely many similar identities. As an application, we obtain a result on non-vanishing of the central critical value of L-functions associated to a cusp Hecke eigenform. Supported by NSF grant DMS-0700933.     

A new look at the Jordan-Hölder theorem for semimodular lattices

We show that in a semimodular lattice L of finite length, from any prime interval we can reach any maximal chain C by an up- and a down-perspectivity. Therefore, C is a congruence-determining sublattice of L.     

Students,mathematics, and technology: assessing the present – challenging the future

This paper is concerned with identifying and assessing the impacts of various technologically enriched approaches to mathematics learning. Its purpose is to address the usefulness of emerging knowledge to enhance practice, and to contribute to theorizing about technology-based learning. Hence the main drivers are intentions to identify and elaborate on obstacles, errors, and unresolved problems on the one hand, and positives and insights on the other. The paper tells a cautionary tale about expectations of technology-enhanced learning, while simultaneously uncovering a rich base from which to theorize and test new appreciations of what is involved when students, technology, and mathematics connect in learning settings. A summary assessment is that this area still very much represents work in progress, but there is now a heightened realization, at least among those not transfixed by technological blindness, that the search for ultimate answers will require much greater exploration of machine–mathematics–learner relationships.     

A new look at the John–Nirenberg and John–Strömberg theorems for BMO

We develop some techniques for studying various versions of the function space BMO. Special cases of one of our results give alternative proofs of the celebrated John–Nirenberg inequality and of related inequalities due to John and to Wik. Our approach enables us to pose a simply formulated “geometric” question, for which an affirmative answer would lead to a version of the John–Nirenberg inequality with dimension free constants.     

GIS-based optimization for the locations of sewage treatment plants and sewage outfalls – A case study of Nansha District in Guangzhou City,China

Based on GIS technology, eco-suitability evaluation method integrating economic, social and ecological factors is employed to optimize the locations of the sewage treatment plants and outfalls in this paper. The ecological indices considering eco-sensitivity areas as key elements of the integrated evaluation system are allotted to the water subsystem, riparian zone subsystem, and land subsystem. A novel integrated eco-suitability evaluation index system encompassing ten criteria and fifteen indices is established to generate the distributed eco-suitability map of the concerned areas and determine the possible locations of sewage treatment plants and sewage outfalls according to the eco-suitability levels. With the case study of Nansha District in Guangzhou City, China, 212 km² areas of land are found to be suitable for locating the sewage treatment plants, 87 km² areas of water suitable for sewage release, and 6 km² area of riparian zone unsuitable for sewage outfalls.     

Cooperative dispatching – exploiting the flexibility of an FMS by means of incremental optimization

We present a new control scheme for releasing parts into a flexible manufacturing system (FMS) that is based on incremental optimization. Our objective is to exploit the available routing flexibility of parts in an enhanced manner by viewing part release as an assignment problem using system status information. In particular, we propose an “intelligent” part release mechanism with some look-ahead and optimization features in order to allow for optimization-based “cooperation” of workcenters. The cooperative dispatching concept is implemented in an object-oriented computer simulation model, and experiments with a varying degree of average routing flexibility are performed. The experimental results are used for a statistical analysis of the benefits of cooperative dispatching versus the common approach of standard dispatching. Finally, we investigate the robustness of the presented FMS control scheme in the case of random machine breakdowns.     

A simple and elementary proof of the Karush–Kuhn–Tucker theorem for inequality-constrained optimization

We present an elementary proof of the Karush–Kuhn–Tucker Theorem for the problem with nonlinear inequality constraints and linear equality constraints. Most proofs in the literature rely on advanced optimization concepts such as linear programming duality, the convex separation theorem, or a theorem of the alternative for systems of linear inequalities. By contrast, the proof given here uses only basic facts from linear algebra and the definition of differentiability.     

Computer use in mathematics education – present situation,intentions, and problems in the Federal Republic of Germany†

Computers, and computer‐related thinking structures, are only gradually influencing mathematics education. On the one hand, there is a discrepancy between involved teachers who already have changed their own classroom teaching to a great extent, and a majority of mathematics teachers who have not yet taken notice of the computer for teaching purposes. On the other hand, knowledge of the computer and of algorithms is frequently merely added to the mathematical subject matter. As opposed to that, the authors argue that it is necessary to genuinely integrate such subject matter, and to include general topics such as social impact and changed attitudes toward application. With regard to implementation, they develop concrete ideas which are aligned in a differentiated manner to the specific situation and the opportunities offered in the Federal Republic of Germany. The rationale for that is that only such reference to a specific situation will provide an opportunity for readers abroad to usefully apply approaches and ideas to the situation given in their own cultural environment.

An abbreviated version of this paper for cursory reading or other purposes has been marked by bold lines on the margin.

     

A new look at some basic concepts in arbitrage pricing theory

The notion of No Free Lunch with Vanishing Risk (or NFLVR in short) w.r.t. admissible strategies depends on the choice of numeraire. Yan introduced the notion of allowable strategy and showed that condition of NFLVR w.r.t. allowable strategies is independent of the choice of numeraire and is equivalent to the existence of an equivalent martingale measure for the deflated price process. In this paper we establish a version of the Kramkov's optional decomposition theorem in the setting of equivalent martingale measures. Based on this theorem, we have a new look at some basic concepts in arbitrage pricing theory: superhedging, fair price, attainable contingent claims, complete markets and etc.     

An epsilon-based measure of efficiency in DEA – A third pole of technical efficiency

In DEA, we have two measures of technical efficiency with different characteristics: radial and non-radial. In this paper we compile them into a composite model called “epsilon-based measure (EBM).” For this purpose we introduce two parameters which connect radial and non-radial models. These two parameters are obtained from the newly defined affinity index between inputs or outputs along with principal component analysis on the affinity matrix. Thus, EBM takes into account diversity of input/output data and their relative importance for measuring technical efficiency.     

Shape optimization of an airfoil in the presence of compressible and viscous flows

The objective of this article is to present a step-by-step problem-solving procedure of shape optimization. The procedure is carried out to design an airfoil in the presence of compressible and viscous flows using a control theory approach based on measure theory. An optimal shape design (OSD) problem governed by full Navier-Stokes equations is given. Then, a weak variational form is derived from the linearized governing equations. During the procedure, because the measure theory (MT) approach is implemented using fixed geometry versus moving geometry, a proper bijective transformation is introduced. Finally, an approximating linear programming (LP) problem of the original shape optimization problem is obtained by means of MT approach that is not iterative and does not need any initial guess to proceed. Illustrative examples are provided to demonstrate efficiency of the proposed procedure.     

A hybrid of genetic algorithm and particle swarm optimization for solving bi-level linear programming problem – A case study on supply chain model

The main goal of supply chain management is to coordinate and collaborate the supply chain partners seamlessly. On the other hand, bi-level linear programming is a technique for modeling decentralized decision. It consists of the upper level and lower level objectives. Thus, this paper intends to apply bi-level linear programming to supply chain distribution problem and develop an efficient method based on hybrid of genetic algorithm (GA) and particle swarm optimization (PSO). The performance of the proposed method is ascertained by comparing the results with GA and PSO using four problems in the literature and a supply chain distribution model.