Assessing Elementary Understanding of Multiplication Concepts

This article summarizes the basic concepts of multiplication and provides some evidence that the traditional third‐grade curriculum and instruction emphasizing memorization of multiplication facts produces much less understanding of the basic concepts of multiplication than a standards‐based curriculum and instruction emphasizing construction of number sense and meaning for operations. This study also describes a collection of assessment tasks that provided meaningful evidence of children's understandings of basic multiplication concepts, including understandings of the relationships between multiplication and addition.     

The semi‐iterative method applied to the hyper‐power iteration

The semi‐iterative method (SIM) is applied to the hyper‐power (HP) iteration, and necessary and sufficient conditions are given for the convergence of the semi‐iterative–hyper‐power (SIM–HP) iteration. The root convergence rate is computed for both the HP and SIM–HP methods, and the quotient convergence rate is given for the HP iteration. Copyright © 2005 John Wiley & Sons, Ltd.     

Spying on the cost structure of naive bidding competitors via linear programming models

A linear programming model is constructed which enables a firm to estimate its competitor's cost structure when the competitor adheres to a non-randomized strategy. The existence of the model makes randomized bidding prudent apart from game theory considerations. The need for increased curriculum attention to optimal bidding is highlighted.     

The tetrahedron algebra and its finite-dimensional irreducible modules

Recently Terwilliger and the present author found a presentation for the three-point sl₂ loop algebra via generators and relations. To obtain this presentation we defined a Lie algebra ? by generators and relations and displayed an isomorphism from ? to the three-point sl₂ loop algebra. In this paper we classify the finite-dimensional irreducible ?-modules.     

On the Role of Mixed Strategies in Elementary Decision Analysis and Related Decision-Support-System Treatments

We first demonstrate that mixed strategies are relevant in decision analysis for a maximin decision-maker quite apart from any game-theory considerations. This rectifies the apparent misconception that results from MS/OR textbooks which discuss mixed strategies only in the game-theory setting. Next we show an example of an implementable mixed strategy, by which we mean a mixed decision strategy which does not require randomization for its implementation. This application is to portfolio construction.     

A topological approach to the theory of groups acting on trees

Bass and Serre recast the foundations of combinatorial group theory in [7]. Here we apply the allied notions of fundamental group and covering space to redevelop their theory in a less combinatorial fashion; for example the Bass—Serre Structure Theorem is proved with no a priori knowledge of the group theoretic structure of the fundamental group of a graph of groups. Van Kampen's Theorem is used only once, in its simplest form (in the proof of Theorem 7). Cancellation arguments and normal form theorems, such as Britton's Lemma, are completely avoided; indeed they are incidental corollaries from our viewpoint. The tree which plays a central role in [7] appears in Theorem 2 as the natural analogue of the “strecken komplexe” introduced by A. Speiser [8], and subsequently also employed by R. Nevanlinna [5], to describe certain simply connected Riemann surfaces occurring in value distribution theory.     

Pseudo-unitarizable weight modules over generalized Weyl algebras

We define a notion of pseudo-unitarizability for weight modules over a generalized Weyl algebra (of rank one, with commutative coefficient ring R), which is assumed to carry an involution of the form X^∗=Y, R^∗⊆R. We prove that a weight module V is pseudo-unitarizable iff it is isomorphic to its finitistic dual V^?. Using the classification of weight modules by Drozd, Guzner and Ovsienko, we obtain necessary and sufficient conditions for an indecomposable weight module to be isomorphic to its finitistic dual, and thus to be pseudo-unitarizable. Some examples are given, including U_q(sl₂) for q a root of unity.     

DEA based dimensionality reduction for classification problems satisfying strict non-satiety assumption

This study shows how data envelopment analysis (DEA) can be used to reduce vertical dimensionality of certain data mining databases. The study illustrates basic concepts using a real-world graduate admissions decision task. It is well known that cost sensitive mixed integer programming (MIP) problems are NP-complete. This study shows that heuristic solutions for cost sensitive classification problems can be obtained by solving a simple goal programming problem by that reduces the vertical dimension of the original learning dataset. Using simulated datasets and a misclassification cost performance metric, the performance of proposed goal programming heuristic is compared with the extended DEA-discriminant analysis MIP approach. The holdout sample results of our experiments shows that the proposed heuristic approach outperforms the extended DEA-discriminant analysis MIP approach.