Methodological reflections on a three-step-design combining observation,stimulated recall and interview

In this paper a tripartite qualitative design combining abservation, stimulated recall and interview is presented and discussed. This three-step-design makes it possible to get insight into the interaction of internal and external processes when solving mathematical tasks. The data analysis depends on the research question and the methodological approach. In the light of two research projects in mathematics education two different methods of data analysis are presented and methodologically reflected.     

The stable forking conjecture and generic structures

 We prove that for any simple theory which is constructed via Fr?issé-Hrushovski method, if the forking independence is the same as the d-independence then the stable forking property holds. Received: 22 January 2001 / Published online: 19 December 2002 This article is part of the author's D-Phil thesis, written at the University of Oxford and supported by the Ministry of Higher Education of Iran. The author would like to thank the Institute for Studies in Theoretical Physics and Mathematics (IPM), Tehran, Iran, for its financial support whilst working on this article. Mathematics Subject Classification (2000): 03C45 Key words or phrases: Generic structures – Fr?issé-Hrushovski method – Predimension – Simple theories – Stable theories – Stable forking conjecture     

An EOQ model with process reliability considerations

The classical economic order quantity (EOQ) model assumes that items produced are of perfect quality and that the unit cost of production is independent of demand. However, in realistic situations, product quality is never perfect, but is directly affected by the reliability of the production process. In this paper, we consider an EOQ model with imperfect production process and the unit production cost is directly related to process reliability and inversely related to the demand rate. In addition, a numerical example is given to illustrate the developed model. Sensitivity analysis is also performed and discussed.     

Near-Optimal Controls of Discrete-Time Dynamic Systems Driven by Singularly-Perturbed Markov Chains

This work is devoted to near-optimal controls of large-scale discrete-time nonlinear dynamic systems driven by Markov chains; the underlying problem is to minimize an expected cost function. Our main goal is to reduce the complexity of the underlying systems. To achieve this goal, discrete-time control models under singularly-perturbed Markov chains are introduced. Using a relaxed control representation, our effort is devoted to finding near-optimal controls. Lumping the states in each irreducible class into a single state gives rise to a limit system. Applying near-optimal controls of the limit system to the original system, near-optimal controls of the original system are derived.     

Approaches to sharing knowledge in group problem structuring

Problem-structuring techniques are an integral aspect of ‘Soft-OR’. SSM, SAST, Strategic Choice, and JOURNEY Making, all depend for their success on a group developing a shared view of a problem through some form of explicit modelling. The negotiated problem structure becomes the basis for problem resolution. Implicit to this process is an assumption that members of the group share and build their knowledge about the problem domain. This paper explores the extent to which this assumption is reasonable. The research is based on detailed records from the use of JOURNEY Making, where it has used special purpose Group Support software to aid the group problem structuring. This software continuously tracks the contributions of each member of the group and thus the extent to which they appear to be ‘connecting’ and augmenting their own knowledge with that of other members of the group. Software records of problem resolution in real organisational settings are used to explore the sharing of knowledge among senior managers. These explorations suggest a typology of knowledge sharing. The implications of this typology for problem structuring and an agenda for future research are considered.     

Inner Radius of Univalence for a Strongly Starlike Domain

 The inner radius of univalence of a domain D with Poincaré density ρ _D is the possible largest number σ such that the condition ∥ S _f ∥ _D  = sup _{w∈ D} ρ _D (w) ⁻²∥ S _f (z) ∥ ≤ σ implies univalence of f for a nonconstant meromorphic function f on D, where S _f is the Schwarzian derivative of f. In this note, we give a lower bound of the inner radius of univalence for strongly starlike domains of order α in terms of the order α. The author was partially supported by the Ministry of Education, Grant-in-Aid for Encouragement of Young Scientists, 11740088. A part of this work was carried out during his visit to the University of Helsinki under the exchange programme of scientists between the Academy of Finland and the JSPS. Received November 26, 2001; in revised form September 24, 2002 Published online May 9, 2003