CAS-supported multiple representations in elementary linear algebra

Usually the Gaussian algorithm (GA) is presented at school as a method of solving a given system of linear equations by reducing it to a “triangular form”. In contrast to this technically oriented view, I will demonstrate a CAS-supported learning environment which includes a visual representation of GA and an activity-oriented «Gauss-game». This game stresses the concept of elementary matrices and leads directly to a partial, implementation of GA in the form of a «semi-automatic” functional CAS-program. These multiple representation of GA tries to take into consideration the research results on mental representations, to design rich variations of student activities and thereby to lead leading to webbeb concepts around GA. The CAS MuPAD is used.     

Die Entwicklung kausaler Hypothesen in der qualitativen Sozialforschung

The disregard of causal inference in the methodological literature about qualitative research is highly problematic, since the category of causality is closely linked to the concept of social action. However, it is also clear that causal analysis is burdened with certain difficulties and methodological challenges in the realm of social research. Some of these problems are discussed in this article using Mackie—s concept of 3 “INUS”-conditions. Thereby it will be shown that strategies of causal analysis based on comparative methods proposed for qualitative research, namely “Analytic Induction” and “Qualitative Comparative Analysis” have great difficulties in dealing adequately with these problems. They can only be solved, if case-comparative methods are combined with explorative research strategies which support the researcher in gaining access to the local knowledge of the research field.     

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.     

CAS based approach for discussing subset construction

This paper continues the discussion of teaching nondeterminism (see [6]) where we presented a didactic approach introducing the notion of nondeterministic automata. Although in this paper we use the same methodology we have to face up to new didactic challenges. Namely, teaching the subset construction requires answers to the question how can CAS be used in teaching the different phases of mathematical problem solving so that we can reach higher cognitive efficiency.     

<Emphasis Type="Italic">L</Emphasis><Subscript>1</Subscript> linear interpolator for missing values in time series

We propose a minimum mean absolute error linear interpolator (MMAELI), based on theL ₁ approach. A linear functional of the observed time series due to non-normal innovations is derived. The solution equation for the coefficients of this linear functional is established in terms of the innovation series. It is found that information implied in the innovation series is useful for the interpolation of missing values. The MMAELIs of the AR(1) model with innovations following mixed normal andt distributions are studied in detail. The MMAELI also approximates the minimum mean squared error linear interpolator (MMSELI) well in mean squared error but outperforms the MMSELI in mean absolute error. An application to a real series is presented. Extensions to the general ARMA model and other time series models are discussed. This research was supported by a CityU Research Grant and Natural Science Foundation of China.     

Invariant ideals of twisted crossed products

We will prove a result concerning the inclusion of non-trivial invariant ideals inside non-trivial ideals of a twisted crossed product. We will also give results concerning the primeness and simplicity of crossed products of twisted actions of locally compact groups on -algebras. Received: 25 January 2002; in final form: 22 May 2002/Published online: 2 December 2002 This work is partially supported by Hong Kong RGC Direct Grant.     

Integral kašin splittings

Forx ∈ ℝ ⁿ andp≥1 put ‖x‖ _p :=(n ⁻¹Σ|x _i| ^p )^1/p . An orthogonal direct sum decomposition ℝ^2k =E⊕E ^⊥ where dimE=k and ‖x‖₂/‖x‖₁≤C is called here a (k, C)-splitting. By a theorem of Kašin there existsC>0 such that (k, C)-splittings exist for allk, and by the volume ratio method of Szarek one can takeC=32eπ. All proofs of existence of (k, C)-splittings heretofore given are nonconstructive. Here we investigate the representation of (k, C)-splittings by matrices with integral entries. For everyC>8e _1/2 π _−1/2 and positive integerk we specify a positive integerN(k, C) such that in the set ofk by 2k matrices with integral entries of absolute value not exceedingN(k, C) there exists a matrix with row span a summand in a (k, C)-splitting. We haveN(k, C)≤2^18k fork large enough depending onC. We explain in detail how to test a matrix for the property of representing a (k, C)-splitting. Taken together our results yield an explicit (if impractical) construction of (k, C)-splittings.