Towards a framework for developing students' fraction proficiency

The importance of the knowledge of fractions in mathematical learning, coupled with the difficulties students have with them, has prompted researchers to focus on this particular area of mathematics. The term ‘fraction proficiency' used in this article refers to a person's conceptual comprehension, procedural skills and the ability to approach daily situations involving fractions. In the area of fractions, there has been a call for more research to study how, and where, efforts should be focused in order to integrate the various aspects of fraction knowledge for students, and even for teachers, to help them develop proficiency in fractions. Thus, the article presents a theoretical synthesis of the specialized literature in the learning and teaching of fractions, with the aim of proposing a framework for developing students' fraction proficiency. The frameworks presented in the article may shed light upon the implications for the design of fraction instruction, which should focus on developing a multi-faceted knowledge of fractions, rather than simply isolating one facet from the others.     

Towards a new framework for evaluating systemic problem structuring methods

Operational researchers and social scientists often make significant claims for the value of systemic problem structuring and other participative methods. However, when they present evidence to support these claims, it is usually based on single case studies of intervention. There have been very few attempts at evaluating across methods and across interventions undertaken by different people. This is because, in any local intervention, contextual factors, the skills of the researcher and the purposes being pursued by stakeholders affect the perceived success or failure of a method. The use of standard criteria for comparing methods is therefore made problematic by the need to consider what is unique in each intervention. So, is it possible to develop a single evaluation approach that can support both locally meaningful evaluations and longer-term comparisons between methods? This paper outlines a methodological framework for the evaluation of systemic problem structuring methods that seeks to do just this.     

Evaluating Web Services: Towards a framework for emergent contexts

The paper develops an evaluation framework with specific reference to Web Services. It is argued that the essential characteristics for such an approach, noted as qualitative, are captured in these constructs through an augmentation of theoretical considerations and empirical findings. A review of the innovation and diffusion literature indicates a considerable amount of research where attention is given to a range of features which may support Web Service adoption. It is argued that the framework proposed in this paper is of value in highlighting the specific situations for an effective evaluation in this respect.The empirical analysis was undertaken within five UK firms. Each of these organizations represent different business sectors including, Telecommunication (two firms), Independent Software Package Vendor (ISV), Software Engineering and Food Services industries. The study involved a cross-section organizational case approach within each situation for factors affecting adoption. The evidence suggests that the empirical data complements traditional innovation and diffusion theories. However, emergent context is identified as a significant influence on potential web evaluation processes to ensure appropriate investment targeting. The findings are noted to guide analysts in determining critical aspects of the complex issues involved and present suggestions for further valid research.     

Algorithms for two-dimensional transfer function computation

Algorithms for the computation of two-dimensional transfer functions from state-space configurations are derived. The principal steps in the programming procedure are provided.     

Towards an auction-driven gas supply: a simulation-based optimization framework for utilities

A platform for the study of the whole transmission problem (arrival of ships, regasification, transportation and distribution) faced by gas utilities companies is proposed. The main objective of this research is to develop a platform that includes the analysis of the new capacity auctions (and not the traditional commodity auctions) that will govern the supply chain in the near future. A simulation-optimization approach has been used to favour the more realistic abstraction of the system. The discrete-event model includes a genetic algorithm to reach the solution in a satisfactory short time, a requisite in auction markets. Design and optimization studies for the utilities are addressed using the platform, which has been validated with real data for one of the main zones in the Spanish market.     

A lower estimate for the computation complexity of a ternary linear function

The computation complexity of a ternary linear function of n variables is found to be at least $n^2 + \tfrac{3} {2}n - o(n)$ .     

Algorithm for the computation of Bessel function integrals

A fortran subroutine is given for the computation of integrals of the form ∫^c₀f(x)J_v(αx)dx, where v = 0, 1,…,10.     

Chebyshev lattices,a unifying framework for cubature with Chebyshev weight function

We present a multivariate extension to Clenshaw-Curtis quadrature based on Sloan’s hyperinterpolation theory. At the centre of it, a cubature rule for integrals with Chebyshev weight function is needed. We introduce so called Chebyshev lattices as a generalising framework for the multitude of point sets that have been discussed in this context. This framework provides a uniform notation that extends easily to higher dimensions. In this paper we describe many known point sets as Chebyshev lattices.     

Modified clenshaw-curtis method for the computation of Bessel function integrals

The numerical evaluation of Bessel function integrals may be difficult when the Bessel function is rapidly oscillating in the interval of integration. In the method presented here, the smooth factor of the integrand is replaced by a truncated Chebyshev series approximation and the resulting integral is computed exactly. The numerical aspects of this exact integration are discussed.     

Towards data driven selection of a penalty function for data driven Neyman tests

The data driven Neyman statistic consists of two elements: a score statistic in a finite dimensional submodel and a selection rule to determine the best fitted submodel. For instance, Schwarz BIC and Akaike AIC rules are often applied in such constructions. For moderate sample sizes AIC is sensitive in detecting complex models, while BIC works well for relatively simple structures. When the sample size is moderate, the choice of selection rule for determining a best fitted model from a number of models has a substantial influence on the power of the related data driven Neyman test. This paper proposes a new solution, in which the type of penalty (AIC or BIC) is chosen on the basis of the data. The resulting refined data driven test combines the advantages of these two selection rules.     

Towards the computation of minimum drag profiles in viscous laminar flow

A numerical method is given for the solution of certain optimum design problems of fluid mechanics. The profile of given area and smallest drag in a uniform laminar flow is computed. This profile is long and slim, its front end is shaped like a wedge of angle 90° and its rear end is shaped like a cusp. Owing to the numerical complexity of the problem the precision of the results is average (around 5%). However, this work is a good illustration of the theoretical method exposed previously and it shows how good precision can be obtained if one is prepared to pay for it. A numerical solution of the adjoint system of the stationary Navier-Stokes equation is also given; this equation will play an important role in optimum design in fluid mechanics.     

On the computation of an element of Clarke generalized Jacobian for a vector-valued max function

In this paper, we present an algorithm for calculating an element of Clarke generalized Jacobian for a vector-valued max-type function. The algorithm reduces the computational cost of an existing algorithm.     

A stochastic framework for recursive computation of spline functions: Part II,smoothing splines

Many of the optimal curve-fitting problems arising in approximation theory have the same structure as certain estimation problems involving random processes. We develop this structural correspondence for the problem of smoothing inaccurate data with splines and show that the smoothing spline is a sample function of a certain linear least-squares estimate. Estimation techniques are then used to derive a recursive algorithm for spline smoothing.     

Some new Jacobi elliptic function solutions for the short-pulse equation via a direct symbolic computation method

Applying a direct symbolic computation method combined with variable transformations, some new Jacobi elliptic function solutions are obtained to the short-pulse equation in nonlinear optics. When Jacobi elliptic function modulus m??1 or?0, the travelling wave solutions degenerate to two types of solutions, namely, the loop-like soliton solution and the trigonometric function solution.     

Scale characterizations in a DEA directional technology distance function framework

The present study extends prior research on data envelopment analysis (DEA)-based returns to scale in view of the fact that the standard methods are very restrictive. The extension is made in the context of the directional technology distance function that generalizes the Shephard input and output distance functions. The two main results in this extended setting are as follows. First, we show how Banker's most productive scale size (MPSS) concept is characterized. Second, we obtain scale elasticity measures and examine the properties and the relationships with standard scale elasticity measures.     

On the numerical computation of the Mittag-Leffler function

Recently simple limiting functions establishing upper and lower bounds on the Mittag-Leffler function were found. This paper follows those expressions to design an efficient algorithm for the approximate calculation of expressions usual in fractional-order control systems. The numerical experiments demonstrate the superior efficiency of the proposed method.     

Uniqueness and computation of Gaussian interval quadrature formula for Jacobi weight function

Summary. In this paper we prove the uniqueness of the Gaussian interval quadrature formula for the Jacobi weight function. An algorithm for numerical construction has also investigated and some suitable solutions are proposed. For the special case of Chebyshev weight of the first kind and the special set of lengths we give an analytic solution.Mathematics Subject Classification (2000): 65D32, 65D30, 41A55The authors were supported in part by the Serbian Ministry of Science and Environmental Protection (Project #2002: Applied Orthogonal Systems, Constructive Approximation and Numerical Methods).     

An algorithm for the computation of the reversible Randles-Sevcik function in electrochemistry

An algorithm for the numerical approximation of the reversible Randles-Sevcik function has been developed. The algorithm makes use of a piecewise nonlinear approximation that can be employed on a digital computer using FORTRAN.