Using Dynamic Geometry Software to Add Contrast to Geometric Situations – A Case Study

There is no lack of suggestions concerning how Dynamic Geometry Software (DGS) may support heuristic approaches to problem solving. However, uses of DGS are often limited purely to a verifying role, in the sense that students are expected to vary or confirm empirically at the computer geometric data which are more or less given. By contrast, it seems worthwhile to seek other uses of DGS which go beyond mere confirmation so that the geometric situation is recognised in its particularity. This paper provides a case study that emerged from a project in which DGS formed an integral part of the pedagogical arrangement. The study is intended to show how the contrasting power of DGS might be utilised in a guided discovery setting.This revised version was published online in September 2005 with corrections to the Cover Date.     

A Further Study of the Choice Between Two Hedging Strategies–the Continuous Case

In our previous work, the choice between two popular hedging strategies was studied under the assumption that the hedge position of the underlying portfolio follows a discrete-time Markov chain with boundary conditions. This paper aims to investigate the same problem for the continuous case. We first assume that the underlying hedge position follows an arbitrary continuous-time Markov process; we give the general formulas for long-run cost per unit time under two cost structures: (1) a fixed transaction cost (2) a non-fixed transaction cost. Then we consider the case where the underlying hedge position follows a Brownian motion with drift; we show that (i) re-balancing the hedge position to the initial position is always more cost-efficient than re-balancing it to the boundary for a fixed transaction cost; (ii) when the cost function satisfies certain conditions, re-balancing the hedge position to the initial position is more cost-efficient than re-balancing it to the boundary for a non-fixed transaction cost.     

Uncertainty Evaluation in Multivariate Analysis – A Test Case Study

We have used different multivariate analysis methods to estimate quantities in the fields of food control and atmospheric remote sensing. In order to estimate the uncertainties in these estimates we studied analytical as well as non-parametric numerical methods. The methods have been evaluated by comparison between obtained results and independent sets of measurements. We present one test case from each field, including results, where these methods have been applied. For the food control test case reduced chi-squared of approximately unity indicate that both the analytical and numerical methods used for uncertainty estimation produce uncertainties of reasonable size. In the atmospheric remote sensing test case, a indicated that the uncertainties from the numerical method were far too small, whereas a indicate that the size of the analytically determined uncertainties can represent the size of the “true” errors.     

A Dynamic Analysis of Schelling’s Binary Corruption Model: A Competitive Equilibrium Approach

Schelling (in Micromotives and Macrobehavior, Norton, New York, 1978) suggested a simple binary choice model to explain the variation of corruption levels across societies. His basic idea was that the expected profitability of engaging in corruption depends on its prevalence. The key result of the so-called Schelling diagram is the existence of multiple equilibria and a tipping point. The present paper puts Schelling’s essentially static approach into an intertemporal setting. We show how the existence of an unstable interior steady state leads to thresholds such that history alone or history in addition to expectations (or coordination) is necessary to determine the long-run outcome. In contrast to the related literature, which classifies these two cases according to whether the unstable equilibrium is a node or a focus, the actual differentiation is more subtle because even a node can lead to an overlap of solution paths such that the initial conditions alone are insufficient to uniquely determine the competitive equilibrium. Another insight is that a (transiently) cycling competitive equilibrium can dominate the direct and monotonic route to a steady state, even if the direct route is feasible.     

Stability Tests for Two Common Classes of Linear Time‐Delay and Hybrid Systems

The stability of linear timedelay systems with point internal delays is difficult to deal with in practice because of the fact that their characteristic equation is usually of transcendent type rather than of polynomial type. This feature usually causes the system to possess an infinite number of poles. In this paper, stability tests for this class of systems are obtained based either on extensions of classical tests applicable to delayfree systems or on approaches within the framework of twodimensional digital filters. Some of those twodimensional stability tests are also proved to be useful for stability testing of a common class of linear hybrid systems which involve coupled continuous and digital substates after a slightadhoc adaptation of the tests for that situation.     

The Stock Control of Engineering Spares—A Case Study

This paper discusses the experience of an operational research group in developing a practical scheme for the control of spares in an organization with many central and subsidiary stores. It covers the development of the scheme from the choice of the method of control to the publication of control tables for use throughout the organization; it is complementary to earlier papers on the methods of cost measurement and the mathematical analysis.Reasons are given for choosing a re-order level scheme rather than a cyclical review scheme, and for using run out costs rather than levels of service. The effect of errors in estimating physical and economic parameters is discussed; in practice not all the parameters need to be considered explicitly. Several modifications made to meet practical objections are described. The paper ends with some comments on possible future developments.     

The Geographical Allocation of Community Care Resources — A Case Study

The paper presents a case study of methods of allocating a given amount of home help resource between a number of geographical areas within a County Social Services Department. Questionnaires were completed for a sample of clients covering such information as:

1. i)
   the type of client,
    
2. ii)
   the amount of home help actually received, and
    
3. iii)
   the "ideal" amount of help judged necessary.
    

Various allocations were presented to the decision makers, ranging from the simple assumption that the number of hours provided to each type of client should be the same throughout the County, to more "sophisticated" allocations which took account of such geographical differences as the level of provision of other support services and the current and forecast population structure.Although developed within the context of the home help service, the approach is general enough to be applicable to the allocation of other such services to clients, e.g. home nursing.     

Model Based Control of a Parallel Robot – A Comparison of Control Algorithms

In this contribution the control behavior of a special construction of a parallel robot, called multi-axes test facility, is investigated. After a brief discussion of the different tasks of the robot the construction of the robot is briefly presented. To solve the tasks, different control algorithms are derived based on model equations of different complexity of the robot. Depending on the task to be performed by the robot, the controllers compensate the kinematic and/or kinetic coupling of the degrees of freedom of the robot, stabilize the system and achieve the desired spatial motion of each degree of freedom as well as sufficient robustness with respect to parameter uncertainties and load variations. A few results obtained in computer simulations and laboratory experiments are presented and judged with respect to the quality of control, the closeness to reality of the computer simulations, and the amount of costs and work needed to realize the different solutions.     

Evaluation of Transport Costs for Alternative Factory Sites — A Case Study

The problem of siting one or more factories so as to minimize transport costs is considered in the context of other factors of interest to management. The method of solution is described. This involved the solution of simple transportation problems, the determination of rules by which costs could be computed, the construction of an analogue machine to assist with the problem of site selection and the writing of a programme for a digital computer to cope with the heavy load of calculation. The way in which the work of the operational research team was organized to meet a deadline is described and a number of technical difficulties are discussed. The solution consisted of the presentation of minimum transport costs for each of a large number of possible combinations of factories so that management could assess the relative importance of transport costs together with other factors when deciding where to build factories for the manufacture of a new product.     

Dynamic flow modelling in precipitator vessels – A study of turbulence modelling approaches

Precipitation is a fundamental unit process within the Bayer circuit for the production of smelter-grade alumina, with the predominant technology used in this unit process being the mechanically-agitated draft-tube precipitator. Previous physical modelling studies have identified dynamic behaviour in these vessels which could potentially impact fluid residence time and the ability to obtain adequate solids suspension. In the current study the commercial CFD code ANSYS CFX14 is used to simulate the dynamic, single phase flow behaviour in a laboratory-scale replica of a full-scale precipitator. Simulations are conducted to investigate the impact of mesh refinement and turbulence model selection, with two-equation, Reynolds Stress and the Scale Adaptive Simulation (SST–SAS) models being investigated. The impact of the different modelling choices on the accuracy of the simulations is assessed through comparison of the CFD results with high-quality Laser Doppler Velocimetry (LDV) data obtained in a laboratory-scale vessel.     

Fatigue crack growth under variable-amplitude loading: Part I – Model formulation in state-space setting

This two-part paper presents formulation and validation of a non-linear dynamical model of fatigue crack growth in ductile alloys under variable-amplitude loading including single-cycle overloads, irregular sequences, and random loads. The model is formulated in the state-space setting based on the crack closure concept and captures the effects of stress overload and reverse plastic flow. The state variables of the model are crack length and crack opening stress. This paper, which is the first part, presents formulation of the state-space model that can be restructured as an autoregressive moving average (ARMA) model for real-time applications such as health monitoring and life extending control. The second part is the companion paper that is dedicated to model validation with fatigue test data under different types of variable-amplitude and spectrum loading.     

Allocation of Doctors to Health Centres in Haryana State of India — A Case Study

Medical Administrators are usually confronted with the problem of determining the number of doctors to be posted at different health centres within their jurisdiction. In India the number of doctors allocated to a health centre is normally decided without any proper study of the health needs of the area served by the centre. In certain areas the number of doctors posted is considerably different from the requirement of the area. Also, in certain health centres situated in villages lacking in modern amenities, absenteeism among doctors poses a very serious problem in day to day running of the health centres. The problem of allocation is formulated and a heuristic method is suggested for determining the optimum number of doctors to be posted at each health centre in order to maximise the number of patients seen by the doctor per day. The heuristic method is applied to nine health centres of Haryana state of India in order to demonstrate the potential benefits.     

A Numerical Study of Traveling Wave Fronts for a Reaction-Diffusion-Advection Model

This paper concerns with the traveling wave solutions of a nonlinear reaction-diffusion-advection model for describing the spatiotemporal evolution of bacterial colony pattern. We use different methods for computing the traveling wave fronts of the model equations. One of the methods involves the traveling wave equations. Numerical solutions of these equations as an initial-value problem lead to accurate computations of the wave profiles and speeds. The second method is to construct the time-dependent solutions by solving an initial-moving boundary-value problem for the PDE system, showing an approximation for such wave fronts, in particular, the minimum speed traveling wave.     

Exterior Differential Systems Prolongations and Application to a Study of Two Nonlinear Partial Differential Equations

A generalized Korteweg-de Vries equation is formulated as an exterior differential system, which is used to determine the prolongation structure of the equation. The prolongation structure is obtained for several cases of the variable powers, and nontrivial algebras are determined. The analysis is extended to a differential system which gives the Camassa-Holm equation as a particular case. The subject of conservation laws is briefly discussed for each of the equations. A Bäcklund transformation is determined using one of the prolongations.     

A Hybrid Micro­Macro­Model for the Description of Evolving Anisotropy in Finite Polycrystal Plasticity

The paper outlines recent developments of an efficient computational micro-macro modeling of evolving anisotropies in metallic polycrystals. Main focus is put onto large strain deformation processes where the anisotropy is caused by developments of crystallographic texture. We construct a hybrid micro-macro framework that mixes ingredients of a purely macroscopic modeling with scale bridging operations of selected micromechanisms. On the micromechanical side, we develop a new algorithmic procedure to capture the crystal reorientation for evolving fcc and bcc textures based on a parametrization of rotations in the Rodigues space. The computational model provides a fast and robust method for the estimation of evolving textures. This computational tool for texture estimation is incorporated in a modular format into a micro-macro-model, where it governs the evolution of macrostructural tensors due to texture development. The general framework for the hybrid embedding is a purely phenomenological setting of anisotropic finite plasticity in the logarithmic strain space. The model provides an efficient and computationally handable two-scale approach for the prediction of effects caused by complex microstructural changes in polycrystals. The capability of the proposed method is demonstrated by means of representative numerical examples. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On a Modification of Olver’s Method: A Special Case

We consider the asymptotic method designed by Olver (Asymptotics and special functions. Academic Press, New York, 1974) for linear differential equations of the second order containing a large (asymptotic) parameter \(\Lambda \): \(x^my''-\Lambda ^2y=g(x)y\), with \(m\in \mathbb {Z}\) and g continuous. Olver studies in detail the cases \(m\ne 2\), especially the cases \(m=0, \pm 1\), giving the Poincaré-type asymptotic expansions of two independent solutions of the equation. The case \(m=2\) is different, as the behavior of the solutions for large \(\Lambda \) is not of exponential type, but of power type. In this case, Olver’s theory does not give many details. We consider here the special case \(m=2\). We propose two different techniques to handle the problem: (1) a modification of Olver’s method that replaces the role of the exponential approximations by power approximations, and (2) the transformation of the differential problem into a fixed point problem from which we construct an asymptotic sequence of functions that converges to the unique solution of the problem. Moreover, we show that this second technique may also be applied to nonlinear differential equations with a large parameter.     

Shortcomings of Mathematics Education Reform in The Netherlands: A Paradigm Case?

This article offers a reflection on the findings of three PhD studies, in the domains of, respectively, subtraction under 100, fractions, and algebra, which independently of each other showed that Dutch students' proficiency fell short of what might be expected of reform in mathematics education aiming at conceptual understanding. In all three cases, the disappointing results appeared to be caused by a deviation from the original intentions of the reform, resulting from the textbooks' focus on individual tasks. It is suggested that this “task propensity”, together with a lack of attention for more advanced conceptual mathematical goals, constitutes a general barrier for mathematics education reform. This observation transcends the realm of textbooks, since more advanced conceptual mathematical understandings are underexposed as curriculum goals. It is argued that to foster successful reform, a conscious effort is needed to counteract task propensity and promote more advanced conceptual mathematical understandings as curriculum goals.     

A model for fluid flow in non-fully filled tribological interfaces – Part 1: Basics and Numerics

In literature, most contributions on starved lubrication focus on the occurring pressures in macroscopic devices. Hereby, usually the Reynolds equation is modified in different ways. In contrast to this proceeding, this paper's intention is the general investigation of this tribological regime to get a fundamental comprehension on the transition from boundary lubrication to mixed lubrication. The respective model describes the flow of the fluid through two rough surfaces moving relative to each other. The lack of fluid is regarded by the fact that elements may not be fully filled with the fluid. Only elements where the fluid fully fills the gap, generate a pressure. This effect is considered by a type of unilateral constraint in combination with a penalty function. The fluid flow is computed according to the Navier-Stokes equation. In combination with the continuity equation, a set of implicit nonlinear equations has to be solved. Its potential and basic application fields are finally discussed. A further paper will show applications of the algorithm towards different scenarios. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)