Spanning cactus of a graph: Existence,extension, optimization,and approximation

We show that testing if an undirected graph contains a bridgeless spanning cactus is NP-hard. As a consequence, the minimum spanning cactus problem (MSCP) on an undirected graph with 0–1 edge weights is NP-hard. For any subgraph $S$ of $K_n$, we give polynomially testable necessary and sufficient conditions for $S$ to be extendable to a cactus in $K_n$ and the weighted version of this problem is shown to be NP-hard. A spanning tree is shown to be extendable to a cactus in $K_n$ if and only if it has at least one node of even degree. When $S$ is a spanning tree, we show that the weighted version can also be solved in polynomial time. Further, we give an $O\left(n^3\right)$ algorithm for computing a minimum cost spanning tree with at least one vertex of even degree on a graph on $n$ nodes. Finally, we show that for a complete graph with edge-costs satisfying the triangle inequality, the MSCP is equivalent to a general class of optimization problems that properly includes the traveling salesman problem and they all have the same approximation hardness.     

Systemic evaluation: a participative,multi-method approach

This paper presents some new ideas on systemic evaluation developed in the context of a project to support capacity building for the evaluation of community health services. Emphases are placed on the need for stakeholder participation; dialogue on the boundaries of evaluations; considering multiple values; and ensuring that marginalized people and issues are properly accounted for. Further developing the work of previous authors, three different approaches to evaluation are outlined, each of which can be applied participatively: goal-based (where goals are set and their achievement is measured); stakeholder (where there are no pre-set goals, and different people's experiences and stories are surveyed to reveal significant issues); and organizational (where organizational processes are compared with models of good practice). There is a logical relationship between these: a stakeholder evaluation can lead to the setting of community-sensitive goals, the achievement of which can be measured through goal-based evaluation, and pursuit of the goals can be enhanced by organizational evaluation. A participative, flexible and responsive evaluation practice will often need to draw upon aspects of all three approaches. Initial feedback from health practitioners suggests that this way of thinking about systemic evaluation may prove useful in a range of situations faced by people in the statutory, voluntary and community sectors.     

Researcher,Teacher, Education Researcher: The Evolution of a University Geoscience Instructor

This case study describes a professor's evolution from geoscience researcher to effective teacher to education researcher. The article details his initial beliefs about teaching, looks at the factors that prompted him to seek a different teaching approach, and enumerates the supports and challenges that he had on his journey. Factors essential to this evolution are early career success in discipline research, an institutional climate to reward teaching, mentoring support by colleagues, access to professional development opportunities, and involvement in action research activities. The case study is linked to education literature about teaching and education research and makes recommendations based on the findings of the study.     

Stability,structural stability,and games: Toward a theory of differential hypergames

Differential games (DG's) are investigated from a stability point of view. Several resemblances between the theory of optimal control and that of structural stability suggest a differential game approach in which the operators have conflicting interests regarding the stability of the system only. This qualitative approach adds several interesting new features. The solution of a differential game is defined to be the equilibrium position of a dynamical system in the framework of a given stability theory: this is the differential hypergame (DHG). Three types of DHG are discussed: abstract structural DHG, Liapunov DHG, and Popov DHG. The first makes the connection between DG and the catastrophe theory of Thom; the second makes the connection between the value function approach and Liapunov theory; and the third provides invariant properties for DG's. To illustrate the fact that the theory sketched here may find interesting applications, the up-to-date problem of the world economy is outlined.This research was supported by the National Research Council of Canada.     

Modelling,simulation, animation,and real-time control (MoSART) for a class of electromechanical systems: a system-theoretic approach

This paper describes an interactive modelling, simulation, animation, and real-time control (MoSART) environment for a class of ‘cart-pendulum’ electromechanical systems that may be used to enhance learning within differential equations and linear algebra classes. The environment is useful for conveying fundamental mathematical/systems concepts through computer-aided analysis, design, graphical visualization, and 3D animation. Referred to as Cart-Pendulum Control3D-Lab1Control3D-Lab-short for control 3D animation laboratory. The author to whom correspondence should be addressed., the environment is based on Microsoft Windows/Visual C++/Direct-3D and MATLAB/Simulink2MATLAB and Simulant are trademarks of The Masterworks, Inc.. Pull-down menus provide access to systems/models/control laws, exogenous signals, parameters, animation models and visual indicators, a suite of (easy-to-modify) Simulant diagrams containing models and control laws, MATLAB m-files for detailed analysis and design, and detailed documentation for each of the above. Three blocks enable animation, joystick inputs, and real-time animation within Simulant. Examples are presented to illustrate the utility of the environment as a valuable tool for analysing/visualizing the above class of electromechanical systems and for enhancing mathematics instruction.     

Efficient optimisation of a vehicle suspension system, using a gradient-based approximation method, Part 1: Mathematical modelling

A methodology is proposed for the efficient determination of gradient information, when performing gradient based optimisation of an off-road vehicle’s suspension system. The methodology is applied to a computationally expensive, non-linear vehicle model, that exhibits severe numerical noise. A recreational off-road vehicle is modelled in MSC.ADAMS, and coupled to MATLAB for the execution of the optimisation. The successive approximation method, Dynamic-Q, is used for the optimisation of the spring and damper characteristics. Optimisation is performed for both ride comfort and handling. The determination of the objective function value is performed using computationally expensive numerical simulations.This paper proposes a non-linear pitch-plane model, to be used for the gradient information, when optimising ride comfort. When optimising for handling, a non-linear four wheel model, that includes roll, is used. The gradients of the objective function and constraint functions are obtained through the use of central finite differences, within Dynamic-Q, via numerical simulation using the proposed simplified models. The importance of correctly scaling these simplified models is emphasised. The models are validated against experimental results. The simplified vehicle models exhibit significantly less numerical noise than the full vehicle simulation model, and solve in significantly less computational time.     

Oscillations of a simple valve connected to a pipe: Part II,experiments

Summary Oscillations of a valve connected to a pipe were treated in [1]. The amplification and frequency of the oscillations were calculated with simplifying assumptions. In the present paper, they are compared with experiments. Good agreement is found for a large range of parameters.

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Zusammenfassung In [1] wurde das Schwingungsverhalten von einem Ventil, das an ein Rohr angeschlossen ist, untersucht. Mit vereinfachenden Annahmen wurden Anfachung und Frequenz der Ventilbewegung berechnet. Die Messungen, die in der Zwischenzeit durchgeführt wurden, zeigen eine gute Uebereinstimmung mit der Theorie von [1] für einen grossen Bereich der Parameter.

     

The Structure of a Linear Model: Sufficiency, Ancillarity, Invariance, Equivariance, and the Normal Distribution

Consider a general linear model Y=Xβ+Z where Cov Z may be known only partially. We investigate carefully the notions of sufficiency, ancillarity, invariance, and equivariance and related notions for projectors in a general linear model. In this way we can prove a Basu type theorem. This result can be used to give the relation between the sufficiency of the generalized least-squares estimator and the assumption that Z is normally distributed. So we can generalize the well-known result that the generalized least-squares estimator is sufficient for β if Z is normally distributed. Further we can solve the converse problem as well.     

Nonlinear Oscillations of a Spring Pendulum at the 1 : 1 : 2 Resonance: Theory,Experiment, and Physical Analogies

Nonlinear spatial oscillations of a material point on a weightless elastic suspension are considered. The frequency of vertical oscillations is assumed to be equal to the doubled swinging frequency (the 1 : 1 : 2 resonance). In this case, vertical oscillations are unstable, which leads to the transfer of the energy of vertical oscillations to the swinging energy of the pendulum. Vertical oscillations of the material point cease, and, after a certain period of time, the pendulum starts swinging in a vertical plane. This swinging is also unstable, which leads to the back transfer of energy to the vertical oscillation mode, and again vertical oscillations occur. However, after the second transfer of the energy of vertical oscillations to the pendulum swinging energy, the apparent plane of swinging is rotated through a certain angle. These phenomena are described analytically: the period of energy transfer, the time variations of the amplitudes of both modes, and the change of the angle of the apparent plane of oscillations are determined. The analytic dependence of the semiaxes of the ellipse and the angle of precession on time agrees with high degree of accuracy with numerical calculations and is confirmed experimentally. In addition, the problem of forced oscillations of a spring pendulum in the presence of friction is considered, for which an asymptotic solution is constructed by the averaging method. An analogy is established between the nonlinear problems for free and forced oscillations of a pendulum and for deformation oscillations of a gas bubble. The transfer of the energy of radial oscillations to a resonance deformation mode leads to an anomalous increase in its amplitude and, as a consequence, to the break-up of a bubble.     

On forests, trees, elephants, and classrooms: a brief for the study of learning ecologies

The classroom intervention studies in this volume, ranging from the study of gestures to that of systemic implementation, are at very different grain sizes. A challenge is to see the forest for the trees – to see how these studies, focusing on different aspects of mathematical activity at different grain sizes, can be seen as aspects of a coherent whole. I propose an ecological metaphor for the study of mathematical activity. In ecological terms, the biosphere is comprised of interconnected and interrelated ecosystems. I argue that, analogously, there are nested and interrelated mathematical activity systems and structures in which “mathematical sense-making” plays the role of “health” in ecosystems. Moreover, what happens in the classroom environment shapes and is shaped by what happens in sub-ecologies of the classroom (e.g., sociomathematical norms, participation structures, communicational forms such as gesture, and representational tools and their use) and the larger social and organizational ecologies of which it is a part (building culture, support structures for teachers and teaching, external pressures such as testing and accountability systems etc).     

Steady-states for shear flows of a liquid-crystal model: Multiplicity, stability, and hysteresis

In this work, we study shear flows of a fluid layer between two solid blocks via a liquid-crystal type model proposed in [C.H.A. Cheng, L.H. Kellogg, S. Shkoller, D.L. Turcotte, A liquid-crystal model for friction, Proc. Natl. Acad. Sci. USA 21 (2007) 1-5] for an understanding of frictions. A characterization on the existence and multiplicity of steady-states is provided. Stability issue of the steady-states is examined mainly focusing on bifurcations of zero eigenvalues. The stability result suggests that this simple model exhibits hysteresis, and it is supported by a numerical simulation.