Pre-service mathematics teachers become full participants in inquiry investigations

Research is described concerning the effectiveness of inquiry-based laboratory environments created in US mathematics/science education programme courses. Laboratory projects were conducted using a framework that allowed pre-service teachers to explore, analyse, and communicate ‘investigable’ realms of physical phenomena. Goals were for pre-service teachers to experience the value of learning in an inquiry-enhanced environment and to engage in contextualized mathematics so they would utilize this instruction in their future classrooms. It is proposed that inquiry-based laboratories are needed within the mathematics classroom in order to allow students the opportunity to contextualize, to connect to other disciplines, and to experience mathematical concepts. Pre-service teachers were expected to pursue conjectures, collect data, think critically, and communicate findings. This qualitative research shows how the use of inquiry can complement the learning of mathematical content and educational strategies for pre-service teachers. Results provide detailed information for teacher educators regarding instructional design of contextualized mathematical inquiry.     

Modelling alcohol problems: total recovery

Binge drinking in the UK is an increasing problem, resulting in negative health, social and economic effects. Mathematical modelling allows for future predictions to be made and may provide valuable information regarding how to approach solving the problem of binge drinking in the UK. We develop a 3-equation model for alcohol problems, specifically binge drinking, which allows for total recovery. Individuals are split into those that are susceptible to developing an alcohol problem, those with an alcohol problem and those in treatment. We find that the model has two equilibrium points: one without alcohol problems and one where alcohol problems are endemic in the population. We compare our results with those of an existing model that does not allow for total recovery. We show that without total recovery, the threshold for alcohol problems to become endemic in the population is lowered. The endemic equilibrium solution is also affected, with an increased proportion of the population in the treatment class and a decreased proportion in the susceptible class. Including totally recovery does not determine whether the proportion of individuals with alcohol problems increases or decreases, however it does effect the size of the change. Parameter estimates are made from information regarding binge drinking where we find an increase in the recovery rate decreases the proportion of binge drinkers in the population.     

Nordhaus-Gaddum results for the sum of the induced path number of a graph and its complement

The induced path number ρ(G) of a graph G is defined as the minimum number of subsets into which the vertex set of G can be partitioned so that each subset induces a path.Broere et al.proved that if G is a graph of order n,then n~(1/2) ≤ρ(G) + ρ(■) ≤ [3n/2].In this paper,we characterize the graphs G for which ρ(G) + ρ(■) = [3n/2],improve the lower bound on ρ(G) + ρ(■) by one when n is the square of an odd integer,and determine a best possible upper bound for ρ(G) + ρ(■) when neither G nor ■ has isolated vertices.     

Lion and man—can both win?

This paper is concerned with continuous-time pursuit and evasion games. Typically, we have a lion and a man in a metric space: they have the same speed, and the lion wishes to catch the man while the man tries to evade capture. We are interested in questions of the following form: is it the case that exactly one of the man and the lion has a winning strategy?As we shall see, in a compact metric space at least one of the players has a winning strategy. We show that, perhaps surprisingly, there are examples in which both players have winning strategies. We also construct a metric space in which, for the game with two lions versus one man, neither player has a winning strategy. We prove various other (positive and negative) related results, and pose some open problems.     

Investigation of a Dynamic Hybrid RANS/LES Modelling Methodology for Finite-Volume CFD Simulations

This paper investigates a recently proposed dynamic hybrid RANS-LES framework using a general-purpose finite-volume flow solver. The new method is highly generalized, allowing coupling of any selected RANS model with any selected LES model and containing no explicit grid dependence in its formulation. Selected results are presented for three test cases: two-dimensional channel flow, backward facing step, and a nozzle flow relevant to biomedical applications. Comparison with experimental and DNS data, and with other hybrid RANS-LES approaches, highlights the advantages of the new method and suggests that further investigation is warranted.     

System size and centrality dependence of charged hadron transverse momentum spectra in Au + Au and Cu + Cu collisions at square root of SNN = 62.4 and 200 GeV

We present transverse momentum distributions of charged hadrons produced in Cu + Cu collisions at square root of SNN = 62.4 and 200 GeV. The spectra are measured for transverse momenta of 0.25 < pT < 5.0 GeV/c at square root of SNN = 62.4 GeV and 0.25 < pT < 7.0 GeV/c at square root of SNN = 200 GeV, in a pseudorapidity range of 0.2 < eta < 1.4. The nuclear modification factor R(AA) is calculated relative to p + p data at both collision energies as a function of collision centrality. At a given collision energy and fractional cross section, R(AA) is observed to be systematically larger in Cu + Cu collisions compared to Au + Au. However, for the same number of participating nucleons, R(AA) is essentially the same in both systems over the measured range of pT, in spite of the significantly different geometries of the Cu + Cu and Au + Au systems.     

Volumetric three-component velocimetry measurements of the turbulent flow around a Rushton turbine

Volumetric three-component velocimetry measurements have been taken of the flow field near a Rushton turbine in a stirred tank reactor. This particular flow field is highly unsteady and three-dimensional, and is characterized by a strong radial jet, large tank-scale ring vortices, and small-scale blade tip vortices. The experimental technique uses a single camera head with three apertures to obtain approximately 15,000 three-dimensional vectors in a cubic volume. These velocity data offer the most comprehensive view to date of this flow field, especially since they are acquired at three Reynolds numbers (15,000, 107,000, and 137,000). Mean velocity fields and turbulent kinetic energy quantities are calculated. The volumetric nature of the data enables tip vortex identification, vortex trajectory analysis, and calculation of vortex strength. Three identification methods for the vortices are compared based on: the calculation of circumferential vorticity; the calculation of local pressure minima via an eigenvalue approach; and the calculation of swirling strength again via an eigenvalue approach. The use of two-dimensional data and three-dimensional data is compared for vortex identification; a ‘swirl strength’ criterion is less sensitive to completeness of the velocity gradient tensor and overall provides clearer identification of the tip vortices. The principal components of the strain rate tensor are also calculated for one Reynolds number case as these measures of stretching and compression have recently been associated with tip vortex characterization. Vortex trajectories and strength compare favorably with those in the literature. No clear dependence of trajectory on Reynolds number is deduced. The visualization of tip vortices up to 140° past blade passage in the highest Reynolds number case is notable and has not previously been shown.     

THz spectrum of monodeuterated methane

We report new measurements of the rotational spectrum of monodeuterated methane (CH₃D) in the range of 690-1200 GHz which allow for an accurate prediction of all lines in the range of the high-resolution spectrometer of the Herschel Space Observatory. Comparison is also made with the previous analysis based on infrared combination differences. Three lines of ¹³CH₃D were measured in natural abundance.     

Dynamic coefficient evaluation for an algebraic subgrid stress model

Static model coefficients for an algebraic subgrid stress (SGS) model are determined using a dynamic approach, based on results from simulations of isotropic decaying turbulence. The study was motivated by the discrepancies in energy transfer predictions using the previously documented coefficients (Bhushan and Warsi, Int. J. Numer. Meth. Fluids 2005; 49 : 489–519). The discrepancies are identified to be due to inconsistent filter functions used in the analytic estimates and the simulations. The study emphasizes that SGS model development should use filter functions compatible with those inherent in CFD application solvers. The dynamic approach predicts consistent model and transfer coefficients for different grid resolutions and is judged to be a reliable basis for model coefficient adjustments. The predicted Leonard's stress coefficient and associated energy transfer coefficients agree very well with the analytic estimates using a Gaussian/cutoff combination filter. This suggests that the modeling of Leonard's stress term using a truncated Taylor series expansion is robust and may not benefit significantly from dynamic modeling. Validation simulations were performed for turbulent channel flow at Re_τ = 180 and 590. The dynamic approach was found to be reliable only for the lower log‐layer of the Re_τ = 590 case, where the scale invariance condition was satisfied. Nonetheless, in this narrow range, the model and transfer coefficients compare well with the isotropic case. The static coefficient algebraic model with new adjusted coefficients shows improved predictions compared with the previous coefficients, for both isotropic decaying turbulence and channel flow cases. Copyright © 2013 John Wiley & Sons, Ltd.