Algorithmic contexts and learning potentiality: a case study of students’ understanding of calculus

The study explores the nature of students’ conceptual understanding of calculus. Twenty students of engineering were asked to reflect in writing on the meaning of the concepts of limit and integral. A sub-sample of four students was selected for subsequent interviews, which explored in detail the students’ understandings of the two concepts. Intentional analysis of the students’ written and oral accounts revealed that the students were expressing their understanding of limit and integral within an algorithmic context, in which the very ‘operations’ of these concepts were seen as crucial. The students also displayed great confidence in their ability to deal with these concepts. Implications for the development of a conceptual understanding of calculus are discussed, and it is argued that developing understanding within an algorithmic context can be seen as a stepping stone towards a more complete conceptual understanding of calculus.     

Limit theorems for projections of random walk on a hypersphere

We show that almost any one-dimensional projection of a suitably scaled random walk on a hypercube, inscribed in a hypersphere, converges weakly to an Ornstein–Uhlenbeck process as the dimension of the sphere tends to infinity. We also observe that the same result holds when the random walk is replaced with spherical Brownian motion. This latter result can be viewed as a “functional” generalisation of Poincaré’s observation for projections of uniform measure on high dimensional spheres; the former result is an analogous generalisation of the Bernoulli–Laplace central limit theorem. Given the relation of these two classic results to the central limit theorem for convex bodies, the modest results provided here would appear to motivate a functional generalisation.     

Uniform Convergence of Discrete Curvatures from Nets of Curvature Lines

We study discrete curvatures computed from nets of curvature lines on a given smooth surface and prove their uniform convergence to smooth principal curvatures. We provide explicit error bounds, with constants depending only on properties of the smooth limit surface and the shape regularity of the discrete net.     

Asymptotic analysis of the differences between the Stokes-Darcy system with different interface conditions and the Stokes-Brinkman system

We consider the coupling of the Stokes and Darcy systems with different choices for the interface conditions. We show that, comparing results with those for the Stokes-Brinkman equations, the solutions of Stokes-Darcy equations with the Beavers-Joseph interface condition in the one-dimensional and quasi-two-dimensional (periodic) cases are more accurate than are those obtained using the Beavers-Joseph-Saffman-Jones interface condition and that both of these are more accurate than solutions obtained using a zero tangential velocity interface condition. The zero tangential velocity interface condition is in turn more accurate than the free-slip interface boundary condition. We also prove that the summation of the quasi-two-dimensional solutions converge so that the conclusions are also valid for the two-dimensional case.     

A confocal set‐up for micro‐XRF and XAFS experiments using diamond‐anvil cells

A confocal set‐up is presented that improves micro‐XRF and XAFS experiments with high‐pressure diamond‐anvil cells (DACs). In this experiment a probing volume is defined by the focus of the incoming synchrotron radiation beam and that of a polycapillary X‐ray half‐lens with a very long working distance, which is placed in front of the fluorescence detector. This set‐up enhances the quality of the fluorescence and XAFS spectra, and thus the sensitivity for detecting elements at low concentrations. It efficiently suppresses signal from outside the sample chamber, which stems from elastic and inelastic scattering of the incoming beam by the diamond anvils as well as from excitation of fluorescence from the body of the DAC.     

Device quality ZnO grown using a Filtered Cathodic Vacuum Arc

In this paper we report on the structural, electrical and optical characteristics of unintentionally doped ZnO films grown on a-plane sapphire substrates using the Filtered Cathodic Vacuum Arc (FCVA) technique. The resulting films showed considerable promise for device applications with properties including high transparency, moderate intrinsic carrier concentrations (10¹⁷–10¹⁹ cm⁻³), electron mobilities up to 30 cm²/Vs, low surface roughness (typically <2% of film thickness) and well-structured photoluminescence. Post-annealing in oxygen at temperatures up to 800 °C produced significant improvements in the properties of these films. Silver oxide Schottky diodes fabricated on FCVA ZnO showed ideality factors as low as 1.20 and good sensitivity to ultraviolet light.     

Precision measurements and computations of transition energies in rotationally cold triatomic hydrogen ions up to the midvisible spectral range

First-principles computations and experimental measurements of transition energies are carried out for vibrational overtone lines of the triatomic hydrogen ion H(3)(+) corresponding to floppy vibrations high above the barrier to linearity. Action spectroscopy is improved to detect extremely weak visible-light spectral lines on cold trapped H(3)(+) ions. A highly accurate potential surface is obtained from variational calculations using explicitly correlated Gaussian wave function expansions. After nonadiabatic corrections, the floppy H(3)(+) vibrational spectrum is reproduced at the 0.1 cm(-1) level up to 16600 cm(-1).     

Auxiliary Equations Approach for the Stochastic Unsteady Navier-Stokes Equations with Additive Random Noise

This paper presents a Martingale regularization method for the stochastic Navier-Stokes equations with additive noise. The original system is split into two equivalent parts, the linear stochastic Stokes equations with Martingale solution and the stochastic modified Navier-Stokes equations with relatively-higher regularities. Meanwhile, a fractional Laplace operator is introduced to regularize the noise term. The stability and convergence of numerical scheme for the pathwise modified Navier-Stokes equations are proved.The comparisons of non-regularized and regularized noises for the Navier-Stokes system are numerically presented to further demonstrate the efficiency of our numerical scheme.