Ways to teach modelling—a 50 year study

This article describes a sequence of design research projects, some exploratory others more formal, on the teaching of modelling and the analysis of modelling skills. The initial motivation was the author’s observation that the teaching of applied mathematics in UK high schools and universities involved no active modelling by students, but was entirely focused on their learning standards models of a restricted range of phenomena, largely from Newtonian mechanics. This did not develop the numeracy/mathematical literacy that was so clearly important for future citizens. Early explorations started with modelling workshops with high school teachers and mathematics undergraduates, observed and analysed—in some case using video. The theoretical basis of this work has been essentially heuristic, though the Shell Centre studies included, for example, a detailed analysis of formulation processes that has not, as so often, been directly replicated. Recent work has focused on developing a formative assessment approach to teaching modelling that has proved both successful and popular. Finally, the system-level challenges in trying to establish modelling as an integral part of mathematics curricula are briefly discussed

Studying the scale and q^2 dependence of K^+-->pi^+e^+e^- decay

We extract the K^+-->pi^+e^+e^- amplitude scale at q^2=0 from the recent Brookhaven E865 high-statistics data. We find that the q^2=0 scale is fitted in excellent agreement with the theoretical long-distance amplitude. Lastly, we find that the observed q^2 shape is explained by the combined effect of the pion and kaon form-factor vector-meson-dominance rho, omega and phi poles, and a charged pion loop coupled to a virtual photon-->e^+e^- transition.Comment: 8 pages, 3 figure

Partial survival and inelastic collapse for a randomly accelerated particle

We present an exact derivation of the survival probability of a randomly accelerated particle subject to partial absorption at the origin. We determine the persistence exponent and the amplitude associated to the decay of the survival probability at large times. For the problem of inelastic reflection at the origin, with coefficient of restitution $r$, we give a new derivation of the condition for inelastic collapse, $r<r_c=e^{-\pi/\sqrt{3}}$, and determine the persistence exponent exactly.Comment: 6 page

Beam-size effect and particle losses at SuperBB factory (Italy)

In the colliders, the macroscopically large impact parameters give a substantial contribution to the standard cross section of the $e^+ e^- \to e^+ e^- \gamma$ process. These impact parameters may be much larger than the transverse sizes of the colliding bunches. It means that the standard cross section of this process has to be substantially modified. In the present paper such a beam-size effect is calculated for bremsstrahlung at Super$B$ factory developed in Italy. We find out that this effect reduces beam losses due to bremsstrahlung by about 40%.Comment: 11 pages, 4 figure

First-passage and extreme-value statistics of a particle subject to a constant force plus a random force

We consider a particle which moves on the x axis and is subject to a constant force, such as gravity, plus a random force in the form of Gaussian white noise. We analyze the statistics of first arrival at point $x_1$ of a particle which starts at $x_0$ with velocity $v_0$. The probability that the particle has not yet arrived at $x_1$ after a time $t$, the mean time of first arrival, and the velocity distribution at first arrival are all considered. We also study the statistics of the first return of the particle to its starting point. Finally, we point out that the extreme-value statistics of the particle and the first-passage statistics are closely related, and we derive the distribution of the maximum displacement $m={\rm max}_t[x(t)]$.Comment: Contains an analysis of the extreme-value statistics not included in first versio

Transverse beam tails due to inelastic scattering

Non-Gaussian beam tails producing low beam lifetimes and background to the experimental detectors can be a serious performance limitation in colliding beam facilities. We describe simulations and measurement of non-Gaussian beam tails, performed on the e+ e- collider LEP, that revealed the importance of inelastic particule scattering as launching processes of particules to large amplitude

Observations and simulations of beam tails in LEP

Transverse beam tails have been measured in LEP using scraping collimators and loss monitors. Very significant non-Gaussian tails are present for colliding beams and high beam-beam tune shift. On a lower but still significant level, non-Gaussian tails are also present in the horizontal plane for a single beam. Comparison of measurements with detailed simulations allowed us to identify off-momentum particles produced by scattering processes as a source of significant transverse tails

Particle Background at LEP with Head-On colliding Bunch Trains

The vertical closed orbit bumps around the interaction points in LEP, which are needed to separate counter rotating bunches at their parasitic collision points, generate additional particle backgrounds at the LEP detectors. Monte Carlo simulations of photon and electron backgrounds have been performed and their predictions compared with experimental data. This has led to a good understanding of the bunch train specific background sources and allowed efficient protection measures to be devised

Surface Critical Behavior of Binary Alloys and Antiferromagnets: Dependence of the Universality Class on Surface Orientation

The surface critical behavior of semi-infinite (a) binary alloys with a continuous order-disorder transition and (b) Ising antiferromagnets in the presence of a magnetic field is considered. In contrast to ferromagnets, the surface universality class of these systems depends on the orientation of the surface with respect to the crystal axes. There is ordinary and extraordinary surface critical behavior for orientations that preserve and break the two-sublattice symmetry, respectively. This is confirmed by transfer-matrix calculations for the two-dimensional antiferromagnet and other evidence.Comment: Final version that appeared in PRL, some minor stylistic changes and one corrected formula; 4 pp., twocolumn, REVTeX, 3 eps fig