3,401 research outputs found
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
Effects of pressure, oxygen concentration, and forced convection on flame spread rate of Plexiglas, Nylon and Teflon
Experiments were conducted in which the burning of cylindrical materials in a flowing oxidant stream was studied. Plexiglas, Nylon, and Teflon fuel specimens were oriented such that the flames spread along the surface in a direction opposed to flowing gas. Correlations of flame spread rate were obtained that were power law relations in terms of pressure, oxygen concentration, and gas velocity
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 , we give a new derivation of
the condition for inelastic collapse, , and determine
the persistence exponent exactly.Comment: 6 page
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 of a particle
which starts at with velocity . The probability that the particle
has not yet arrived at after a time , 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 .Comment: Contains an analysis of the extreme-value statistics not included in
first versio
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
Sylvester's question and the Random Acceleration Process
Let n points be chosen randomly and independently in the unit disk.
"Sylvester's question" concerns the probability p_n that they are the vertices
of a convex n-sided polygon. Here we establish the link with another problem.
We show that for large n this polygon, when suitably parametrized by a function
r(phi) of the polar angle phi, satisfies the equation of the random
acceleration process (RAP), d^2 r/d phi^2 = f(phi), where f is Gaussian noise.
On the basis of this relation we derive the asymptotic expansion log p_n = -2n
log n + n log(2 pi^2 e^2) - c_0 n^{1/5} + ..., of which the first two terms
agree with a rigorous result due to Barany. The nonanalyticity in n of the
third term is a new result. The value 1/5 of the exponent follows from recent
work on the RAP due to Gyorgyi et al. [Phys. Rev. E 75, 021123 (2007)]. We show
that the n-sided polygon is effectively contained in an annulus of width \sim
n^{-4/5} along the edge of the disk. The distance delta_n of closest approach
to the edge is exponentially distributed with average 1/(2n).Comment: 29 pages, 4 figures; references added and minor change
Simulation of a semiflexible polymer in a narrow cylindrical pore
The probability that a randomly accelerated particle in two dimensions has
not yet left a simply connected domain after a time decays as
for long times. The same quantity also determines the
confinement free energy per unit length of a
semiflexible polymer in a narrow cylindrical pore with cross section . From simulations of a randomly accelerated particle we estimate the
universal amplitude of for both circular and rectangular cross
sections.Comment: 10 pages, 2 eps figure
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