409 research outputs found
Inclusion of Diffraction Effects in the Gutzwiller Trace Formula
The Gutzwiller trace formula is extended to include diffraction effects. The
new trace formula involves periodic rays which have non-geometrical segments as
a result of diffraction on the surfaces and edges of the scatter.Comment: 4 pages, LaTeX, 1 ps figur
Two novel approaches to the content analysis of school mathematics textbooks
The analysis of the content of school textbooks, particularly in a time of cross-cultural borrowing, is a growing field restricted by the tools currently available. In this paper, drawing on the analyses of three English year-one mathematics textbooks, we show how two approaches to the analysis of sequential data not only supplement conventional frequency analyses but highlight trends in the content of such textbooks hidden from frequency analyses alone. The first, moving averages, is conventionally used in science to eliminate noise and demonstrate trends in data. The second, Lorenz curves, is typically found in the social sciences to compare different forms of social phenomena. Both, as we show, extend the range of questions that can be meaningfully asked of textbooks. Finally, we speculate as to how both approaches can be used with other forms of ordered classroom data
Small Disks and Semiclassical Resonances
We study the effect on quantum spectra of the existence of small circular
disks in a billiard system. In the limit where the disk radii vanish there is
no effect, however this limit is approached very slowly so that even very small
radii have comparatively large effects. We include diffractive orbits which
scatter off the small disks in the periodic orbit expansion. This situation is
formally similar to edge diffraction except that the disk radii introduce a
length scale in the problem such that for wave lengths smaller than the order
of the disk radius we recover the usual semi-classical approximation; however,
for wave lengths larger than the order of the disk radius there is a
qualitatively different behaviour. We test the theory by successfully
estimating the positions of scattering resonances in geometries consisting of
three and four small disks.Comment: Final published version - some changes in the discussion and the
labels on one figure are correcte
Classical, semiclassical, and quantum investigations of the 4-sphere scattering system
A genuinely three-dimensional system, viz. the hyperbolic 4-sphere scattering
system, is investigated with classical, semiclassical, and quantum mechanical
methods at various center-to-center separations of the spheres. The efficiency
and scaling properties of the computations are discussed by comparisons to the
two-dimensional 3-disk system. While in systems with few degrees of freedom
modern quantum calculations are, in general, numerically more efficient than
semiclassical methods, this situation can be reversed with increasing dimension
of the problem. For the 4-sphere system with large separations between the
spheres, we demonstrate the superiority of semiclassical versus quantum
calculations, i.e., semiclassical resonances can easily be obtained even in
energy regions which are unattainable with the currently available quantum
techniques. The 4-sphere system with touching spheres is a challenging problem
for both quantum and semiclassical techniques. Here, semiclassical resonances
are obtained via harmonic inversion of a cross-correlated periodic orbit
signal.Comment: 12 pages, 5 figures, submitted to Phys. Rev.
Spectral statistics in chaotic systems with a point interaction
We consider quantum systems with a chaotic classical limit that are perturbed
by a point-like scatterer. The spectral form factor K(tau) for these systems is
evaluated semiclassically in terms of periodic and diffractive orbits. It is
shown for order tau^2 and tau^3 that off-diagonal contributions to the form
factor which involve diffractive orbits cancel exactly the diagonal
contributions from diffractive orbits, implying that the perturbation by the
scatterer does not change the spectral statistic. We further show that
parametric spectral statistics for these systems are universal for small
changes of the strength of the scatterer.Comment: LaTeX, 21 pages, 7 figures, small corrections, new references adde
Swedish parents’ perspectives on homework: manifestations of principled pragmatism
Motivated by earlier research highlighting Swedish teachers’ beliefs that the setting of homework compromises deep-seated principles of educational equity, this paper presents an exploratory study of Swedish parents’ perspectives on homework in their year-one children’s learning. Twenty-five parents, drawn from three demographically different schools in the Stockholm region, participated in semi-structured interviews. The interviews, broadly focused on how parents support their children’s learning and including questions about homework in general and mathematics homework in particular, were transcribed and data subjected to a constant comparison analytical process. This yielded four broad themes, highlighting considerable variation in how parents perceive the relationship between homework and educational equity. First, all parents spoke appreciatively of their children receiving reading homework and, in so doing, indicated a collective construal that reading homework is neither homework nor a threat to equity. Second, four parents, despite their enthusiasm for reading homework, opposed the setting of any homework due to its potential compromise of family life. Third, seven parents indicated that they would appreciate mathematics homework where it were not a threat to equity. Finally, fourteen parents, despite acknowledging homework’s potential compromise to equity, were unequivocally in favour of mathematics homework being set to their children
Estimation: An inadequately operationalised national curriculum competence
Research has highlighted the importance of estimation, in various forms, as both an essential life-skill and a significant underpinning of other forms of mathematical learning. It has also highlighted a lack of opportunities for learners to acquire estimational competence. In this paper, we present a review of the literature that identified four forms of estimation. These are measurement, computational, quantity (or numerosity) and number line estimation. In addition to summarising the characteristics and significance of each form of estimation, we examine critically the estimation-related expectations of the English national curriculum for primary mathematics to highlight a problematic lack of opportunity
Geometrical theory of diffraction and spectral statistics
We investigate the influence of diffraction on the statistics of energy
levels in quantum systems with a chaotic classical limit. By applying the
geometrical theory of diffraction we show that diffraction on singularities of
the potential can lead to modifications in semiclassical approximations for
spectral statistics that persist in the semiclassical limit . This
result is obtained by deriving a classical sum rule for trajectories that
connect two points in coordinate space.Comment: 14 pages, no figure, to appear in J. Phys.
Periodic Orbit Quantization beyond Semiclassics
A quantum generalization of the semiclassical theory of Gutzwiller is given.
The new formulation leads to systematic orbit-by-orbit inclusion of higher
contributions to the spectral determinant. We apply the theory to
billiard systems, and compare the periodic orbit quantization including the
first contribution to the exact quantum mechanical results.Comment: revte
Three disks in a row: A two-dimensional scattering analog of the double-well problem
We investigate the scattering off three nonoverlapping disks equidistantly
spaced along a line in the two-dimensional plane with the radii of the outer
disks equal and the radius of the inner disk varied. This system is a
two-dimensional scattering analog to the double-well-potential (bound state)
problem in one dimension. In both systems the symmetry splittings between
symmetric and antisymmetric states or resonances, respectively, have to be
traced back to tunneling effects, as semiclassically the geometrical periodic
orbits have no contact with the vertical symmetry axis. We construct the
leading semiclassical ``creeping'' orbits that are responsible for the symmetry
splitting of the resonances in this system. The collinear three-disk-system is
not only one of the simplest but also one of the most effective systems for
detecting creeping phenomena. While in symmetrically placed n-disk systems
creeping corrections affect the subleading resonances, they here alone
determine the symmetry splitting of the 3-disk resonances in the semiclassical
calculation. It should therefore be considered as a paradigm for the study of
creeping effects. PACS numbers: 03.65.Sq, 03.20.+i, 05.45.+bComment: replaced with published version (minor misprints corrected and
references updated); 23 pages, LaTeX plus 8 Postscript figures, uses
epsfig.sty, espf.sty, and epsf.te
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