129 research outputs found
Exploring Predictors of First Appointment Attendance at a Pain Management Service
Background: Individual characteristics such as gender, employment and age have been shown to predict attendance at pain management services (PMS). The characteristics of those who drop out of pain management programmes have also been explored, but as yet no studies have analysed the characteristics of those who do not attend the service following referral. Purpose: To explore the characteristics and predictors of those who attend and those who do not attend their first appointment with a PMS. Method: Predictive factors in the two groups – attenders (n = 425) and non-attenders (n = 69) – were explored using logistic regression. Results: Non-attendance was significantly predicted by the patient being a smoker and the appointment being in the morning. Non-attenders also scored higher on the Modified Somatic Perception Questionnaire, indicating higher levels of somatic pain. Discussion: Predictors of non-attendance were different from those for individuals who drop out of pain services. Implications and recommendations are made for PMS
Planning and Teaching for Student Learning in Mathematics: How Graduate Student Instructors Develop and Implement Instruction
The purpose of this study was to explore undergraduate mathematics teaching from the perspective of a graduate student serving as a first-time precalculus instructor of record. A multiple case study was designed to follow three mathematics graduate student instructors (MGSIs) through one semester of teaching to understand their goals for student learning, efforts to achieve these goals, influences on planning, and to identify challenges they encountered. For each MGSI, data collection included four interviews, three observations of teaching, weekly journal entries, and written assignments from a pedagogy course. A focus group, field notes from the pedagogy course, audio from mentor meetings, and mentor interviews also informed the data analysis.
Dramaturgical coding was utilized to arrive at common themes across MGSIs related to goals for student learning and challenges. Findings indicated MGSIs aimed to prepare students for their future, develop students’ reasoning, sense making and understanding of mathematics, help students develop productive dispositions, and procedural skills. MGSIs challenges related to implementing lesson plans as intended, preparing to teach, and interpreting student’s course performance and preparation. Individual case studies describe each MGSIs tactics used in the classroom, perceived lesson strengths, additional objectives, and key influences. Findings illustrate MGSIs planning, identify their needs, and may be informative for mathematics departments and individuals working to support graduate students
Universality in the flooding of regular islands by chaotic states
We investigate the structure of eigenstates in systems with a mixed phase
space in terms of their projection onto individual regular tori. Depending on
dynamical tunneling rates and the Heisenberg time, regular states disappear and
chaotic states flood the regular tori. For a quantitative understanding we
introduce a random matrix model. The resulting statistical properties of
eigenstates as a function of an effective coupling strength are in very good
agreement with numerical results for a kicked system. We discuss the
implications of these results for the applicability of the semiclassical
eigenfunction hypothesis.Comment: 11 pages, 12 figure
The Statistics of the Points Where Nodal Lines Intersect a Reference Curve
We study the intersection points of a fixed planar curve with the
nodal set of a translationally invariant and isotropic Gaussian random field
\Psi(\bi{r}) and the zeros of its normal derivative across the curve. The
intersection points form a discrete random process which is the object of this
study. The field probability distribution function is completely specified by
the correlation G(|\bi{r}-\bi{r}'|) = .
Given an arbitrary G(|\bi{r}-\bi{r}'|), we compute the two point
correlation function of the point process on the line, and derive other
statistical measures (repulsion, rigidity) which characterize the short and
long range correlations of the intersection points. We use these statistical
measures to quantitatively characterize the complex patterns displayed by
various kinds of nodal networks. We apply these statistics in particular to
nodal patterns of random waves and of eigenfunctions of chaotic billiards. Of
special interest is the observation that for monochromatic random waves, the
number variance of the intersections with long straight segments grows like , as opposed to the linear growth predicted by the percolation model,
which was successfully used to predict other long range nodal properties of
that field.Comment: 33 pages, 13 figures, 1 tabl
Nodal domain distributions for quantum maps
The statistics of the nodal lines and nodal domains of the eigenfunctions of
quantum billiards have recently been observed to be fingerprints of the
chaoticity of the underlying classical motion by Blum et al. (Phys. Rev. Lett.,
Vol. 88 (2002), 114101) and by Bogomolny and Schmit (Phys. Rev. Lett., Vol. 88
(2002), 114102). These statistics were shown to be computable from the random
wave model of the eigenfunctions. We here study the analogous problem for
chaotic maps whose phase space is the two-torus. We show that the distributions
of the numbers of nodal points and nodal domains of the eigenvectors of the
corresponding quantum maps can be computed straightforwardly and exactly using
random matrix theory. We compare the predictions with the results of numerical
computations involving quantum perturbed cat maps.Comment: 7 pages, 2 figures. Second version: minor correction
Casimir force between integrable and chaotic pistons
We have computed numerically the Casimir force between two identical pistons
inside a very long cylinder, considering different shapes for the pistons. The
pistons can be considered as quantum billiards, whose spectrum determines the
vacuum force. The smooth part of the spectrum fixes the force at short
distances, and depends only on geometric quantities like the area or perimeter
of the piston. However, correcting terms to the force, coming from the
oscillating part of the spectrum which is related to the classical dynamics of
the billiard, are qualitatively different for classically integrable or chaotic
systems. We have performed a detailed numerical analysis of the corresponding
Casimir force for pistons with regular and chaotic classical dynamics. For a
family of stadium billiards, we have found that the correcting part of the
Casimir force presents a sudden change in the transition from regular to
chaotic geometries.Comment: 13 pages, 10 figure
Wavefunctions, Green's functions and expectation values in terms of spectral determinants
We derive semiclassical approximations for wavefunctions, Green's functions
and expectation values for classically chaotic quantum systems. Our method
consists of applying singular and regular perturbations to quantum
Hamiltonians. The wavefunctions, Green's functions and expectation values of
the unperturbed Hamiltonian are expressed in terms of the spectral determinant
of the perturbed Hamiltonian. Semiclassical resummation methods for spectral
determinants are applied and yield approximations in terms of a finite number
of classical trajectories. The final formulas have a simple form. In contrast
to Poincare surface of section methods, the resummation is done in terms of the
periods of the trajectories.Comment: 18 pages, no figure
Geometric characterization of nodal domains: the area-to-perimeter ratio
In an attempt to characterize the distribution of forms and shapes of nodal
domains in wave functions, we define a geometric parameter - the ratio
between the area of a domain and its perimeter, measured in units of the
wavelength . We show that the distribution function can
distinguish between domains in which the classical dynamics is regular or
chaotic. For separable surfaces, we compute the limiting distribution, and show
that it is supported by an interval, which is independent of the properties of
the surface. In systems which are chaotic, or in random-waves, the
area-to-perimeter distribution has substantially different features which we
study numerically. We compare the features of the distribution for chaotic wave
functions with the predictions of the percolation model to find agreement, but
only for nodal domains which are big with respect to the wavelength scale. This
work is also closely related to, and provides a new point of view on
isoperimetric inequalities.Comment: 22 pages, 11 figure
Wavepacket Dynamics in Nonlinear Schr\"odinger Equations
Coherent states play an important role in quantum mechanics because of their
unique properties under time evolution. Here we explore this concept for
one-dimensional repulsive nonlinear Schr\"odinger equations, which describe
weakly interacting Bose-Einstein condensates or light propagation in a
nonlinear medium. It is shown that the dynamics of phase-space translations of
the ground state of a harmonic potential is quite simple: the centre follows a
classical trajectory whereas its shape does not vary in time. The parabolic
potential is the only one that satisfies this property. We study the time
evolution of these nonlinear coherent states under perturbations of their
shape, or of the confining potential. A rich variety of effects emerges. In
particular, in the presence of anharmonicities, we observe that the packet
splits into two distinct components. A fraction of the condensate is
transferred towards uncoherent high-energy modes, while the amplitude of
oscillation of the remaining coherent component is damped towards the bottom of
the well
Nodal domains on quantum graphs
We consider the real eigenfunctions of the Schr\"odinger operator on graphs,
and count their nodal domains. The number of nodal domains fluctuates within an
interval whose size equals the number of bonds . For well connected graphs,
with incommensurate bond lengths, the distribution of the number of nodal
domains in the interval mentioned above approaches a Gaussian distribution in
the limit when the number of vertices is large. The approach to this limit is
not simple, and we discuss it in detail. At the same time we define a random
wave model for graphs, and compare the predictions of this model with analytic
and numerical computations.Comment: 19 pages, uses IOP journal style file
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