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The role of the harmonic vector average in motion integration
The local speeds of object contours vary systematically with the cosine of the angle between the normal component of the local velocity and the global object motion direction. An array of Gabor elements whose speed changes with local spatial orientation in accordance with this pattern can appear to move as a single surface. The apparent direction of motion of plaids and Gabor arrays has variously been proposed to result from feature tracking, vector addition and vector averaging in addition to the geometrically correct global velocity as indicated by the intersection of constraints (IOC) solution. Here a new combination rule, the harmonic vector average (HVA), is introduced, as well as a new algorithm for computing the IOC solution. The vector sum can be discounted as an integration strategy as it increases with the number of elements. The vector average over local vectors that vary in direction always provides an underestimate of the true global speed. The HVA, however, provides the correct global speed and direction for an unbiased sample of local velocities with respect to the global motion direction, as is the case for a simple closed contour. The HVA over biased samples provides an aggregate velocity estimate that can still be combined through an IOC computation to give an accurate estimate of the global velocity, which is not true of the vector average. Psychophysical results for type II Gabor arrays show perceived direction and speed falls close to the IOC direction for Gabor arrays having a wide range of orientations but the IOC prediction fails as the mean orientation shifts away from the global motion direction and the orientation range narrows. In this case perceived velocity generally defaults to the HVA
Space-Time Evolution of the Oscillator, Rapidly moving in a random media
We study the quantum-mechanical evolution of the nonrelativistic oscillator,
rapidly moving in the media with the random vector fields. We calculate the
evolution of the level probability distribution as a function of time, and
obtain rapid level diffusion over the energy levels. Our results imply a new
mechanism of charmonium dissociation in QCD media.Comment: 32 pages, 13 figure
Linear frictional forces cause orbits to neither circularize nor precess
For the undamped Kepler potential the lack of precession has historically
been understood in terms of the Runge-Lenz symmetry. For the damped Kepler
problem this result may be understood in terms of the generalization of Poisson
structure to damped systems suggested recently by Tarasov[1]. In this
generalized algebraic structure the orbit-averaged Runge-Lenz vector remains a
constant in the linearly damped Kepler problem to leading order in the damping
coeComment: 16 pages. 1 figure, Rewrite for resubmissio
Geometry of the ergodic quotient reveals coherent structures in flows
Dynamical systems that exhibit diverse behaviors can rarely be completely
understood using a single approach. However, by identifying coherent structures
in their state spaces, i.e., regions of uniform and simpler behavior, we could
hope to study each of the structures separately and then form the understanding
of the system as a whole. The method we present in this paper uses trajectory
averages of scalar functions on the state space to: (a) identify invariant sets
in the state space, (b) form coherent structures by aggregating invariant sets
that are similar across multiple spatial scales. First, we construct the
ergodic quotient, the object obtained by mapping trajectories to the space of
trajectory averages of a function basis on the state space. Second, we endow
the ergodic quotient with a metric structure that successfully captures how
similar the invariant sets are in the state space. Finally, we parametrize the
ergodic quotient using intrinsic diffusion modes on it. By segmenting the
ergodic quotient based on the diffusion modes, we extract coherent features in
the state space of the dynamical system. The algorithm is validated by
analyzing the Arnold-Beltrami-Childress flow, which was the test-bed for
alternative approaches: the Ulam's approximation of the transfer operator and
the computation of Lagrangian Coherent Structures. Furthermore, we explain how
the method extends the Poincar\'e map analysis for periodic flows. As a
demonstration, we apply the method to a periodically-driven three-dimensional
Hill's vortex flow, discovering unknown coherent structures in its state space.
In the end, we discuss differences between the ergodic quotient and
alternatives, propose a generalization to analysis of (quasi-)periodic
structures, and lay out future research directions.Comment: Submitted to Elsevier Physica D: Nonlinear Phenomen
Evolution of squeezed states under the Fock-Darwin Hamiltonian
We develop a complete analytical description of the time evolution of
squeezed states of a charged particle under the Fock-Darwin Hamiltonian and a
time-dependent electric field. This result generalises a relation obtained by
Infeld and Pleba\'nski for states of the one-dimensional harmonic oscillator.
We relate the evolution of a state-vector subjected to squeezing to that of
state which is not subjected to squeezing and for which the time-evolution
under the simple harmonic oscillator dynamics is known (e.g. an eigenstate of
the Hamiltonian). A corresponding relation is also established for the Wigner
functions of the states, in view of their utility in the analysis of cold-ion
experiments. In an appendix, we compute the response functions of the FD
Hamiltonian to an external electric field, using the same techniques as in the
main text
Semi-classical limitations for photon emission in strong external fields
The semi-classical heuristic emission formula of Baier-Katkov [Sov. Phys.
JETP \textbf{26}, 854 (1968)] is well-known to describe radiation of an
ultrarelativistic electron in strong external fields employing the electron's
classical trajectory. To find the limitations of the Baier-Katkov approach, we
investigate electron radiation in a strong rotating electric field quantum
mechanically using the Wentzel-Kramers-Brillouin approximation. Except for an
ultrarelativistic velocity, it is shown that an additional condition is
required in order to recover the widely used semi-classical result. A violation
of this condition leads to two consequences. First, it gives rise to
qualitative discrepancy in harmonic spectra between the two approaches. Second,
the quantum harmonic spectra are determined not only by the classical
trajectory but also by the dispersion relation of the effective photons of the
external field
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