2,326 research outputs found
Enhanced quantum tunnelling induced by disorder
We reconsider the problem of the enhancement of tunnelling of a quantum
particle induced by disorder of a one-dimensional tunnel barrier of length ,
using two different approximate analytic solutions of the invariant imbedding
equations of wave propagation for weak disorder. The two solutions are
complementary for the detailed understanding of important aspects of numerical
results on disorder-enhanced tunnelling obtained recently by Kim et al. (Phys.
rev. B{\bf 77}, 024203 (2008)). In particular, we derive analytically the
scaled wavenumber -threshold where disorder-enhanced tunnelling of an
incident electron first occurs, as well as the rate of variation of the
transmittance in the limit of vanishing disorder. Both quantities are in good
agreement with the numerical results of Kim et al. Our non-perturbative
solution of the invariant imbedding equations allows us to show that the
disorder enhances both the mean conductance and the mean resistance of the
barrier.Comment: 10 page
A linear theory for control of non-linear stochastic systems
We address the role of noise and the issue of efficient computation in
stochastic optimal control problems. We consider a class of non-linear control
problems that can be formulated as a path integral and where the noise plays
the role of temperature. The path integral displays symmetry breaking and there
exist a critical noise value that separates regimes where optimal control
yields qualitatively different solutions. The path integral can be computed
efficiently by Monte Carlo integration or by Laplace approximation, and can
therefore be used to solve high dimensional stochastic control problems.Comment: 5 pages, 3 figures. Accepted to PR
Algorithms for response adaptive sampling designs
An experimental design is a formula or algorithm that specifies how resources are to be utilized throughout a study. The design is considered to be good or even optimal if it allows for sufficiently precise and accurate data analysis with the least output of resources such as time, money and experimental units. Most experimental designs use fixed sampling procedures in which the sample sizes and order of allocations to different study groups are known in advance. Copyright © 2009 John Wiley & Sons, Inc.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/64301/1/25_ftp.pd
Stabilizing unstable periodic orbits in the Lorenz equations using time-delayed feedback control
For many years it was believed that an unstable periodic orbit with an odd
number of real Floquet multipliers greater than unity cannot be stabilized by
the time-delayed feedback control mechanism of Pyragus. A recent paper by
Fiedler et al uses the normal form of a subcritical Hopf bifurcation to give a
counterexample to this theorem. Using the Lorenz equations as an example, we
demonstrate that the stabilization mechanism identified by Fiedler et al for
the Hopf normal form can also apply to unstable periodic orbits created by
subcritical Hopf bifurcations in higher-dimensional dynamical systems. Our
analysis focuses on a particular codimension-two bifurcation that captures the
stabilization mechanism in the Hopf normal form example, and we show that the
same codimension-two bifurcation is present in the Lorenz equations with
appropriately chosen Pyragus-type time-delayed feedback. This example suggests
a possible strategy for choosing the feedback gain matrix in Pyragus control of
unstable periodic orbits that arise from a subcritical Hopf bifurcation of a
stable equilibrium. In particular, our choice of feedback gain matrix is
informed by the Fiedler et al example, and it works over a broad range of
parameters, despite the fact that a center-manifold reduction of the
higher-dimensional problem does not lead to their model problem.Comment: 21 pages, 8 figures, to appear in PR
Practical Open-Loop Optimistic Planning
We consider the problem of online planning in a Markov Decision Process when
given only access to a generative model, restricted to open-loop policies -
i.e. sequences of actions - and under budget constraint. In this setting, the
Open-Loop Optimistic Planning (OLOP) algorithm enjoys good theoretical
guarantees but is overly conservative in practice, as we show in numerical
experiments. We propose a modified version of the algorithm with tighter
upper-confidence bounds, KLOLOP, that leads to better practical performances
while retaining the sample complexity bound. Finally, we propose an efficient
implementation that significantly improves the time complexity of both
algorithms
The kink Casimir energy in a lattice sine-Gordon model
The Casimir energy of quantum fluctuations about the classical kink
configuration is computed numerically for a recently proposed lattice
sine-Gordon model. This energy depends periodically on the kink position and is
found to be approximately sinusoidal.Comment: 10 pages, 4 postscript figure
Gigantic transmission band edge resonance in periodic stacks of anisotropic layers
We consider Fabry-Perot cavity resonance in periodic stacks of anisotropic
layers with misaligned in-plane anisotropy at the frequency close to a photonic
band edge. We show that in-plane dielectric anisotropy can result in a dramatic
increase in field intensity and group delay associated with the transmission
resonance. The field enhancement appears to be proportional to forth degree of
the number N of layers in the stack. By contrast, in common periodic stacks of
isotropic layers, those effects are much weaker and proportional to N^2. Thus,
the anisotropy allows to drastically reduce the size of the resonance cavity
with similar performance. The key characteristic of the periodic arrays with
the gigantic transmission resonance is that the dispersion curve omega(k)at the
photonic band edge has the degenerate form Delta(omega) ~ Delta(k)^4, rather
than the regular form Delta(omega) ~ Delta(k)^2. This can be realized in
specially arranged stacks of misaligned anisotropic layers. The degenerate band
edge cavity resonance with similar outstanding properties can also be realized
in a waveguide environment, as well as in a linear array of coupled multimode
resonators, provided that certain symmetry conditions are in place.Comment: To be submitted to Phys. Re
Path integrals and symmetry breaking for optimal control theory
This paper considers linear-quadratic control of a non-linear dynamical
system subject to arbitrary cost. I show that for this class of stochastic
control problems the non-linear Hamilton-Jacobi-Bellman equation can be
transformed into a linear equation. The transformation is similar to the
transformation used to relate the classical Hamilton-Jacobi equation to the
Schr\"odinger equation. As a result of the linearity, the usual backward
computation can be replaced by a forward diffusion process, that can be
computed by stochastic integration or by the evaluation of a path integral. It
is shown, how in the deterministic limit the PMP formalism is recovered. The
significance of the path integral approach is that it forms the basis for a
number of efficient computational methods, such as MC sampling, the Laplace
approximation and the variational approximation. We show the effectiveness of
the first two methods in number of examples. Examples are given that show the
qualitative difference between stochastic and deterministic control and the
occurrence of symmetry breaking as a function of the noise.Comment: 21 pages, 6 figures, submitted to JSTA
Two-parameter deformations of logarithm, exponential, and entropy: A consistent framework for generalized statistical mechanics
A consistent generalization of statistical mechanics is obtained by applying
the maximum entropy principle to a trace-form entropy and by requiring that
physically motivated mathematical properties are preserved. The emerging
differential-functional equation yields a two-parameter class of generalized
logarithms, from which entropies and power-law distributions follow: these
distributions could be relevant in many anomalous systems. Within the specified
range of parameters, these entropies possess positivity, continuity, symmetry,
expansibility, decisivity, maximality, concavity, and are Lesche stable. The
Boltzmann-Shannon entropy and some one parameter generalized entropies already
known belong to this class. These entropies and their distribution functions
are compared, and the corresponding deformed algebras are discussed.Comment: Version to appear in PRE: about 20% shorter, references updated, 13
PRE pages, 3 figure
Time-Energy Tradeoffs for Evacuation by Two Robots in the Wireless Model
Two robots stand at the origin of the infinite line and are tasked with
searching collaboratively for an exit at an unknown location on the line. They
can travel at maximum speed and can change speed or direction at any time.
The two robots can communicate with each other at any distance and at any time.
The task is completed when the last robot arrives at the exit and evacuates. We
study time-energy tradeoffs for the above evacuation problem. The evacuation
time is the time it takes the last robot to reach the exit. The energy it takes
for a robot to travel a distance at speed is measured as . The
total and makespan evacuation energies are respectively the sum and maximum of
the energy consumption of the two robots while executing the evacuation
algorithm.
Assuming that the maximum speed is , and the evacuation time is at most
, where is the distance of the exit from the origin, we study the
problem of minimizing the total energy consumption of the robots. We prove that
the problem is solvable only for . For the case , we give an
optimal algorithm, and give upper bounds on the energy for the case .
We also consider the problem of minimizing the evacuation time when the
available energy is bounded by . Surprisingly, when is a
constant, independent of the distance of the exit from the origin, we prove
that evacuation is possible in time , and this is optimal up
to a logarithmic factor. When is linear in , we give upper bounds
on the evacuation time.Comment: This is the full version of the paper with the same title which will
appear in the proceedings of the 26th International Colloquium on Structural
Information and Communication Complexity (SIROCCO'19) L'Aquila, Italy during
July 1-4, 201
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