332 research outputs found
On the Hamiltonian structure of Ermakov systems
A canonical Hamiltonian formalism is derived for a class of Ermakov systems
specified by several different frequency functions. This class of systems
comprises all known cases of Hamiltonian Ermakov systems and can always be
reduced to quadratures. The Hamiltonian structure is explored to find exact
solutions for the Calogero system and for a noncentral potential with dynamic
symmetry. Some generalizations of these systems possessing exact solutions are
also identified and solved
Thresholded Covering Algorithms for Robust and Max-Min Optimization
The general problem of robust optimization is this: one of several possible
scenarios will appear tomorrow, but things are more expensive tomorrow than
they are today. What should you anticipatorily buy today, so that the
worst-case cost (summed over both days) is minimized? Feige et al. and
Khandekar et al. considered the k-robust model where the possible outcomes
tomorrow are given by all demand-subsets of size k, and gave algorithms for the
set cover problem, and the Steiner tree and facility location problems in this
model, respectively.
In this paper, we give the following simple and intuitive template for
k-robust problems: "having built some anticipatory solution, if there exists a
single demand whose augmentation cost is larger than some threshold, augment
the anticipatory solution to cover this demand as well, and repeat". In this
paper we show that this template gives us improved approximation algorithms for
k-robust Steiner tree and set cover, and the first approximation algorithms for
k-robust Steiner forest, minimum-cut and multicut. All our approximation ratios
(except for multicut) are almost best possible.
As a by-product of our techniques, we also get algorithms for max-min
problems of the form: "given a covering problem instance, which k of the
elements are costliest to cover?".Comment: 24 page
Heisenberg-picture approach to the exact quantum motion of a time-dependent forced harmonic oscillator
In the Heisenberg picture, the generalized invariant and exact quantum
motions are found for a time-dependent forced harmonic oscillator. We find the
eigenstate and the coherent state of the invariant and show that the
dispersions of these quantum states do not depend on the external force. Our
formalism is applied to several interesting cases.Comment: 15 pages, two eps files, to appear in Phys. Rev. A 53 (6) (1996
Test of Quantum Action for Inverse Square Potential
We present a numerical study of the quantum action previously introduced as a
parametrisation of Q.M. transition amplitudes. We address the questions: Is the
quantum action possibly an exact parametrisation in the whole range of
transition times ()? Is the presence of potential terms beyond
those occuring in the classical potential required? What is the error of the
parametrisation estimated from the numerical fit? How about convergence and
stability of the fitting method (dependence on grid points, resolution, initial
conditions, internal precision etc.)? Further we compare two methods of
numerical determination of the quantum action: (i) global fit of the Q.M.
transition amplitudes and (ii) flow equation. As model we consider the inverse
square potential, for which the Q.M. transition amplitudes are analytically
known. We find that the relative error of the parametrisation starts from zero
at T=0 increases to about at and then decreases to zero
when . Second, we observe stability of the quantum action under
variation of the control parameters. Finally, the flow equation method works
well in the regime of large giving stable results under variation of
initial data and consistent with the global fit method.Comment: Text (LaTeX), Figures(ps
p-Adic and Adelic Harmonic Oscillator with Time-Dependent Frequency
The classical and quantum formalism for a p-adic and adelic harmonic
oscillator with time-dependent frequency is developed, and general formulae for
main theoretical quantities are obtained. In particular, the p-adic propagator
is calculated, and the existence of a simple vacuum state as well as adelic
quantum dynamics is shown. Space discreteness and p-adic quantum-mechanical
phase are noted.Comment: 10 page
Field theoretic description of charge regulation interaction
In order to find the exact form of the electrostatic interaction between two
proteins with dissociable charge groups in aqueous solution, we have studied a
model system composed of two macroscopic surfaces with charge dissociation
sites immersed in a counterion-only ionic solution. Field-theoretic
representation of the grand canonical partition function is derived and
evaluated within the mean-field approximation, giving the Poisson-Boltzmann
theory with the Ninham-Parsegian boundary condition. Gaussian fluctuations
around the mean-field are then analyzed in the lowest order correction that we
calculate analytically and exactly, using the path integral representation for
the partition function of a harmonic oscillator with time-dependent frequency.
The first order (one loop) free energy correction gives the interaction free
energy that reduces to the zero-frequency van der Waals form in the appropriate
limit but in general gives rise to a mono-polar fluctuation term due to charge
fluctuation at the dissociation sites. Our formulation opens up the possibility
to investigate the Kirkwood-Shumaker interaction in more general contexts where
their original derivation fails.Comment: 12 pages, 9 figures, submitted to EPJ
Estimating Temperature Fluctuations in the Early Universe
A lagrangian for the essence field is constructed for a constant scalar
potential and its form determined when the scale factor was very small compared
to the present epoch but very large compared to the inflationary epoch. This
means that one is already in an expanding and flat universe. The form is
similar to that of an oscillator with time-dependent frequency. Expansion is
naturally built into the theory with the existence of growing classical
solutions of the scale factor. The formalism allows one to estimate
fluctuations of the temperature of the background radiation in these early
stages (compared to the present epoch) of the universe. If the temperature at
time is and at time the temperature is
(), then for small times, the probability for the logarithm of
inverse temperature evolution can be estimated to be given by
where
, is the Planck mass and Planck's constant and the
speed of light has been put equal to unity. There is the further possibility
that a single scalar field may suffice for an inflationary scenario as well as
the dark matter and dark energy realms.Comment: 8 pages, Revtex, title,abstract and format changed for journal
publication,no change in basic results, clarifications and a figure added.
Keywords: physics of the early universe,inflation, dark matter theory, dark
energy theory. PACS: 95.35.+d ; 95.36.+x ; 98.80.Cq ; 98.80.-
Free energy of the Fr\"ohlich polaron in two and three dimensions
We present a novel Path Integral Monte Carlo scheme to solve the Fr\"ohlich
polaron model. At intermediate and strong electron-phonon coupling, the polaron
self-trapping is properly taken into account at the level of an effective
action obtained by a preaveraging procedure with a retarded trial action. We
compute the free energy at several couplings and temperatures in three and two
dimensions. Our results show that the accuracy of the Feynman variational upper
bound for the free energy is always better than 5% although the thermodynamics
derived from it is not correct. Our estimates of the ground state energies
demonstrate that the second cumulant correction to the variational upper bound
predicts the self energy to better than 1% at intermediate and strong coupling.Comment: RevTeX 7 pages 3 figures, revised versio
Unitary relation between a harmonic oscillator of time-dependent frequency and a simple harmonic oscillator with and without an inverse-square potential
The unitary operator which transforms a harmonic oscillator system of
time-dependent frequency into that of a simple harmonic oscillator of different
time-scale is found, with and without an inverse-square potential. It is shown
that for both cases, this operator can be used in finding complete sets of wave
functions of a generalized harmonic oscillator system from the well-known sets
of the simple harmonic oscillator. Exact invariants of the time-dependent
systems can also be obtained from the constant Hamiltonians of unit mass and
frequency by making use of this unitary transformation. The geometric phases
for the wave functions of a generalized harmonic oscillator with an
inverse-square potential are given.Comment: Phys. Rev. A (Brief Report), in pres
Quantum mechanical path integrals and thermal radiation in static curved spacetimes
The propagator of a spinless particle is calculated from the quantum
mechanical path integral formalism in static curved spacetimes endowed with
event-horizons. A toy model, the Gui spacetime, and the 2D and 4D Schwarzschild
black holes are considered. The role of the topology of the coordinates
configuration space is emphasised in this framework. To cover entirely the
above spacetimes with a single set of coordinates, tortoise coordinates are
extended to complex values. It is shown that the homotopic properties of the
complex tortoise configuration space imply the thermal behaviour of the
propagator in these spacetimes. The propagator is calculated when end points
are located in identical or distinct spacetime regions separated by one or
several event-horizons. Quantum evolution through the event-horizons is shown
to be unitary in the fifth variable.Comment: 22 pages, 10 figure
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