72,752 research outputs found
Solution to the Equations of the Moment Expansions
We develop a formula for matching a Taylor series about the origin and an
asymptotic exponential expansion for large values of the coordinate. We test it
on the expansion of the generating functions for the moments and connected
moments of the Hamiltonian operator. In the former case the formula produces
the energies and overlaps for the Rayleigh-Ritz method in the Krylov space. We
choose the harmonic oscillator and a strongly anharmonic oscillator as
illustrative examples for numerical test. Our results reveal some features of
the connected-moments expansion that were overlooked in earlier studies and
applications of the approach
Parametrized Post-Newtonian Theory of Reference Frames, Multipolar Expansions and Equations of Motion in the N-body Problem
We discuss a covariant generalization of the parametrized post-Newtonian
(PPN) formalism in a class of scalar-tensor theories of gravity. It includes an
exact construction of a set of global and local (Fermi-like) references frames
for an isolated N-body astronomical system as well as PPN multipolar
decomposition of gravitational field in these frames. We derive PPN equations
of translational and rotational motion of extended bodies taking into account
all gravitational multipoles and analyze the body finite-size effects in
relativistic dynamics that can be important at the latest stages of orbital
evolution of coalescing binary systems. We also reconcile the IAU 2000
resolutions on the general relativistic reference frames in the solar system
with the PPN equations of motion of the solar system bodies used in JPL
ephemerides.Comment: 121 pages, 5 figures, references added, improvements made in response
to referee's repor
Nonlinear Photonic Crystals: IV. Nonlinear Schrodinger Equation Regime
We study here the nonlinear Schrodinger Equation (NLS) as the first term in a
sequence of approximations for an electromagnetic (EM) wave propagating
according to the nonlinear Maxwell equations (NLM). The dielectric medium is
assumed to be periodic, with a cubic nonlinearity, and with its linear
background possessing inversion symmetric dispersion relations. The medium is
excited by a current producing an EM wave. The wave nonlinear
evolution is analyzed based on the modal decomposition and an expansion of the
exact solution to the NLM into an asymptotic series with respect to some three
small parameters , and . These parameters are
introduced through the excitation current to scale respectively
(i) its amplitude and consequently the magnitude of the nonlinearity; (ii) the
range of wavevectors involved in its modal composition, with
scaling its spatial extension; (iii) its frequency bandwidth, with scaling its time extension. We develop a consistent theory of
approximations of increasing accuracy for the NLM with its first term governed
by the NLS. We show that such NLS regime is the medium response to an almost
monochromatic excitation current . The developed approach not only
provides rigorous estimates of the approximation accuracy of the NLM with the
NLS in terms of powers of , and , but it also
produces new extended NLS (ENLS) equations providing better approximations.
Remarkably, quantitative estimates show that properly tailored ENLS can
significantly improve the approximation accuracy of the NLM compare with the
classical NLS
Post-Newtonian Gravitational Radiation
1 Introduction 2 Multipole Decomposition 3 Source Multipole Moments 4
Post-Minkowskian Approximation 5 Radiative Multipole Moments 6 Post-Newtonian
Approximation 7 Point-Particles 8 ConclusionComment: 46 pages, in Einstein's Field Equations and Their Physical
Implications, B. Schmidt (Ed.), Lecture Notes in Physics, Springe
Asymptotic safety in the f(R) approximation
In the asymptotic safety programme for quantum gravity, it is important to go
beyond polynomial truncations. Three such approximations have been derived
where the restriction is only to a general function f(R) of the curvature R>0.
We confront these with the requirement that a fixed point solution be smooth
and exist for all non-negative R. Singularities induced by cutoff choices force
the earlier versions to have no such solutions. However, we show that the most
recent version has a number of lines of fixed points, each supporting a
continuous spectrum of eigen-perturbations. We uncover and analyse the first
five such lines. Sensible fixed point behaviour may be achieved if one
consistently incorporates geometry/topology change. As an exploratory example,
we analyse the equations analytically continued to R<0, however we now find
only partial solutions.We show how these results are always consistent with,
and to some extent can be predicted from, a straightforward analysis of the
constraints inherent in the equations.Comment: Latex, 66 pages, published version, typos correcte
Basic Concepts Underlying Singular Perturbation Techniques
In many singular perturbation problems multiple scales are used. For instance, one may use both the coordinate x and the coordinate x^* = ε^(-1)x. In a secular-type problem x and x^* are used simultaneously. This paper discusses layer-type problems in which x^* is used in a thin layer and x outside this layer. Assume one seeks approximations to a function f(x,ε), uniformly valid to some order in ε for x in a closed interval D. In layer-type problems one uses (at least) two expansions (called inner and outer) neither of which is uniformly valid but whose domains of validity together cover the interval D. To define "domain of validity" one needs to consider intervals whose endpoints depend on epsilon. In the construction of the inner and outer expansions, constants and functions of e occur which are determined by comparison of the two expansions "matching." The comparison is possible only in the domain of overlap of their regions of validity. Once overlap is established, matching is easily carried out. Heuristic ideas for determining domains of validity of approximations by a study of the corresponding equations are illustrated with the aid of model equations. It is shown that formally small terms in an equation may have large integrated effects. The study of this is of central importance for understanding layer-type problems. It is emphasized that considering the expansions as the result of applying limit processes can lead to serious errors and, in any case, hides the nature of the expansions
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