596 research outputs found
A Lanczos Method for Approximating Composite Functions
We seek to approximate a composite function h(x) = g(f(x)) with a global
polynomial. The standard approach chooses points x in the domain of f and
computes h(x) at each point, which requires an evaluation of f and an
evaluation of g. We present a Lanczos-based procedure that implicitly
approximates g with a polynomial of f. By constructing a quadrature rule for
the density function of f, we can approximate h(x) using many fewer evaluations
of g. The savings is particularly dramatic when g is much more expensive than f
or the dimension of x is large. We demonstrate this procedure with two
numerical examples: (i) an exponential function composed with a rational
function and (ii) a Navier-Stokes model of fluid flow with a scalar input
parameter that depends on multiple physical quantities
Superradiant instabilities of rotating black branes and strings
Black branes and strings are generally unstable against a certain sector of
gravitational perturbations. This is known as the Gregory-Laflamme instability.
It has been recently argued that there exists another general instability
affecting many rotating extended black objects. This instability is in a sense
universal, in that it is triggered by any massless field, and not just
gravitational perturbations. Here we investigate this novel mechanism in
detail. For this instability to work, two ingredients are necessary: (i) an
ergo-region, which gives rise to superradiant amplification of waves, and (ii)
``bound'' states in the effective potential governing the evolution of the
particular mode under study. We show that the black brane Kerr_4 x R^p is
unstable against this mechanism, and we present numerical results for
instability timescales for this case. On the other hand, and quite
surprisingly, black branes of the form Kerr_d x R^p are all stable against this
mechanism for d>4. This is quite an unexpected result, and it stems from the
fact that there are no stable circular orbits in higher dimensional black hole
spacetimes, or in a wave picture, that there are no bound states in the
effective potential. We also show that it is quite easy to simulate this
instability in the laboratory with acoustic black branes.Comment: 19 pages, 10 figures. v2: Enlarged discussion on the necessary
conditions for the existence of instabilit
On the Spectrum of Field Quadratures for a Finite Number of Photons
The spectrum and eigenstates of any field quadrature operator restricted to a
finite number of photons are studied, in terms of the Hermite polynomials.
By (naturally) defining \textit{approximate} eigenstates, which represent
highly localized wavefunctions with up to photons, one can arrive at an
appropriate notion of limit for the spectrum of the quadrature as goes to
infinity, in the sense that the limit coincides with the spectrum of the
infinite-dimensional quadrature operator. In particular, this notion allows the
spectra of truncated phase operators to tend to the complete unit circle, as
one would expect. A regular structure for the zeros of the Christoffel-Darboux
kernel is also shown.Comment: 16 pages, 11 figure
Quasiclassical Approach to Transport in the Vortex State and the Hall Effect
We derive generalized quasiclassical transport equations which include the
terms responsible for the Hall Effect in the vortex state of a clean type-II
superconductor, and calculate the conductivity tensor for an s-wave
superconductor in the high-field regime. We find that below the superconducting
transition the contribution to the transverse conductivity due to dynamical
fluctuations of the order parameter is compensated by the modification of the
quasiparticle contribution. In this regime the nonlinear behaviour of the Hall
angle is governed by the change in the effective quasiparticle scattering rate
due to the reduction in the density of states at the Fermi level. The
connection with experimental results is discussed.Comment: 32 pages, 3 postscript figure
Degenerate Rotating Black Holes, Chiral CFTs and Fermi Surfaces I - Analytic Results for Quasinormal Modes
In this work we discuss charged rotating black holes in
that degenerate to extremal black holes with zero entropy. These black holes
have scaling properties between charge and angular momentum similar to those of
Fermi surface operators in a subsector of SYM. We add a
massless uncharged scalar to the five dimensional supergravity theory, such
that it still forms a consistent truncation of the type IIB ten dimensional
supergravity and analyze its quasinormal modes. Separating the equation of
motion to a radial and angular part, we proceed to solve the radial equation
using the asymptotic matching expansion method applied to a Heun equation with
two nearby singularities. We use the continued fraction method for the angular
Heun equation and obtain numerical results for the quasinormal modes. In the
case of the supersymmetric black hole we present some analytic results for the
decay rates of the scalar perturbations. The spectrum of quasinormal modes
obtained is similar to that of a chiral 1+1 CFT, which is consistent with the
conjectured field-theoretic dual. In addition, some of the modes can be found
analytically.Comment: 41 pages, 1 figure, LaTeX; v2: typos corrected, references adde
Sparse Pseudospectral Approximation Method
Multivariate global polynomial approximations - such as polynomial chaos or
stochastic collocation methods - are now in widespread use for sensitivity
analysis and uncertainty quantification. The pseudospectral variety of these
methods uses a numerical integration rule to approximate the Fourier-type
coefficients of a truncated expansion in orthogonal polynomials. For problems
in more than two or three dimensions, a sparse grid numerical integration rule
offers accuracy with a smaller node set compared to tensor product
approximation. However, when using a sparse rule to approximately integrate
these coefficients, one often finds unacceptable errors in the coefficients
associated with higher degree polynomials.
By reexamining Smolyak's algorithm and exploiting the connections between
interpolation and projection in tensor product spaces, we construct a sparse
pseudospectral approximation method that accurately reproduces the coefficients
of basis functions that naturally correspond to the sparse grid integration
rule. The compelling numerical results show that this is the proper way to use
sparse grid integration rules for pseudospectral approximation
Non-iterative computation of Gauss-Jacobi quadrature
Asymptotic approximations to the zeros of Jacobi polynomials are given, with methods to obtain the coefficients in the expansions. These approximations can be used as standalone methods for the noniterative computation of the nodes of Gauss--Jacobi quadratures of high degree (). We also provide asymptotic approximations for functions related to the first-order derivative of Jacobi polynomials which are used for computing the weights of the Gauss--Jacobi quadrature. The performance of the asymptotic approximations is illustrated with numerical examples, and it is shown that nearly double precision relative accuracy is obtained for both the nodes and the weights when and . For smaller degrees the approximations are also useful as they provide relative accuracy for the nodes when , and just one Newton step would be sufficient to guarantee double precision accuracy in that cases
The impact of Stieltjes' work on continued fractions and orthogonal polynomials
Stieltjes' work on continued fractions and the orthogonal polynomials related
to continued fraction expansions is summarized and an attempt is made to
describe the influence of Stieltjes' ideas and work in research done after his
death, with an emphasis on the theory of orthogonal polynomials
Acceleration of generalized hypergeometric functions through precise remainder asymptotics
We express the asymptotics of the remainders of the partial sums {s_n} of the
generalized hypergeometric function q+1_F_q through an inverse power series z^n
n^l \sum_k c_k/n^k, where the exponent l and the asymptotic coefficients {c_k}
may be recursively computed to any desired order from the hypergeometric
parameters and argument. From this we derive a new series acceleration
technique that can be applied to any such function, even with complex
parameters and at the branch point z=1. For moderate parameters (up to
approximately ten) a C implementation at fixed precision is very effective at
computing these functions; for larger parameters an implementation in higher
than machine precision would be needed. Even for larger parameters, however,
our C implementation is able to correctly determine whether or not it has
converged; and when it converges, its estimate of its error is accurate.Comment: 36 pages, 6 figures, LaTeX2e. Fixed sign error in Eq. (2.28), added
several references, added comparison to other methods, and added discussion
of recursion stabilit
Mass hierarchy, mass gap and corrections to Newton's law on thick branes with Poincare symmetry
We consider a scalar thick brane configuration arising in a 5D theory of
gravity coupled to a self-interacting scalar field in a Riemannian manifold. We
start from known classical solutions of the corresponding field equations and
elaborate on the physics of the transverse traceless modes of linear
fluctuations of the classical background, which obey a Schroedinger-like
equation. We further consider two special cases in which this equation can be
solved analytically for any massive mode with m^2>0, in contrast with numerical
approaches, allowing us to study in closed form the massive spectrum of
Kaluza-Klein (KK) excitations and to compute the corrections to Newton's law in
the thin brane limit. In the first case we consider a solution with a mass gap
in the spectrum of KK fluctuations with two bound states - the massless 4D
graviton free of tachyonic instabilities and a massive KK excitation - as well
as a tower of continuous massive KK modes which obey a Legendre equation. The
mass gap is defined by the inverse of the brane thickness, allowing us to get
rid of the potentially dangerous multiplicity of arbitrarily light KK modes. It
is shown that due to this lucky circumstance, the solution of the mass
hierarchy problem is much simpler and transparent than in the (thin)
Randall-Sundrum (RS) two-brane configuration. In the second case we present a
smooth version of the RS model with a single massless bound state, which
accounts for the 4D graviton, and a sector of continuous fluctuation modes with
no mass gap, which obey a confluent Heun equation in the Ince limit. (The
latter seems to have physical applications for the first time within braneworld
models). For this solution the mass hierarchy problem is solved as in the
Lykken-Randall model and the model is completely free of naked singularities.Comment: 25 pages in latex, no figures, content changed, corrections to
Newton's law included for smooth version of RS model and an author adde
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