384 research outputs found
Self-similarity under inflation and level statistics: a study in two dimensions
Energy level spacing statistics are discussed for a two dimensional
quasiperiodic tiling. The property of self-similarity under inflation is used
to write a recursion relation for the level spacing distributions defined on
square approximants to the perfect quasiperiodic structure.
New distribution functions are defined and determined by a combination of
numerical and analytical calculations.Comment: Latex, 13 pages including 6 EPS figures, paper submitted to PR
An explicit calculation of the Ronkin function
We calculate the second order derivatives of the Ronkin function in the case
of an affine linear polynomial in three variables and give an expression of
them in terms of complete elliptic integrals and hypergeometric functions. This
gives a semi-explicit expression of the associated Monge-Amp\`ere measure, the
Ronkin measure.Comment: 22 pages, 13 figure
Numerical wave optics and the lensing of gravitational waves by globular clusters
We consider the possible effects of gravitational lensing by globular
clusters on gravitational waves from asymmetric neutron stars in our galaxy. In
the lensing of gravitational waves, the long wavelength, compared with the
usual case of optical lensing, can lead to the geometrical optics approximation
being invalid, in which case a wave optical solution is necessary. In general,
wave optical solutions can only be obtained numerically. We describe a
computational method that is particularly well suited to numerical wave optics.
This method enables us to compare the properties of several lens models for
globular clusters without ever calling upon the geometrical optics
approximation, though that approximation would sometimes have been valid.
Finally, we estimate the probability that lensing by a globular cluster will
significantly affect the detection, by ground-based laser interferometer
detectors such as LIGO, of gravitational waves from an asymmetric neutron star
in our galaxy, finding that the probability is insignificantly small.Comment: To appear in: Proceedings of the Eleventh Marcel Grossmann Meetin
Computing hypergeometric functions rigorously
We present an efficient implementation of hypergeometric functions in
arbitrary-precision interval arithmetic. The functions , ,
and (or the Kummer -function) are supported for
unrestricted complex parameters and argument, and by extension, we cover
exponential and trigonometric integrals, error functions, Fresnel integrals,
incomplete gamma and beta functions, Bessel functions, Airy functions, Legendre
functions, Jacobi polynomials, complete elliptic integrals, and other special
functions. The output can be used directly for interval computations or to
generate provably correct floating-point approximations in any format.
Performance is competitive with earlier arbitrary-precision software, and
sometimes orders of magnitude faster. We also partially cover the generalized
hypergeometric function and computation of high-order parameter
derivatives.Comment: v2: corrected example in section 3.1; corrected timing data for case
E-G in section 8.5 (table 6, figure 2); adjusted paper siz
On Klein's Icosahedral Solution of the Quintic
We present an exposition of the icosahedral solution of the quintic equation
first described in Klein's classic work "Lectures on the icosahedron and the
solution of equations of the fifth degree". Although we are heavily influenced
by Klein we follow a slightly different approach which enables us to arrive at
the solution more directly.Comment: 29 pages, 5 figure
A fast semi-direct least squares algorithm for hierarchically block separable matrices
We present a fast algorithm for linear least squares problems governed by
hierarchically block separable (HBS) matrices. Such matrices are generally
dense but data-sparse and can describe many important operators including those
derived from asymptotically smooth radial kernels that are not too oscillatory.
The algorithm is based on a recursive skeletonization procedure that exposes
this sparsity and solves the dense least squares problem as a larger,
equality-constrained, sparse one. It relies on a sparse QR factorization
coupled with iterative weighted least squares methods. In essence, our scheme
consists of a direct component, comprised of matrix compression and
factorization, followed by an iterative component to enforce certain equality
constraints. At most two iterations are typically required for problems that
are not too ill-conditioned. For an HBS matrix with
having bounded off-diagonal block rank, the algorithm has optimal complexity. If the rank increases with the spatial dimension as is
common for operators that are singular at the origin, then this becomes
in 1D, in 2D, and
in 3D. We illustrate the performance of the method on
both over- and underdetermined systems in a variety of settings, with an
emphasis on radial basis function approximation and efficient updating and
downdating.Comment: 24 pages, 8 figures, 6 tables; to appear in SIAM J. Matrix Anal. App
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