The method of finite-product extraction and an application to Wiener-Hopf theory

Copyright @ The Author, 2011. The publisher version of the article can be accessed at the link below.In this work we describe a simple method for finding approximate representations for special functions which are entire transcendental functions that can be represented by infinite products. This method replaces the infinite product by a finite polynomial and Gamma functions. This approximate representation is shown in the case of Bessel functions to be very accurate over a large range of parameter values. These approximate expressions can be useful for finding the roots of a transcendental equation and the Wiener-Hopf factorization of functions involving such Bessel functions.The method is shown to be potentially useful for other transcendental andWiener-Hopf problems, which involve other entire functions that have infinite product representations

Asymptotic optimality of the cross-entropy method for Markov chain problems

The correspondence between the cross-entropy method and the zero-variance approximation to simulate a rare event problem in Markov chains is shown. This leads to a sufficient condition that the cross-entropy estimator is asymptotically optimal.Comment: 13 pager; 3 figure

Constraints from precision measurements on the hadron-molecule interpretation of X,Y,Z resonances

The precise measurement of binding energy, total width and DDpi branching fraction, constrains, in a model independent way, the loosely bound hadron molecule interpretation of X(3872). A full discernment is not yet possible on the basis of present data. Similar considerations might be extended to all those Y,Z resonances susceptible of a molecular description.Comment: 6 pages, 1 figur

Matrix Wiener-Hopf-Hilbert factorisation

A method is described for effecting the explicit Wiener-Hopf factorisation of a class of (2 x 2)-matrices. The class is determined such that the factorisation problem can be reduced to a matrix Hilbert problem which involves an upper or lower triangular matrix. Then the matrix Hilbert problem can be further reduced to three scalar Hilbert problems on a half-line, which are solvable in the standard manner

Approximate boundary conditions for diffraction by thin transmissive media

The object of this note is to describe a method that can be used to obtain useful boundary conditions to model various situations that arise in diffraction theory. In particular when wanting to apply the Wiener-Hopf technique to diffraction problems that involve thin transmissive media. Transmissive here means that the thin layer medium suffers a change in the physical quantities of density, acoustic velocity, and wave number from the surrounding medium. The present approach can be used to obtain approximate boundary conditions for other physical applications where thin strata of transmissive material arise

A note on uniform asymptotic wave diffraction by a wedge

New expressions for asymptotically uniform Green’s functions for high-frequency wave diffraction when a plane, cylindrical or point wave field is incident on an ideal wedge are derived. They are useful for deriving a uniform asymptotic expression for the exact solution in terms of the high-frequency diffracted and geometrical optics far field. The present method is simple and consists of differentiating out the singularities of the integral representations and using new representations for trigonometrical sums that arise when the wedge angle is a rational multiple of π. The new results make explicit the continuity of the fields across shadow and reflection boundaries

A note on point source diffraction by a wedge

The object of this paper is to give new expressions for the wave field produced when a time harmonic point source is diffracted by a wedge with Dirichlet or Neumann boundary conditions on its faces. The representation of the total field is expressed in terms of quadratures of elementary functions, rather than Bessel functions, which is usual in the literature. An analogous expression is given for the three-dimensional free-space Green's function

A bifurcated circular waveguide problem

This is a pre-copy-editing, author-produced PDF of an article accepted for publication in IMA Journal of Applied Mathematics following peer review. The definitive publisher-authenticated version A D Rawlins. A bifurcated circular waveguide problem. J.I.M.A. 54 (1995) 59-81. Oxford University press is available online at: http://imamat.oxfordjournals.org/cgi/reprint/54/1/59.pdfA rigorous and exact solution is obtained for the problem of the radiation of sound from a semi-infinite rigid duct inserted axially into a larger acoustically lined tube of infinite length. The solution to this problem is obtained by the Wiener-Hopf technique. The transmission and reflection coefficients, when the fundamental mode propagates in the semi-infinite tube, are obtained. The present results could be of use for exhaust design, and as a possible instrument for impedance measurement