Method for the reconstruction of a function from its line integrals

A new method for the determination of a two-dimensional function from its line integrals is presented. Compared with the standard solutions of this problem, it has the advantage of not using series expansions and also it provides an alternative way of collecting the experimentally measured line integrals.     

Causes and cures for the infinities in slow-motion expansions in general relativity

The causes of the divergent integrals arising in slow-motion expansions of the general relativistic field equations are studied and a remedy for them is suggested. This is done within the context of a model problem involving a coupled nonlinear scalar field and isotropic oscillator. The model is shown to give rise to divergent integrals directly attributable to the nonlinearity when the field is assumed to be analytic in a slowness parameter. Application of a nonregular perturbation approach which includes the method of matched asymptotic expansions is shown to eliminate the infinite contributions.Supported in part by The National Science Foundation under grant no. PH 79-15.     

The failure of specular limit formulae for Kirchhoff integrals associated with Gaussian surfaces with ocean-like spectra

We analyse the use of asymptotic expansions for Kirchhoff and related integrals for scattering from Gaussian surfaces, for the limit of high Rayleigh parameter (high frequency, near-normal incidence). An extension of the traditional analysis of this type has been given very recently by Tatarskii and Tatarskii. We show that such expansions are invalid for typical frequencies of interest (L-, C-, X-band radar) when the correlation function arises from a class of spectra with ocean-like characteristics, and present alternative numerical results for such integrals. The near-normal-incidence behaviour of these integrals can in fact be fitted quite well by a parametrized Gaussian, but with parameters not directly related to the distribution of slopes. This suggests that measurements of the cross section near normal incidence may not be the reliable indicator of surface-slope parameters that they are thought to have been. These results also illustrate the dangers of using asymptotic expansions in cases where the relevant parameters are insufficiently large for the asymptotic method to be valid.     

Shorter Expansions of Molecular Integrals in Hylleraas-CI Calculations

Over Cartesian Gaussian orbitals, Clementi and his co-workers have recently derived the formulas of molecular integrals appearing in the Hylleraas-CI method. There are many terms which exactly cancel one another in their formulas. By using the approach of generating functions, these terms are removed, giving rise to shorter expansions and simpler expressions for the molecular integrals.     

On Analytical Treatment of SU(3) Lattice Gauge Theory

The order parameters of SU(3) lattice gauge theory Ep at zero temperature and (L) at finitetemperature are calculated by the variational cumulant expansion method. The SU(3) singlelinkgroup integrals are treated as double definite integrals. Two plaquettes with oppositeorientations are combined into a single one to simplify the count of equivalent connectedgraphs. Three different approaches to choose variational parameters are compared. Theagreement with the Monte-Carlo simulations is good in the strong and weak coupling regions.The necessity of higher order expansions in the intermediate coupling region is discussed.     

Cluster expansion for the dielectric constant of a polarizable suspension

We derive a cluster expansion for the electric susceptibility kernel of a dielectric suspension of spherically symmetric inclusions in a uniform background. This also leads to a cluster expansion for the effective dielectric constant. It is shown that the cluster integrals of any order are absolutely convergent, so that the dielectric constant is well defined and independent of the shape of the sample in the limit of a large system. We compare with virial expansions derived earlier in statistical mechanics for the dielectric constant of a nonpolar gas. In these expansions the virial coefficients are given by integrals which are only conditionally convergent.     

Analytical expressions for momentum autocorrelation function of a classic diatomic chain

We use recurrence relations method to study a classical harmonic diatomic chain. The momentum autocorrelation function results from contributions of acoustic and optical branches. By use of convolution theorem, analytical expressions for the acoustic and optical branches are derived as even-order Bessel function expansions. The expansion coefficients are given in terms of integrals of real and complex elliptic functions for the acoustic and optical branches, respectively. Double convolution results respectively in integrals of trigonometric and hyperbolic functions for expansion coefficients of acoustic and optical branches.     

HIGH-FREQUENCY ASYMPTOTICS FOR THE HELMHOLTZ EQUATION IN A HALF-PLANE

Base on the integral representations of the solution being derived via Fokas' transform method, the high-frequency asymptotics for the solution of the Helmholtz equation, in a half-plane and subject to the Neumann condition is discussed. For the case of piecewise constant boundary data, full asymptotic expansions of the solution are obtained by using Watson's lemma and the method of steepest descents for definite integrals.     

Master integrals for four-loop massless propagators up to weight twelve

We evaluate a Laurent expansion in dimensional regularization parameter ?=(4−d)/2 of all the master integrals for four-loop massless propagators up to weight twelve, using a recently developed method of one of the present coauthors (R.L.) and extending thereby results by Baikov and Chetyrkin obtained at weight seven. We observe only multiple zeta values in our results. Therefore, we conclude that all the four-loop massless propagator integrals, with any integer powers of numerators and propagators, have only multiple zeta values in their epsilon expansions up to weight twelve.     

Twistor diagram equalities for generalized mass-scattering integrals for Dirac fields

A method which enables one to convert the integral of a holomorphic projective twistor one-form into the integral of a holomorphic projective twistor three-form is used to modify the end-vertex structures of certain twistor diagrams that represent mass-scattering integrals for Dirac fields. Each term of the twistorial diagrammatic expansions recovering the entire fields is then re-expressed appropriately. This gives rise to four sets of explicit twistor-diagram equalities for the mass-scattering formulae.I am most grateful to Professor Roger Penrose for making a key point concerning the introduction of the basic auxiliary-twistor vertices. My warmest thanks go to the World Laboratory for supporting this work financially. I wish to acknowledge the Third World Academy of Sciences for a relevant travel grant. I am grateful, also, to Dr. Asghar Quadir for his invaluable suggestions.     

Disappearance of time integrals of exact memory functions in time-convolution generalized master equations

Memory functions in time-convolution Generalized Master Equations (GME) for probabilities of finding a general system (interacting by a general coupling with a true thermodynamic bath) in individual states are considered without resorting to any approximation. After taking the thermodynamic bath limit, time integrals from zero to infinite times of the memories are considered. It is argued that these integrals entering, e.g., the usual naive Markov approximation converting GME the Pauli master (PME) equations are exactly zero. This implies long-time tails of memories (unobtainable by perturbational expansions) and slower-than-exponential long-time asymptotics of relaxation.     

Asymptotic analysis of Gaussian integrals,II: Manifold of minimum points

This paper derives the asymptotic expansions of a wide class of Gaussian function space integrals under the assumption that the minimum points of the action form a nondegenerate manifold. Such integrals play an important role in recent physics. This paper also proves limit theorems for related probability measures, analogous to the classical law of large numbers and central limit theorem.Alfred P. Sloan Research Fellow. Research supported in part by NSF Grant MCS-80-02149Research supported in part by NSF Grant MCS-80-02140     

High order quadratures for the evaluation of interfacial velocities in axi-symmetric Stokes flows

We propose new high order accurate methods to compute the evolution of axi-symmetric interfacial Stokes flow. The velocity at a point on the interface is given by an integral over the surface. Quadrature rules to evaluate these integrals are developed using asymptotic expansions of the integrands, both locally about the point of evaluation, and about the poles, where the interface crosses the axis of symmetry. The local expansions yield methods that converge to the chosen order pointwise, for fixed evaluation point. The pole expansions yield corrections that remove maximal errors of low order, introduced by singular behaviour of the integrands as the evaluation point approaches the poles. An interesting example of roundoff error amplification due to cancellation is also addressed. The result is a uniformly accurate fifth order method. Second order, pointwise fifth order, and uniform fifth order methods are applied to compute three sample flows, each of which presents a different computational difficulty: an initially bar-belled drop that pinches in finite time, a drop in a strain flow that approaches a steady state, and a continuously extending drop. In each case, the fifth order methods significantly improve the ability to resolve the flow. The examples furthermore give insight into the effect of the corrections needed for uniformity. We determine conditions under which the pointwise method is sufficient to obtain resolved results, and others under which the corrections significantly improve the results.     

The diffraction of radio waves by the curvature of the earth

This paper gives a simplified approximate derivation of the field at moderate heights and distances due to an oscillating electric dipole at moderate height above the earth, taking into account the curvature and electrical constants of the earth. The method applies to both vertically and horizontally polarized waves. It is closely connected with treating the earth as flat, the propagation of the waves being curved upwards. In this way the expansions in Legendre functions can be replaced by Fourier integrals, which are simpler to handle. The results of previous workers are reproduced, but are obtained more directly in a form suitable for easy computation.     

Functional integral mean field expansions for nuclear many fermion systems

We present a careful analysis of the auxiliary field functional integral formalism for many fermion systems. We examine the limiting procedure used in construction of such integrals and show that a wide flexibility exists with respect to the choice of the one-body field representation upon which mean field expansions are made. We demonstrate the utility of this flexibility in the context of the evaluation of the grand canonical partition function. We examine the zero order. RPA and certain higher-order terms. The above-mentioned flexibility is reflected in the dependence of the results on a trial two-body interaction, different choices of which produce Hartree, Fock, HartreeFock or other forms of the mean field expansions. A standard variational procedure selects the Hartree-Fock as the optimal choice. With this choice we find certain corrections to previously reported RPA contribution for the Hartree mean field. We also indicate the relevance of our formulation for the recent applications of the functional integral mean field approach to nuclear dynamical problems.     

Adiabatic dilations and essential Regge spectrum

Adiabatic perturbation theory works for non-conserved dilation operators. At each effective coupling, all renormalization group functions are dilation eigenvalues of a “tangent” field theory, which preserves scale invariance instead of translation invariance. The adiabatic expansion reconstructs the true field theory from this tangent bundle. For complex angular momentum, the short and long distance expansions mix indecomposably at certain turning points. By connecting the UV and IR limits across them, the adiabatic method determines Regge intercepts from integrals over the renormalization group. Fixed Regge cuts or accumulations of poles are insensitive to the IR region.     

The influence of solid bodies on low Mach number vortex sound

The mechanisms of sound generation by unsteady, subsonic flows in the presence of solid boundaries are investigated. For this purpose an alternative integral representation for the radiated pressure field is applied which is different from the generally used integral representation introduced by Lighthill and Curle. The main advantage of the method is that there is a linear dependence of the integrand on the time derivative of the vorticity fluctuations in the hydrodynamic near field; instead of the ordinary Green function a “vector Green function” is used. This vector Green function can be chosen for a given flow field in such a way that surface integrals do not appear. Finally, the theory is illustrated by two- and three-dimensional model flows. Analytical solutions are determined by applying the method of matched asymptotic expansions.     

On the sound field of a shallow spherical shell in an infinite baffle

A method is presented for calculating the far field sound radiation from a shallow spherical shell in an acoustic medium. The shell has a concentrated ring mass boundary condition at its perimeter representing a loudspeaker voice coil and is excited by a concentrated ring force exerted by the end of the voice coil. A Green's function is developed for a shallow spherical shell, which is based upon Reissner's solution to the shell wave equation [Q. Appl. Math. 13, 279-290 (1955)]. The shell is then coupled to the surrounding acoustic medium using an eigenfunction expansion, with unknown coefficients, for its deflection. The resulting surface pressure distribution is solved using the King integral together with the free space Green's function in cylindrical coordinates. In order to eliminate the need for numerical integration, the radiation (coupling) integrals are solved analytically to yield fast converging expansions. Hence, a set of simultaneous equations is obtained which is solved for the coefficients of the eigenfunction expansion. These coefficients are finally used in formulas for the far field sound radiation.     

Asymptotics of Tracy-Widom Distributions and the Total Integral of a Painlevé II Function

The Tracy-Widom distribution functions involve integrals of a Painlevé II function starting from positive infinity. In this paper, we express the Tracy-Widom distribution functions in terms of integrals starting from minus infinity. There are two consequences of these new representations. The first is the evaluation of the total integral of the Hastings-McLeod solution of the Painlevé II equation. The second is the evaluation of the constant term of the asymptotic expansions of the Tracy-Widom distribution functions as the distribution parameter approaches minus infinity. For the GUE Tracy-Widom distribution function, this gives an alternative proof of the recent work of Deift, Its, and Krasovsky. The constant terms for the GOE and GSE Tracy-Widom distribution functions are new.     

Exciton states in alkali halides

Recent experiments based on modulation spectroscopy have shown that it is possible to detect exciton levels in alkali halides up to n = 4. Therefore we worked out numerical calculations in order to predict the whole exciton series in KI and RbI. In our calculations the deep exciton levels are treated by considering the actual hole-electron interaction, whereas the effective mass approximation is used for the shallow exciton levels. The direct and exchange terms of the hole-electron interaction have been evaluated by performing three and four center integrals, the Wannier wave functions appearing in such integrals being approximated by suitable gaussian expansions of atomic orbitals.It is shown that by allowing the exciton state to extend up to 42 shells of neighbors it is possible to predict the exciton levels up to n = 2, the n = 3, 4 excitons being accounted for by the effective mass approximation. Similar computations performed for excitons in solid rare gases were found in excellent agreement with the experimental data and confirmed the reliability of our method.