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Bellman, R. 1959 Asymptotic behaviour of solutions of differential-difference equations. Mem. Am. Math. Soc. 35, 123-134.

Bellman, R. & Cooke, K. L. 1963 Differential-difference equations, 2nd edn, pp. 342-348. New York: Academic Press.

Bellman, R., Dyson, A. & Cooke, K. L. 1964 The deficiency index problems. Lecture Notes in Mathematics, vol. 621. Heidelberg: Springer-Verlag.

Bellman, R., Cooke, K. L. & Dyson, A. 1965 Shock dynamics in metals. In Proc. Int. Conf. on Mechanical Behaviour of Materials (ed. J. L. Miller & R. F. Smith), vol. 2, pp. 173-183. Oxford: Pergamon Press.

Proc. R. Soc. Lond. A (1994)

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For Te}X authors, the source code for these references is:

\begin{thebibliography>{>\bibitem[] {}Bellman, R. 1959Asymptotic behaviour of solutions of differential-difference equations. {\em Mem. Am. Math. Soc.} {\bf 35}, 123-134.

% Author and year 7. Title

7, Journal details\bibitem [] {}Bellman, R. \& Cooke, K. L. 1963{\em Differential-difference equations}, 7» Title and book details 2nd edn, pp.~342— 348.

\bibitem[] {}Bellman, R., Dyson, A. \& Cooke, K. L. 1964 {\em The deficiency index problems.}Lecture Notes in Mathematics, vol.~621.Heidelberg: Springer-Verlag.\bibitem[] {}Bellman, R., Cooke, K. L. \& Dyson, A. 1965 Shock dynamics in metals. % Title of paperIn {\em Proc. Int. Conf. on Mechanical 7» Title of conferenceBehaviour of Materials'^}(ed. J. L. Miller \& R. F. Smith), 7. Editorsvol.~2, pp.~173— 183.Oxford: Pergamon Press.

\end{thebibliography}

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New York: Academic Press. 7, Place and publisher

[November 1994]

Proc. R. Soc. bond. A (1994)

Indexes to Volume 447 (A)

Author index

Augousti, M. See Sureshkumar et al.

Bau, H. H. See Hu & Bau.; see also Tang & Bau.Benveniste, Y. Exact connections between polycrystal and crystal properties in two-dimensional

polycrystalline aggregates, 1.Beris, A. N. See Sureshkumar et al.Berry, M. V. & Howls, C. J. High orders of the Weyl expansion for quantum billiards: resurgence

of periodic orbits, and the Stokes phenomenon, 527.Boyd, W. G. C. Gamma function asymptotics by an extension of the method of steepest descents,

' 609.Bruno, O. P. & Reitich, F. Maxwell equations in a nonlinear Kerr medium, 65.

Carleton, F. B. & Weinberg, F. J. Correlating minimum ignition criteria, 513.Cates, A. T. See Sionoid & Cates.Chandler, J. M. See Lane et al.Cheng, A. See Desroches et al.Cowpertwait, P. S. P. A generalized point process model for rainfall, 23.

Desroches, J., Detournay, E., Lenoach, B., Papanastasiou, P., Pearson, J. R. A., Thiercelin, M. & Cheng, A. The crack tip region in hydraulic fracturing, 39.

Detournay, E. See Desroches et al.Doinikov, A. A. Acoustic radiation pressure on a rigid sphere in a viscous fluid, 447.Dolan, P. & Kim, C. W. The wave equation for the Lanczos potential. I, 557.Dolan, P. & Kim, C. W. Some solutions of the Lanczos vacuum wave equation, 577.

Howls, C. J. See Berry & Howls.Hoyle, J. M. & Smith, F. T. On finite-time break-up in three-dimensional interacting boundary

layers, 467.Hu, H. H. & Bau, H. H. Feedback control to delay or advance linear loss of stability in planar

Poiseuille flow, 299.

Jensen, H. M. See Thouless et al.

Kailas, S. V. & Narasim ha, R. Sim ilarity in viTA-detected events in a nearly neutral atmospheric boundary layer, 211.

Keating, J. P. & Sieber, M. Calculation of spectral determinants, 413.Keller, J. B. See Wolf & Keller.Kevlahan, N. K.-R. & Vassilicos, J. C. The space and scale dependencies of the self-similar

structure of turbulence, 341.Kim, C. W. See Dolan & Kim.Kornev, K. & Mukhamadullina, G. Mathematical theory of freezing for flow in porous media, 281.

Lane, S. N., Richards, K. S. & Chandler, J. M. Distributed sensitivity analysis in modelling environmental systems, 49.

Lenoach, B. See Desroches et al.Liniger, E. G. See Thouless et al.

McGeehan, J. P. See Wijayasuriya et al.Mackaplow, M. B., Shaqfeh, E. S. G. & Schiek, R. L. A numerical study of heat and mass transport

in fibre suspensions, 77.

651

652 Author index

Malakondaiah, G. See Srinivas et al.Merkin, J. H. See Metcalf et al.Metcalf, M. J., Merkin, J. H. & Scott, S. K. Oscillating wave fronts in isothermal chemical systems

with arbitrary powers of autocatalysis, 155.Mukhamadullina, G. See Kornev & Mukhamadullina.

Narasimha, R. See Kailas & Narasimha.Navarro, A. See Rios et al.Norris, A. N. Dynamic Green’s functions in anisotropic piezoelectric, thermoelastic and poroelastic

solids, 175.Norton, G. H. & Wijayasuriya, S. S. H. New properties of convolutional codes and some

applications to communication systems, 439; see also Wijayasuriya et al.

Papanastasiou, P. See Desroches et al.Pearson, J. R. A. See Desroches et al.Penrose, P. Is there a ‘canonical’ non-equilibrium ensemble?, 631.Percival, I. C. Primary state diffusion, 189.

Rama Rao, P. See Srinivas et al.Reitich, F. See Bruno & Reitich.Richards, K. S. See Lane et al.Rios, E. R. de los, Xin, X. J. & Navarro, A. Modelling microstructurally sensitive fatigue short

crack growth, 111.Roulstone, I. See Sewell & Roulstone.

Schiek, R. L. See Mackaplow et al.Scott, S. K. See Metcalf et al.Sewell, M. J. & Roulstone, I. Families of lift and contact transformations, 493.Shaqfeh, E. S. G. See Mackaplow et al.Sieber, M. See Keating & Sieber.Sionoid, P. N. & Cates, A. T. The generalized Burgers and Zabolotskaya-Kholov equations:

transformations, exact solutions and qualitative properties, 253.Smith, F. T. See Hoyle & Smith.Srinivas, M., Malakondaiah, G. & Rama Rao, P. Influence of solute additions on the fracture

behaviour of Armco iron, 223.Srinivas, M., Sundararajan, G., Malakondaiah, G. & Rama Rao, P. An analysis of ductile fracture

initiation toughness in iron, its binary alloys and nickel, 237.Sundararajan, G. See Srinivas et al.Sureshkumar, R., Beris, A. N. & Augousti, M. Non-axisymmetric subcritical bifurcations in

viscoelastic Taylor-Couette flow, 135.

Talbot, D. R. S. & Willis, J. R. Upper and lower bounds for the overall properties of a nonlinear composite dielectric. I. Random microgeometry, 365.

Talbot, D. R. S. & Willis, J. R. Upper and lower bounds for the overall properties of a nonlinear composite dielectric. II. Periodic microgeometry, 385.

Tang, J. & Bau, H. H. Stabilization of the no-motion state in the Rayleigh-Benard problem, 587. Thiercelin, M. See Desroches et al.Thouless, M. D., Jensen, H. M. & Liniger, E. G. Delamination from edge flaws, 271.

Vassilicos, J. C. See Kevlahan & Vassilicos.

Weinberg, F. J. See Carleton & Weinberg.Wijayasuriya, S. S. H., Norton, G. H. & McGeehan, J. P. Analysis of sliding window decorrelation

in DS-CDMA mobile radio, 313; see also Norton & Wijayasuriya.Willis, J. R. See Talbot & Willis.Wolf, S. A. & Keller, J. B. Range of the first two eigenvalues of the Laplacian, 397.

Xin, X. J. See Rios et al.

Subject index

acoustic radiation pressure, 447. alloy softening, 223. anisotropy, 175. arrow of time, 631. asymptotic expansions, 609. asymptotics, 527. atmospheric boundary layer, 211. autocatalysis, 155. axisymmetric sound field, 447.

boundary layers, 467. bounded mean oscillation, 365. bounds, 365, 385. buckling, 271.

caustic, 253. chaos, 155. cluster models, 23. clustered point processes, 23. composite dielectric, 365, 385. composite materials, 1. compressive stress, 271. computational hydraulics, 49. continuum mechanics, 281. control, 587.control of bifurcation, 299. convolutional code, 439. crack initiation micromechanisms, 223. crack initiation toughness, 237. crack instability, 111. crystal-poly crystal connections, 1.

decorrelation, 313. diffraction, 253. diffusion, 253.diffusion-controlled reaction, 77. diffusion equations, 189. divergent series, 527. duality, 493.ductile crack initiation, 237. dynamical systems, 631.

eddy structure, 341. eigenvalue lower bounds, 397. eigenvalue upper bounds, 397. envelopes, 493. environmental modelling, 49. equalizer, 439. ergodic theory, 631. experimental tests, 189. explicit approximation, 65. explosion initiation, 513.

Fabry-Perot resonator, 65.feedback control, 299, 587.fibre composites, 77.fibre optics, hazards, 513.finite sequence length, 313.flammable atmospheres, hazards, 513.flow events, 211.focusing, 253.fractals, 341.fracture mechanics, 39.front instabilities, 155.

gamma function, 609.Green’s functions, 175.

hamiltonian systems, 493. hazards, ignition, 513. heat transfer, 281. high Reynolds numbers, 467. hydraulic fracture, 39. hydrothermal stability, 587.

ideal fluids, 281. ignition criteria, 513. initiation fracture toughness, 223. interaction, 467. interface crack, 271. intrinsic theory, 189. iron-solid solutions, 223. irreversibility, 631. isothermal flames, 155.

Jlc of alloys, 237.Jlc of iron, 237.

Kerr effect, 65.

laminates, 271.Lanczos potential, 557, 577.Laplacian eigenvalues, 397.Legendre transformations, 493. localized failure, 271.

measurement, quantum, 189. membrane frequencies, 397. meteorology, 493. micromechanical modelling, 111. micromechanisms of cracking, 237. microstructural-continuum transition, 111. microstructural-sensitive fatigue, 111. mixing, 631. model for JIC, 237.

653

Subject indexmomentum events, 211. multiple access, 313. multi-site time series, 23. multi-user detectors, 313.

near-far resistance, 313. non-axisymmetric, 135. non-equilibrium statistical mechanics, 631. nonlinear acoustics, 253. nonlinear composite dielectric, 365, 385. nonlinear Maxwell equations, 65. nonlinear optics, 65.

optical sensors, hazards, 513. oscillatory chemical waves, 155.

partial differential equations, 493. particle interactions, 77. periodic microgeometry, 385. periodic orbit formulae, 413. piezoelasticity, 175. piezoelectric aggregates, 1. planar domain eigenvalues, 397.Poiseuille flow, 299. polycrystals, 1. poroelasticity, 175. predictive property, 439.

quantum theory foundations, 189.

rain cells, 23. rainfall time series, 23. random microgeometry, 365. range of eigenvalues, 397. Rayleigh-Benard convection, 587. reflective mirrors, 65. resonators, 527. resultant, 439.Riemann-Siegel resummation, 413. river channel hydraulics, 49. roughness, 49.

654screening length, 77. semiclassical expansions, 527. semiclassical quantization, 413. semiclassical zeta-function expansions, 413 semidilute fibre suspensions, 77. sensitivity analysis, 49. shear flow, 299. shocks, 253.short crack growth, 111.similarity in turbulence events, 211.singularities, 467.singularity, 341.solute effects on fracture, 223.spinor wave equation, 557, 577.spirals, 341.stability, 135, 299.steepest descents, method of, 609.stochastic models, 23.strain/precipitation hardening, 111.subcritical bifurcations, 135.

Taylor-Couette flow, 135. tensor wave equation, 557, 577. thermal conductivity, effective, 77. thermal convection, 587. thermoelasticity, 175. thermo-hydrodynamics, 281. thin films, 271. three dimensions, 467. transition, 467. trellis path uniqueness, 439. turbulence, 211, 341.

vacuum wave equation, 577. viscoelastic flow, 135. viscous fluid, 447.Viterbi decoder, 439.

wave equation, 527. wave propagation, 155. wavelet transform, 341.Weyl field, 577.

End of the four hundred and forty-seventh volume ( A)

Proceedings ofThe Royal Society of London

Series AMathematical and Physical Sciences

VOLUME 447 1994

Published byT H E ROYAL SO CIETY

6 Carlton House Terrace, London SW1Y 5AG

ISSN 0962-8444

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resultant surface pH is in excess of 7.5, which gives maximum practical permanence.

Printed in G reat Britain by the University Press, Cambridge

ContentsSeries A Volume 446

No. 1929 8 October 1994

Exact connections between polycrystal and crystal properties in two-dimensional polycrystalline aggregates

By Y. Benveniste page 1A generalized point process model for rainfall

By P. S. P. Cowpertwait 23The crack tip region in hydraulic fracturing

By J. Desroches, E. Detournay, B. Lenoach, P. Papanastasiou,J. R. A. Pearson, M. Thiercelin and A. Cheng 39

Application of distributed sensitivity analysis to a model of turbulent open channel flow in a natural river channelBy S. N. Lane, K. S. Richards and J. H. Chandler 49

Maxwell equations in a nonlinear Kerr mediumBy 0. P. Bruno and F. Reitich 65

A numerical study of heat and mass transport in fibre suspensionsBy M. B. Mackaplow, E. S. G. Shaqfeh and R. L. Schiek 77

Modelling microstructurally sensitive fatigue short crack growthBy E. R. de los Rios, X. J. Xin and A. Navarro 111

Non-axisymmetric subcritical bifurcations in viscoelastic Taylor-Couette flowBy R. Sureshkumar, A. N. Beris and M. Avgousti 135

Oscillating wave fronts in isothermal chemical systems with arbitrary powers of autocatalysisBy M. J. Metcalf, J. H. Merkin and S. K. Scott 155

Dynamic Green’s functions in anisotropic piezoelectric, thermoelastic and poroelastic solidsBy A. N. Norris 175

Primary state diffusionBy I. C. Percival 189

No. 1930 8 November 1994Sim ilarity in viTA-detected events in a nearly neutral atm ospheric boundary layer

By S. V. Kailas and R. Narasimha 211Influence of solute additions on the fracture behaviour of Armco iron

By M. Srinivas, G. Malakondaiah and P. Rama Rao 223An analysis of ductile fracture initiation toughness in iron, its binary alloys and nickel

By M. Srinivas, G. Sundararajan, G. Malakondaiah and P. Rama Rao 237The generalized Burgers and Zabolotskaya-Khokhlov equations: transformations, exact

solutions and qualitative propertiesBy P. N. Sionoid and A. T. Cates 253

Delamination from edge flawsBy M. D. Thouless, H. M. Jensen and E. G. Liniger 271

Mathematical theory of freezing for flow in porous mediaBy K. Kornev and G. Mukhamadullina 281

Feedback control to delay or advance linear loss of stability in planar Poiseuille flowBy H. H. Hu and H. H. Bau ' 299

Analysis of sliding window decorrelation in DS-CDMA mobile radioBy S. S. H. Wijayasuriya, G. H. Norton and J. P. McGeehan 313

The space and scale dependencies of the self-similar structure of turbulenceBy N. K. R. Kevlahan and J. C. Vassilicos 341

Upper and lower bounds for the overall properties of a nonlinear composite dielectric.I. Random microgeometryBy D. R. S. Talbot and J. R. Willis 365

iv Contents

Upper and lower bounds for the overall properties of a nonlinear composite dielectric.II. Periodic microgeometryBy D. R. S. Talbot and J. R. Willis 385

Range of the first two eigenvalues of the laplacianBy S. A. Wolf and J. B. Keller 397

Calculation of spectral determinantsBy J. P. Keating and M. Sieber 413

New properties of convolutional codes and some applications to communication systemsBy G. H. Norton and S. S. H. Wijayasuriya 439

No. 1931 8 December 1994Acoustic radiation pressure on a rigid sphere in a viscous fluid

By A. A. Doinikov 447On finite-time break-up in three-dimensional unsteady interacting boundary layers

By J. M. Hoyle and F. T. Smith 467Families of lift and contact transformations

By M. J. Sewell and I. Roulstone 493Correlating minimum ignition criteria

By F. B. Carleton and F. J. Weinberg 513High orders of the Weyl expansion for quantum billiards: resurgence of periodic orbits,

and the Stokes phenomenonBy M. V. Berry and C. J. Howls 527

The wave equation for the Lanczos potential. IBy P. Dolan and C. W. Kim 557

Some solutions of the Lanczos vacuum wave equationBy P. Dolan and C. W. Kim 577

Stabilization of the no-motion state in the Rayleigh Benard problemBy J. Tang and H. H. Bau 587

Gamma function asymptotics by an extension of the method of steepest descentsBy W. G. C. Boyd ' 609

Is there a ‘canonical’ non-equilibrium ensemble?By O. Penrose and P. V. Coveney 631

Instructions to Authors 647Indexes 651

Philosophical Transactions: Physical Sciences and Engineering

Series A Volume 348 Number 1688 15 September 1994

Chaos and forecastingA Discussion organized and edited by B. Grenfell, R. M. May and H. Tong

B. Cheng & H. TongOrthogonal projection, embedding dimension and sample size in chaotic time series from a statistical perspective

C. D. CutlerA theory of correlation dimension for stationary time series

Q. Y ao & H. TongOn prediction and chaos in stochastic systems

L. A. Smith

Local optimal prediction: exploiting strangeness and the variation of sensitivity to initial condition

R. C. L. Wolff

Independence in time series: another look at the BDS test

B. LeBaron

Chaos and nonlinear forecastability in economics and finance

A. S. WeigendParadigm change in prediction

L. F. Olsen, K. R. Valeur, T. Geest, C. W. Tidd & W. M. Schaffer Nonlinear forecasting of non-uniform chaotic attractors in an enzyme reaction

R. Hide

Chaos in geophysical fluids. I. General introduction

N. O. WeissChaotic modulation of the solar cycle

M. Matsuzaki Fractals in earthquakes

T. N. Palmer, R. Buizza, F. Molteni, Y.-C. Chen & S. Corti Singular vectors and the predictability of weather and climate

G. SugiharaNonlinear forecasting for the classification of natural time series

D. A. Rand

Measuring and characterizing spatial patterns, dynamics and chaos in spatially extended dynamical systems and ecologies

B. T. Grenfell, A. Kleczkowski, S. P. Ellner & B. M. Bolker Measles as a case study in nonlinear forecasting and chaos

THE ROYAL SOCIETY

Proceedings: Mathematical and Physical Sciences

Series A Volume 447 Number 1931 8 December 1994

CONTENTS

A. A. Doinikov pages 447-466Acoustic radiation pressure on a rigid sphere in a viscous fluid

J. M. H oyle & F. T. S m ith 467-492On finite-time break-up in three-dimensional unsteady interacting boundary layers

M. J. Sewell & I. Roulstone 493-512Families of lift and contact transformations

F. B. Carleton & F. J. Weinberg 513-526Correlating minimum ignition criteria

M. V. Berry & C. J. Howls 527-555High orders of the Weyl expansion for quantum billiards: resurgence of periodic orbits, and the Stokes phenomenon

P. Dolan & C. W. K im 557-575The wave equation for the Fanczos potential. If

P. Dolan & C. W. K im 577-585Some solutions of the Fanczos vacuum wave equation f

J. T a n g & H. H. Bau 587-607Stabilization of the no-motion state in the Rayletgh-Benard problem f

W. G. C. Boyd * 609-630Gamma function asymptotics by an extension of the method of steepest descents!

O. Penrose & P. V. Coveney 631-646Is there a ‘canonical’ non-equilibrium ensemble?!

Instructions to Authors 647-650

Indexes 651-654* * *

V olum e T itle P age a n d C on ten ts

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