# Showing papers in "Proceedings of The Royal Society A: Mathematical, Physical and Engineering Sciences in 1998"

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TL;DR: In this paper, a new method for analysing nonlinear and nonstationary data has been developed, which is the key part of the method is the empirical mode decomposition method with which any complicated data set can be decoded.

Abstract: A new method for analysing nonlinear and non-stationary data has been developed. The key part of the method is the empirical mode decomposition method with which any complicated data set can be dec...

16,171 citations

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TL;DR: In this article, a common pattern underpinning quantum algorithms can be identified when quantum computation is viewed as multiparticle interference, and an explicit algorithm for generating any prescribed interference pattern with an arbitrary precision is provided.

Abstract: Quantum computers use the quantum interference of different computational paths to enhance correct outcomes and suppress erroneous outcomes of computations. A common pattern underpinning quantum algorithms can be identified when quantum computation is viewed as multiparticle interference. We use this approach to review (and improve) some of the existing quantum algorithms and to show how they are related to different instances of quantum phase estimation. We provide an explicit algorithm for generating any prescribed interference pattern with an arbitrary precision.

976 citations

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TL;DR: In this article, a physically motivated regularization of the Euler equations is proposed to allow topological transitions to occur smoothly, where the sharp interface is replaced by a narrow transition layer across which the fluids may mix.

Abstract: One of the fundamental problems in simulating the motion of sharp interfaces between immiscible fluids is a description of the transition that occurs when the interfaces merge and reconnect. It is well known that classical methods involving sharp interfaces fail to describe this type of phenomena. Following some previous work in this area, we suggest a physically motivated regularization of the Euler equations which allows topological transitions to occur smoothly. In this model, the sharp interface is replaced by a narrow transition layer across which the fluids may mix. The model describes a flow of a binary mixture, and the internal structure of the interface is determined by both diffusion and motion. An advantage of our regularization is that it automatically yields a continuous description of surface tension, which can play an important role in topological transitions. An additional scalar field is introduced to describe the concentration of one of the fluid components and the resulting system of equations couples the Euler (or Navier–Stokes) and the Cahn–Hilliard equations. The model takes into account weak non–locality (dispersion) associated with an internal length scale and localized dissipation due to mixing. The non–locality introduces a dimensional surface energy; dissipation is added to handle the loss of regularity of solutions to the sharp interface equations and to provide a mechanism for topological changes. In particular, we study a non–trivial limit when both components are incompressible, the pressure is kinematic but the velocity field is non–solenoidal (quasi–incompressibility). To demonstrate the effects of quasi–incompressibility, we analyse the linear stage of spinodal decomposition in one dimension. We show that when the densities of the fluids are not perfectly matched, the evolution of the concentration field causes fluid motion even if the fluids are inviscid. In the limit of infinitely thin and well–separated interfacial layers, an appropriately scaled quasi–incompressible Euler–Cahn–Hilliard system converges to the classical sharp interface model. In order to investigate the behaviour of the model outside the range of parameters where the sharp interface approximation is sufficient, we consider a simple example of a change of topology and show that the model permits the transition to occur without an associated singularity.

781 citations

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TL;DR: The field of quantum error correction has developed spectacularly since its origin less than two years ago as discussed by the authors, and it can be classified into two categories: error correction and error correction.

Abstract: The new field of quantum error correction has developed spectacularly since its origin less than two years ago. Encoded quantum information can be protected from errors that arise due to uncontroll...

778 citations

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TL;DR: In this paper, it was shown that arbitrarily accurate quantum computations are possible provided that the error per operation is below a threshold value, which holds under physically realistic assumptions on the errors.

Abstract: Recent research has demonstrated that quantum computers can solve certain types of problems substantially faster than the known classical algorithms. These problems include factoring integers and certain physics simulations. Practical quantum computation requires overcoming the problems of environmental noise and operational errors, problems which appear to be much more severe than in classical computation due to the inherent fragility of quantum superpositions involving many degrees of freedom. Here we show that arbitrarily accurate quantum computations are possible provided that the error per operation is below a threshold value. The result is obtained by combining quantum error–correction, fault–tolerant state recovery, fault–tolerant encoding of operations and concatenation. It holds under physically realistic assumptions on the errors.

400 citations

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TL;DR: It is shown that the time evolution of the wave function of a quantum–mechanical many–particle system can be simulated precisely and efficiently on a quantum computer, and that ultimately the simulation of quantum field theory might be possible on large quantum computers.

Abstract: We show that the time evolution of the wave function of a quantum–mechanical many–particle system can be simulated precisely and efficiently on a quantum computer. The time needed for such a simulation is comparable to the time of a conventional simulation of the corresponding classical system, a performance which can9t be expected from any classical simulation of a quantum system. We then show how quantities of interest, like the energy spectrum of a system, can be obtained. We also indicate that ultimately the simulation of quantum field theory might be possible on large quantum computers.

300 citations

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TL;DR: In this article, it was shown that quantum computation is possible with mixed states instead of pure states as inputs, by embedding within the mixed state a subspace that transforms like a pure state and that can be identified by labelling it based on logical (spin), temporal, or spatial degrees of freedom.

Abstract: We show that quantum computation is possible with mixed states instead of pure states as inputs. This is performed by embedding within the mixed state a subspace that transforms like a pure state and that can be identified by labelling it based on logical (spin), temporal, or spatial degrees of freedom. This permits quantum computation to be realized with bulk ensembles far from the ground state. Experimental results are presented for quantum gates and circuits implemented with liquid nuclear magnetic resonance techniques and verified by quantum state tomography.

263 citations

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TL;DR: In this article, the authors examined the role played by small-scale instabilities and mixing formed initially by the acceleration of subcritical stratified flow over the obstacle crest, and the resulting internal hydraulic response was explained in terms of a theory that accommodates the spatially variable density difference across the sheared interface.

Abstract: Stratified flow over topography is examined in the context of its establishment from rest. A key element of numerical and steady–state analytical solutions for large amplitude topographic flow is the splitting of streamlines, which then enclose a trapped wedge of mixed fluid above the rapidly moving deeper layer. Measurements have been acquired that illustrate the development of this wedge and the role played by small–scale instabilities and mixing formed initially by the acceleration of subcritical stratified flow over the obstacle crest. The volume of trapped fluid progressively increases with time, permitting the primary flow to descend beneath it over the lee face of the obstacle. Throughout the evolution of this flow, small–scale instability and consequent entrainment would seem to be a prime candidate for producing the weakly stratified wedge, thus allowing establishment of the downslope flow to take place. Velocity structure of instabilities within the entrainment zone is observed and the associated entrainment rate determined. The entrainment is sufficient to produce a slow downstream motion within the upper layer and a density step between the layers that decreases with downstream distance. The resulting internal hydraulic response is explained in terms of a theory that accommodates the spatially variable density difference across the sheared interface. The measurements described here were acquired in a coastal inlet subject to gradually changing tidal currents. It is proposed that the observed mechanism for flow establishment also has application to atmospheric flow over mountains.

201 citations

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TL;DR: The quantum algorithms of Deutsch, Simon and Shor are described in a way which highlights their dependence on the Fourier transform and an efficient quantum factoring algorithm based on a general formalism of Kitaev is described.

Abstract: The quantum algorithms of Deutsch, Simon and Shor are described in a way which highlights their dependence on the Fourier transform. The general construction of the Fourier transform on an Abelian group is outlined and this provides a unified way of understanding the efficacy of the algorithms. Finally we describe an efficient quantum factoring algorithm based on a general formalism of Kitaev and contrast its structure to the ingredients of Shor9 algorithm.

189 citations

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TL;DR: In this article, the singular boundary value problem is considered and the existence, uniqueness, regularity and the dependency on parameters of the positive solutions under various assumptions are investigated under the same assumptions.

Abstract: We consider the singular boundary value problemWe study the existence, uniqueness, regularity and the dependency on parameters of the positive solutions under various assumptions.

156 citations

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TL;DR: In this article, new numerical results based upon the concept of solidification were produced which match the intriguing dimple observed by Kaneta in elastohydrodynamic lubrication of point contacts under pure sli...

Abstract: New numerical results, based upon the concept of solidification, are produced which match the intriguing dimple observed by Kaneta in elastohydrodynamic lubrication of point contacts under pure sli...

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TL;DR: In this article, it was shown that the discrete Darboux system admits constraints on the (adjoint) eigenfunctions which may be interpreted as discrete orthogonality conditions on the lattices.

Abstract: It is shown that the discrete Darboux system, descriptive of conjugate lattices in Euclidean spaces, admits constraints on the (adjoint) eigenfunctions which may be interpreted as discrete orthogonality conditions on the lattices. Thus, it turns out that the elementary quadrilaterals of orthogonal lattices are cyclic. Orthogonal lattices on lines, planes and spheres are discussed and the underlying integrable systems in one, two and three dimensions are derived explicitly. A discrete analogue of Bianchi's Ribaucour transformation is set down and particular orthogonal lattices are given. As a by–product, discrete Dini surfaces are obtained.

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TL;DR: In this article, the authors considered the viscous streaming from a spherical bubble undergoing small radial and lateral oscillations simultaneously, using an extension of the method of Davidson & Riley so as to include radial oscillations.

Abstract: Asymmetric patterns of streaming around a single sonoluminescent bubble have been observed by Lepoint-Mullie and co-workers. The present paper considers theoretically the viscous streaming from a spherical bubble undergoing small radial and lateral oscillations simultaneously, using an extension of the method of Davidson & Riley so as to include radial oscillations. On the assumption that the radial and lateral oscillations are of comparable magnitude, it is shown that the presence of the radial oscillations greatly enhances the streaming. The streaming is greatest when the phase difference between the oscillations is ±90°, and reverses in direction when the phase difference passes through zero or 180°.

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IBM

^{1}TL;DR: In this article, a historical review of the emergence of the quantum logic gate from the theory of reversible Boolean gates is given, highlighting the quantum XOR or controlled NOT as the fundamental two-bit gate for quantum computation.

Abstract: A historical review is given of the emergence of the idea of the quantum logic gate from the theory of reversible Boolean gates. I highlight the quantum XOR or controlled NOT as the fundamental two–bit gate for quantum computation. This gate plays a central role in networks for quantum error correction.

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TL;DR: In this article, the linear elastic response of a multicontact interface to a moderate shear force is investigated, i.e. below the threshold for incipient sliding, in the spirit of macroscopic friction laws, which should be of practical interest when evaluating the performances of a built-up system.

Abstract: The macroscopic multicontact between two rough nominally flat surfaces is a common object whose physics is only partially understood. This paper is aimed at giving experimental evidence for the linear elastic response of a multicontact interface to a moderate shear force, i.e. below the threshold for incipient sliding. Non–intuitive properties of the interfacial shear stiffness are exhibited, in the spirit of macroscopic friction laws, which should be of practical interest when evaluating the performances of a built–up system. These are explained qualitively within the random surface framework prevailing in multicontact mechanics, and a numerical treatement of the three–dimensional profile of a real rough surface is proposed, which enables a direct quantitative simulation of the elastic stiffness. This is found to be compatible with experimental data on a polymer glass and an aluminium alloy. The sensitivity of interfacial stiffness measurements is discussed, and illustrated by the experimental evidence of the plastic deformation of aluminium alloy asperities under light nominal pressure. This emphasizes the need for an elastoplastic description of asperity deformation within a multicontact.

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TL;DR: In this article, a perturbative method in critical point theory is proposed for the existence of bound states of a class of elliptic differential equations that branch off from the infimum of the essential spectrum.

Abstract: This paper consists of two main parts. The first deals with a perturbative method in critical point theory and can be seen as the generalisation and completion of some earlier results. The second part is concerned with applications of the abstract setup to the existence of bound states of a class of elliptic differential equations that branch off from the infimum of the essential spectrum.

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TL;DR: In this article, a variational formulation, leading to a pair of nonlinear differential equations subject to integral constraints, describes the post-buckling response of compressed sandwich structures.

Abstract: Compressed sandwich structures, comprising two stiff face plates separated by a softer core material, while designed principally as efficient integral structures, can lose this quality when faces buckle locally. Interaction between overall (Euler) buckling and local buckling of one face suggests that failure will localize into the centre. A variational formulation, leading to a pair of nonlinear differential equations subject to integral constraints, describes the post–buckling response. These are solved by a combination of numerical shooting and continuation techniques, such that the response far into the unstable post–buckling regime can be portrayed. Solutions with both linear and nonlinear constitutive core relations are compared with the results of an engineering (body–force) approach, and with those of earlier (periodic) Rayleigh–Ritz analyses. The latter demonstrate the extra destabilization that comes with localization.

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TL;DR: Contaminant transport in aquifers with spatially variable hydraulic and sorption properties is studied in this article, where the authors consider the case of aquifer aquifer filling.

Abstract: Contaminant transport in aquifers with spatially variable hydraulic and sorption properties

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TL;DR: The potential of quantum computation is assessed, some of the known quantum algorithms and the prospects for finding new ones are reviewed, and the specifications that should be met by future hardware are commented on.

Abstract: I assess the potential of quantum computation. Broad and important applications must be found to justify construction of a quantum computer; I review some of the known quantum algorithms and consider the prospects for finding new ones. Quantum computers are notoriously susceptible to making errors; I discuss recently developed fault–tolerant procedures that enable a quantum computer with noisy gates to perform reliably. Quantum computing hardware is still in its infancy; I comment on the specifications that should be met by future hardware. Over the past few years, work on quantum computation has erected a new classification of computational complexity, has generated profound insights into the nature of decoherence, and has stimulated the formulation of new techniques in high–precision experimental physics. A broad interdisciplinary effort will be needed if quantum computers are to fulfil their destiny as the world9s fastest computing devices. This paper is an expanded version of remarks that were prepared for a panel discussion at the ITP Conference on Quantum Coherence and Decoherence, December 1996.

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TL;DR: In this article, an aluminium single crystal of cube orientation has been rolled to 15, 30 and 50% reductions under controlled homogeneous rolling conditions and the deformation structure of the rolled specimens was investigated by both scanning electron microscopy (SEM) and TEM over several scales of magnification, where the local crystallographic orientations have been measured by an automatic electron back scattering patterns (EBSP) technique and a semiautomatic TEM method.

Abstract: An aluminium single crystal of cube orientation has been rolled to 15, 30 and 50% reductions under controlled homogeneous rolling conditions. The deformation structure of the rolled specimens was investigated by both scanning electron microscopy (SEM) and transmission electron microscopy (TEM) over several scales of magnification. The local crystallographic orientations have been measured by an automatic electron back scattering patterns (EBSP) technique and a semiautomatic TEM method. Orientation image maps based on the local orientation data have been used to reveal the evolution of the deformation structure during rolling. It is observed that by an opposite rotation around transverse direction (TD) the crystal was subdivided into four macroscopic bands, termed matrix bands in the present paper, which are parallel to the rolling plane. Between the four bands there are three transition bands in which the orientation changes continuously from that of a matrix band to that of the adjoining one. A model based on the idea of location–dependent shear strain caused by geometric and friction effects together with a plasticity analysis has been used to explain the macroscopic subdivision of the crystal. In addition to the macroscopic subdivision, a microscopic subdivision by the formation of cell–blocks within the matrix bands and a cell structure within transition bands has also been observed. A difference related to shear amplitude difference between the active slip systems changing continuously across the crystal has been observed. Both the macroscopic orientation of the dislocation boundaries and the misorientation angles and axes across dislocation boundaries are analysed and it is found that Frank9s formula is a useful tool in analysing the dislocation boundaries formed during deformation.

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TL;DR: In this paper, a conceptual stochastic model for rainfall, based on a Poisson-cluster process with rectangular pulses representing rain cells, is further developed, and a method for deriving high-order moments is applied to obtain the third-moment function for the model.

Abstract: A conceptual stochastic model for rainfall, based on a Poisson-cluster process with rectangular pulses representing rain cells, is further developed. A method for deriving high-order moments is applied to obtain the third-moment function for the model. This is used with second-order properties to fit the model to January and July time-series taken from a site in Wellington, New Zealand. It is found that the parameter estimates may follow two solution paths converging on an optimum value over a bounded interval. The parameter estimates are used with the model to simulate 200 years of hourly data, and parametric tests used to compare simulated and observed extreme rainfalls. These show good agreement over a range of sampling intervals. The paper concludes with a discussion of the standard errors of the model parameter estimates which are obtained using a non-parametric bootstrap.

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TL;DR: In this article, the authors formulate the principles of classical statistical inference in a natural geometric setting, and establish generalisations of the classical Cramer-Rao and Bhattacharyya inequalities, described in terms of the geometry of the underlying real Hilbert space.

Abstract: A statistical model M is a family of probability distributions, characterised by a set of continuous parameters known as the parameter space. This possesses natural geometrical properties induced by the embedding of the family of probability distributions into the space of all square–integrable functions. More precisely, by consideration of the square–root density function we can regard M as a submanifold of the unit sphere S in a real Hilbert space H . Therefore, H embodies the ‘state space’ of the probability distributions, and the geometry of the given statistical model can be described in terms of the embedding of M in S . The geometry in question is characterised by a natural Riemannian metric (the Fisher–Rao metric), thus allowing us to formulate the principles of classical statistical inference in a natural geometric setting. In particular, we focus attention on the variance lower bounds for statistical estimation, and establish generalisations of the classical Cramer–Rao and Bhattacharyya inequalities, described in terms of the geometry of the underlying real Hilbert space. As a comprehensive illustration of the utility of the geometric framework, the statistical model M is then specialised to the case of a submanifold of the state space of a quantum mechanical system. This is pursued by introducing a compatible complex structure on the underlying real Hilbert space, which allows the operations of ordinary quantum mechanics to be reinterpreted in the language of real Hilbert space geometry. The application of generalised variance bounds in the case of quantum statistical estimation leads to a set of higher order corrections to the Heisenberg uncertainty relations for canonically conjugate observables.

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TL;DR: In this paper, the authors studied the Cauchy problem in Rn of general parabolic equations which take the form ut = Δum + ts|x|σup with non-negative initial value.

Abstract: In this paper we study the Cauchy problem in Rn of general parabolic equations which take the form ut = Δum + ts|x|σup with non-negative initial value. Here s ≧ 0, m > (n − 2)+/n, p > max (1, m) and σ > − 1 if n = 1 or σ > − 2 if n ≧ 2. We prove, among other things, that for p ≦ pc, where pc ≡ m + s(m − 1) + (2 + 2s + σ)/n > 1, every nontrivial solution blows up in finite time. But for p > pc a positive global solution exists.

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TL;DR: In this article, the thermodynamic cost of error correction is analyzed and it is shown that error correction can be regarded as a kind of 'Maxwell demon', for which there is an entropy cost associated with information obtained from measurements performed during error correction.

Abstract: Quantum operations provide a general description of the state changes allowed by quantum mechanics. The reversal of quantum operations is important for quantum error–correcting codes, teleportation and reversing quantum measurements. We derive information–theoretic conditions and equivalent algebraic conditions that are necessary and sufficient for a general quantum operation to be reversible. We analyse the thermodynamic cost of error correction and show that error correction can be regarded as a kind of `Maxwell demon', for which there is an entropy cost associated with information obtained from measurements performed during error correction. A prescription for thermodynamically efficient error correction is given.

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TL;DR: In this article, a relationship between the periodic and Dirichlet boundary value problems for second-order ODEs with singularities is established, which may be useful in explaining the difference between the nonresonance of singular and nonsingular differential equations.

Abstract: In this paper, a relationship between the periodic and the Dirichlet boundary value problems for second-order ordinary differential equations with singularities is established. This relationship may be useful in explaining the difference between the nonresonance of singular and nonsingular differential equations. Using this relationship, we give in this paper an existence result of positive periodic solutions to singular differential equations when the singular forces satisfy some strong force condition at the singularity 0 and some linear growth condition at infinity.

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TL;DR: In this paper, the authors considered a two-dimensional problem of scattering of a time harmonic electromagnetic plane wave by an inhomogeneous conducting or dielectric layer on a perfectly conducting plate.

Abstract: We consider a two-dimensional problem of scattering of a time harmonic electromagnetic plane wave by an inhomogeneous conducting or dielectric layer on a perfectly conducting plate. The magnetic permeability is assumed to be a fixed positive constant in the media. The material properties of the media are characterized completely by an index of refraction, which is a bounded measurable function in the layer and a positive constant above the layer corresponding to a homogeneous dielectric medium. In this paper, we only examine the TM (transverse magnetic) polarization case. A radiation condition is introduced and equivalence with a second kind, Lippmann-Schwinger-type integral equation is shown. With additional assumptions on the index of refraction in the layer, uniqueness of solution is proved. Existence of solution is then established by employing a form of Fredholm alternative using a general result on the solvability of integral equations on unbounded domains published earlier by Chandler-Wilde and Zhang. An approximate analytic solution for the case of a thin inhomogeneous layer is obtained from the integral equation formulation and is used to show that, if the index of refraction is appropriately chosen, the scattered field can grow with distance from the plate.

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TL;DR: In this article, the local Cauchy problem for the generalized Kadomtsev-Petviashvili equation is discussed in both the periodic and non-periodic settings.

Abstract: We discuss the local Cauchy problem for the generalised Kadomtsev–Petviashvili equation, namely , in the periodic and nonperiodic settings.

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TL;DR: In this paper, the generalized weighted mean values with two parameters are defined; their basic properties and monotonicities are investigated, and their properties are shown to be monotone.

Abstract: The generalized weighted mean values with two parameters are defined; their basic properties and monotonicities are investigated.

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TL;DR: A spatial–temporal model of rainfall is studied in which storms arrive in a Poisson process in time, each storm giving rise to a random number of elliptical rain cells.

Abstract: A spatial–temporal model of rainfall is studied in which storms arrive in a Poisson process in time, each storm giving rise to a random number of elliptical rain cells. Each rain cell moves with a random velocity for a random time before terminating. Rain is deposited by the cell at a random intensity which is constant over the area of the cell and over its lifetime. The main properties of this model are studied analytically where possible. Further properties and the aggregation of model properties over space for direct comparison with rainfall radar data require the numerical evaluation of integrals.

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TL;DR: In this article, a technique for calculating exponentially small terms beyond all orders in singularly perturbed ODEs is presented, based on the application of a WKBJ-type ansatz to the late terms in the naive asymptotic expansion and the identification of Stokes lines.

Abstract: A technique for calculating exponentially small terms beyond all orders in singularly perturbed ordinary differential equations is presented. The approach is based on the application of a WKBJ–type ansatz to the late terms in the naive asymptotic expansion and the identification of Stokes lines, and is closely related to the well–known Stokes line–smoothing phenomenon in linear ordinary differential equations. The method is illustrated by application to examples.