Showing papers in "Proceedings of The Royal Society A: Mathematical, Physical and Engineering Sciences in 1992"

Rapid Solution of Problems by Quantum Computation

Abstract: A class of problems is described which can be solved more efficiently by quantum computation than by any classical or stochastic method. The quantum computation solves the problem with certainty in exponentially less time than any classical deterministic computation.

Properties of Sufficiency and Statistical Tests

Abstract: 1—In a previous paper*, dealing with the importance of properties of sufficiency in the statistical theory of small samples, attention was mainly confined to the theory of estimation. In the present paper the structure of small sample tests, whether these are related to problems of estimation and fiducial distributions, or are of the nature of tests of goodness of fit, is considered further.

Transformation field analysis of inelastic composite materials

Abstract: A new method is proposed for evaluation of local fields and overall properties of composite materials subjected to incremental thermomechanical loads and to transformation strains in the phases. The composite aggregate may consist of many perfectly bonded inelastic phases of arbitrary geometry and elastic material symmetry. In principle, any inviscid or time-dependent inelastic constitutive relation that complies with the additive decomposition of total strains can be admitted in the analysis. The governing system of equations is derived from the representation of local stress and strain fields by novel transformation influence functions and concentration factor tensors, as discussed in the preceding paper by G. J. Dvorak and Y. Benveniste. The concentration factors depend on local and overall thermoelastic moduli, and can be evaluated with a selected micromechanical model. Applications to elastic-plastic, viscoelastic, and viscoplastic systems are discussed. The new approach is contrasted with some presently accepted procedures based on the self-consistent and Mori-Tanaka approximations, which are shown to violate exact relations between local and overall inelastic strains.

Helicity and the Calugareanu Invariant

Abstract: The helicity of a localized solenoidal vector field (i.e. the integrated scalar product of the field and its vector potential) is known to be a conserved quantity under ‘frozen field’ distortion of the ambient medium. In this paper we present a number of results concerning the helicity of linked and knotted flux tubes, particularly as regards the topological interpretation of helicity in terms of the Gauss linking number and its limiting form (the Calugareanu invariant). The helicity of a single knotted flux tube is shown to be intimately related to the Calugareanu invariant and a new and direct derivation of this topological invariant from the invariance of helicity is given. Helicity is decomposed into writhe and twist contributions, the writhe contribution involving the Gauss integral (for definition, see equation (4.8)), which admits interpretation in terms of the sum of signed crossings of the knot, averaged over all projections. Part of the twist contribution is shown to be associated with the torsion of the knot and part with what may be described as ‘intrinsic twist’ of the field lines in the flux tube around the knot (see equations (5.13) and (5.15)). The generic behaviour associated with the deformation of the knot through a configuration with points of inflexion (points at which the curvature vanishes) is analysed and the role of the twist parameter is discussed. The derivation of the Calugareanu invariant from first principles of fluid mechanics provides a good demonstration of the relevance of fluid dynamical techniques to topological problems.

A New Perspective on Constrained Motion

Abstract: The explicit general equations of motion for constrained discrete dynamical systems are obtained. These new equations lead to a simple and new fundamental view of lagrangian mechanics.

Periodic homogenisation of certain fully nonlinear partial differential equations

Abstract: We demonstrate how a fairly simple “perturbed test function” method establishes periodic homogenisation for certain Hamilton-Jacobi and fully nonlinear elliptic partial differential equations. The idea, following Lions, Papanicolaou and Varadhan, is to introduce (possibly nonsmooth) correctors, and to modify appropriately the theory of viscosity solutions, so as to eliminate then the effects of high-frequency oscillations in the coefficients.

On transformation strains and uniform fields in multiphase elastic media

Abstract: The effect of local eigenstrain and eigenstress fields, or transformation fields, on the local strains and stresses is explored in multiphase elastic solids of arbitrary geometry and material symmetry. The residual local fields caused by such transformation fields are sought in terms of certain transformation influence functions and transformation concentration factor tensors. General properties of these functions and concentration factors, and their relation to the analogous mechanical influence functions and concentration factors, are established, in part, with the help of uniform strain fields in multiphase media. Specific estimates of the transformation concentration factor tensors are evaluated by the self-consistent and Mori-Tanaka methods. It is found here that although the two methods use different constraint tensors in solutions of the respective dilute problems, their estimates of the mechanical, thermal, and transformation concentration factor tensors, and of the overall stiffness of multiphase media have a similar structure. Proofs that guarantee that these methods comply with the general properties of the transformation influence functions, and provide diagonally symmetric estimates of the overall elastic stiffness, are given for two-phase and multiphase systems consisting of, or reinforced by, inclusions of similar shape and alignment. One of the possible applications of the results, in analysis of overall instantaneous properties and local fields in inelastic composite materials, is described in the following paper.

Rank-one convexity does not imply quasiconvexity

Abstract: We consider variational integralsdefined for (sufficiently regular) functions u: Ω→Rm. Here Ω is a bounded open subset of Rn, Du(x) denotes the gradient matrix of u at x and f is a continuous function on the space of all real m × n matrices Mm × n. One of the important problems in the calculus of variations is to characterise the functions f for which the integral I is lower semicontinuous. In this connection, the following notions were introduced (see [3], [9], [10]).

The validity of modulation equations for extended systems with cubic nonlinearities

Abstract: Modulation equations play an essential role in the understanding of complicated systems near the threshold of instability. Here we show that the modulation equation dominates the dynamics of the full problem locally, at least over a long time-scale. For systems with no quadratic interaction term, we develop a method which is much simpler than previous ones. It involves a careful bookkeeping of errors and an estimate of Gronwall type.As an example for the dissipative case, we find that the Ginzburg–Landau equation is the modulation equation for the Swift–Hohenberg problem. Moreover, the method also enables us to handle hyperbolic problems: the nonlinear Schrodinger equation is shown to describe the modulation of wave packets in the Sine–Gordon equation.

Water waves for small surface tension : an approach via normal form

Abstract: In this paper we determine the possible crest-forms of permanent waves of small amplitude which exist on the free surface of a two-dimensional fluid layer under the influence of gravity and surface tension when the Froude number is close to 1. The Bond number b, measuring surface tension, is assumed to satisfy b < ⅓. We find one-parameter families of periodic waves of two different types, quasiperiodic waves and solitary waves with oscillations at infinity. The existence of true solitary waves is established for a sequence of systems approximating the full Euler equations in every algebraic order of − 1.

Density of smooth functions in Wk,p(x)(Ω)

Abstract: Kovacik & Rakosnik investigated the spaces L$^{p(x)}$($\Omega $) of functions which are integrable with variable power p(x) and the corresponding counterparts of the Sobolev spaces W$^{k,p(x)}$($\Omega $). We continue that investigation and describe a class of functions p(x) for which the set of smooth functions on $\Omega $ is dense in W$^{k,p(x)}$($\Omega $). As a corollary we obtain in terms of the distance function a condition on elements of W$^{k,p(x)}$($\Omega $) sufficient to ensure that they belong to W$_{0}^{k,p(x)}$($\Omega $).

A New Asymptotic Representation for $\zeta $($\frac{1}{2}$+it) and Quantum Spectral Determinants

Abstract: By analytic continuation of the Dirichlet series for the Riemann zeta function ζ(s) to the critical line s = ½ + i t ( t real), a family of exact representations, parametrized by a real variable K , is found for the real function Z ( t ) = ζ(½ + i t ) exp {iθ( t )}, where θ is real. The dominant contribution Z 0 ( t,K ) is a convergent sum over the integers n of the Dirichlet series, resembling the finite ‘main sum ’ of the Riemann-Siegel formula (RS) but with the sharp cut-off smoothed by an error function. The corrections Z 3 ( t,K ), Z 4 ( t,K )... are also convergent sums, whose principal terms involve integers close to the RS cut-off. For large K , Z 0 contains not only the main sum of RS but also its first correction. An estimate of high orders m ≫ 1 when K t 1/6 shows that the corrections Z k have the ‘factorial/power ’ form familiar in divergent asymptotic expansions, the least term being of order exp { ─½ K 2 t }. Graphical and numerical exploration of the new representation shows that Z 0 is always better than the main sum of RS, providing an approximation that in our numerical illustrations is up to seven orders of magnitude more accurate with little more computational effort. The corrections Z 3 and Z 4 give further improvements, roughly comparable to adding RS corrections (but starting from the more accurate Z 0 ). The accuracy increases with K , as do the numbers of terms in the sums for each of the Z m . By regarding Planck’s constant h as a complex variable, the method for Z ( t ) can be applied directly to semiclassical approximations for spectral determinants ∆( E, h ) whose zeros E = E j ( h ) are the energies of stationary states in quantum mechanics. The result is an exact analytic continuation of the exponential of the semiclassical sum over periodic orbits given by the divergent Gutzwiller trace formula. A consequence is that our result yields an exact asymptotic representation of the Selberg zeta function on its critical line.

Shock tube study of the drag coefficient of a sphere in a non-stationary flow

Abstract: A shock tube facility was used for inducing relatively high acceleration on small spheres laid on the shock tube floor. The acceleration resulted from the drag force imposed by the post shock wave flow. Using double exposure holography, the sphere trajectory could be constructed accurately. Based upon such trajectories, the sphere drag coefficient was evaluated for a relatively wide range of Reynolds numbers (6000≤ Re ≤ 101000). It was found that the value obtained for the sphere drag coefficient were significantly larger than those obtained in a similar steady flow case.

Exact Analytic Solutions for Diffusion Impeded By an Infinite Array of Partially Permeable Barriers

Abstract: We give an exact solution for the macroscopic diffusion of particles in one dimension impeded by any set of plane, parallel partially permeable barriers. Explicit results are given for one barrier, and for an infinite system of uniformly spaced barriers, for any initial position of the particles. This model can be generalized from slits to square tubes, to cubic pores, etc., and is proposed as a convenient model, with disposable parameters, for fitting to experiment. Various convenient and physically illuminating approximations are discussed. An alternative, and sometimes more convenient, way of computing the exact solution for complicated systems of pores is also given.

Invariant Properties of the Stress in Plane Elasticity and Equivalence Classes of Composites

Abstract: Attention is drawn to the invariance of the stress field in a two-dimensional body loaded at the boundary by fixed forces when the compliance tensor $\scr{G}$($\chi $) is shifted uniformly by $\ell^{\text{I}}$($\lambda $, -$\lambda $), where $\lambda $ is an arbitrary constant and $\scr{G}^{\text{I}}$($\kappa $, $\mu $) is the compliance tensor of a isotropic material with two-dimensional bulk and shear moduli $\kappa $ and $\mu $. This invariance is explained from two simple observations: first, that in two dimensions the tensor $\scr{G}^{\text{I}}$($\frac{1}{2}$, -$\frac{1}{2}$) acts to locally rotate the stress by 90 degrees and the second that this rotated field is the symmetrized gradient of a vector field and therefore can be treated as a strain. For composite materials the invariance of the stress field implies that the effective compliance tensor $\ell^{\ast}$ also gets shifted by $\scr{G}^{\text{I}}$($\lambda $, -$\lambda $) when the constituent moduli are each shifted by $\ell^{\text{I}}$($\lambda $, -$\lambda $). This imposes constraints on the functional dependence of $\ell^{\ast}$ on the material moduli of the components. Applied to an isotropic composite of two isotropic components it implies that when the inverse bulk modulus is shifted by the constant 1/$\lambda $ and the inverse shear modulus is shifted by -1/$\lambda $, then the inverse effective bulk and shear moduli undergo precisely the same shifts. In particular it explains why the effective Young's modulus of a two-dimensional media with holes does not depend on the Poisson's ratio of the matrix material.

Universality classes and fluctuations in disordered systems

Abstract: the medium. We apply this result to trace(T TL)M, where TL is the amplitude transmission matrix. The eigenfunctions of TL T define a set of channels through which the current flows, and the eigenvalues are the corresponding transmission coefficients. We prove that these coefficients are either O 0 or / 1. As L increases more channels are shut down. This is the maximal fluctuation theorem: fluctuations cannot be greater than this. We expect that our classification scheme will prove of further value in proving theorems about limiting distributions. We show by numerical simulations that our theorem holds good for a wide variety of systems, in one, two and three dimensions. Disordered media scatter waves incident upon them, and induce in the scattered wavefield a degree of disorder which is far more extreme than the physical disorder of the medium itself. The fluctuations are brought about by multiple scattering of the wavefield and its ability to interfere constructively or destructively with itself. The most extreme instances are found in the absence of absorption, when multiple scattering can have free rein. As a specific example we consider the system shown in figure 1 in which waves are incident on a slab of disordered material of finite thickness in one direction, but effectively infinite in the other directions. We shall assume that the slab is statistically homogeneous. The waves can be either transmitted through or reflected from the slab. Statistics in the transmission coefficient pose particular challenges: as the thickness, L, of the slab is increased fluctuations in the transmission coefficient, far from settling down to some 'average' value, become more extreme. The question is what can we say about the limiting scattering properties of a thick slab of material ? As far as we are aware this question has only been addressed for special cases. The results we derive here apply in very general circumstances,

New results in the theory of elasticity for two-dimensional composites

Abstract: We bring together and discuss a number of exact relationships in two-dimensional (or plane) elasticity, that are useful in studying the effective elastic constants and stress fields in two-dimensional composite materials. The first of these dates back to Michell (1899) and states that the stresses, induced by applied tractions, are independent of the elastic constants in a two-dimensional material containing holes. The second involves the use of Dundurs constants which, for a composite consisting of two isotropic elastic phases, reduce the dependence of stresses on the elastic constants from three independent dimensionless parameters to two. It is shown that these two results are closely related to a recently proven theorem by Cherkaev, Lurie and Milton, which we use to show that the effective Young's modulus of a sheet containing holes is independent of the Poisson's ratio of the matrix material. We also show that the elastic moduli of a composite can be found exactly if the shear moduli of the components are all equal; a previously known result. We illustrate these results with computer simulations, where appropriate. Finally we conjecture on generalizations to multicomponent composite materials and to situations where the bonding between the phases is not perfect.

T-periodic solutions for some second order differential equations with singularities*

Abstract: We study the existence of T-periodic positive solutions of the equationwhere f(t, .) has a singularity of repulsive type near the origin. Under the assumption that f(t, x) lies between two lines of positive slope for large and positive x, we find a non-resonance condition which predicts the existence of one T-periodic solution.Our main result gives a Fredholm alternative-like result for the existence of T-periodic positive solutions for

Solution of a boundary value problem for the Helmholtz equation via variation of the boundary into the complex domain

Abstract: In this paper we deal with the problem of diffraction of electromagnetic waves by a periodic interface between two materials. This corresponds to a two-dimensional quasi-periodic boundary value problem for the Helmholtz equation. We prove that solutions behave analytically with respect to variations of the interface. The interest of this result is both theoretical – the legitimacy of power series expansions in the parameters of the problem has indeed been questioned – and, perhaps more importantly, practical: we have found that the solution can be computed on the basis of this observation. The simple algorithm that results from such boundary variations is described. To establish the property of analyticity of the solution for the grating with respect to the height δ, we present a holomorphic formulation of the problem using surface potentials. We show that the densities entering into the potential theoretic formulation are analytic with respect to variations of the boundary, or, in other words, that the integral operator that results from the transmission conditions at the interface is invertible in a space of holomorphic functions of the variables ( x , y , δ). This permits us to conclude, in particular, that the partial derivatives of u with respect to δ at δ = 0 satisfy certain boundary value problems for the Helmholtz equation, in regions with plane boundaries, which can be solved in a closed form.

Fatigue crack growth modelling by successive blocking of dislocations

Abstract: Work hardening and the study of instability is incorporated into the description of the growth of a crack in terms of the successive blocking of the plastic zone by slip barriers, such as grain boundaries, and the subsequent initiation of the slip in neighbouring grains. A simple equation is derived to determine the critical position of the crack tip in relation to the grain boundary where the plastic zone is blocked at the moment of slip transmission. The intermittent pattern of decelerating and accelerating behaviour of short cracks and the existence of non-propagating cracks is explained. Instability in crack growth is seen to occur when the rate of hardening is insufficient to compensate for the increase in crack driving force in relation to the increase in crack length. This is associated with fracture toughness. The transition point between the short and long crack regimes is seen to occur when the size of the plastic zone is of the order of the microstructural parameter.

The structure of molten and glassy 2:1 binary systems: An approach using the Bhatia-Thornton formalism

Abstract: A systematic analysis of those liquid binary 2:1 systems (denoted MX$\_{2}$), for which experimental partial structure factors are available from the isotopic substitution method in neutron diffraction, is made using the Bhatia-Thornton (BT) formalism. Particular attention is paid to the origin of the first sharp diffraction peak (FSDP), which occurs in the measured diffraction patterns for some of the MX$\_{2}$ systems, since it appears, from recent studies, that this feature is a signature of directional bonding. It is found that FSDPS can occur in all three BT partial structure factors S$\_{\alpha \beta}$(k). A FSDP feature in the concentration-concentration partial structure factor S$\_{\text{CC}}$(k) is not, however, pronounced except in the case of MgCl$\_{2}$ and the glass forming network melts ZnCl$\_{2}$ and GeSe$\_{2}$ To the extent that these systems can be regarded as ionic melts a FSDP in S$\_{\text{CC}}$(k) implies a non-uniformity in the charge distribution on the scale of the intermediate-range order (IRO). The structure of molten GeSe$\_{2}$ is compared with the structures of molten ZnCl$\_{2}$, glassy GeS$\_{2}$ and glassy SiO$\_{2}$. Although the GeSe$\_{2}$ and ZnCl$\_{2}$ melts have different short-range order, there are similarities in the observed IRO which can be attributed to the arrangement of the electropositive species M. The essential features of the measured total structure factor for glassy GeS$\_{2}$ can be reproduced by using the molten GeSe$\_{2}$ S$\_{\alpha \beta}$(k). This result lends support to the notion that the S$\_{\alpha \beta}$(k) for liquid GeSe$\_{2}$ (and ZnCl$\_{2}$) are characteristic of both the liquid and glassy states of other network glass forming systems. The structures of molten GeSe$\_{2}$ (or ZnCl$\_{2}$) and glassy SiO$_{2}$ are, however, found to be different. The observed discrepancies are largest in the region of the FSDP which signifies pronounced differences in the nature of the IRO for these systems.

An Improved Method for Compressive Stress-Strain Measurements at Very High Strain Rates

Abstract: This paper describes a modification of the split Hopkinson pressure bar, to allow compression testing of high strength metals at a strain rate of up to about 10$^{5}$ s$^{-1}$. All dimensions are minimized to reduce effects of dispersion and inertia, with specimens of the order of 1 mm diameter. Strain is calculated from the stress record and calibrated with high-speed photography. Particular attention has been paid to the accuracy of the technique, and errors arising from nonlinearity in the instrumentation, dispersion, frictional restraint and inertia have all been quantitatively assessed. Stress-strain results are presented of Ti 6A14V alloy, a high strength tungsten alloy, and pure copper.

Low-dimensional lattices. VI : Voronoi reduction of three-dimensional lattices

Abstract: The aim of this paper is to describe how the Voronoi cell of a lattice changes as that lattice is continuously varied. The usual treatment is simplified by the introduction of new parameters called the vonorms and conorms of the lattice. The present paper deals with dimensions n $\leq $ 3; a sequel will treat four-dimensional lattices. An elegant algorithm is given for the Voronoi reduction of a three-dimensional lattice, leading to a new proof of Voronoi's theorem that every lattice of dimension n $\leq $ 3 is of the first kind, and of Fedorov's classification of the three-dimensional lattices into five types. There is a very simple formula for the determinant of a three-dimensional lattice in terms of its conorms.

Potential flow and forces for incompressible viscous flow

Abstract: Forces on a finite body in an incompressible viscous flow are shown to be contributed by a potential flow and fluid elements of non-zero vorticity in a revealing formulation. The present study indicates that the potential flow play also a geometric role in determining the contribution of the fluid elements. Consideration is given to a solid body moving through a fluid, fluid accelerating past a solid body and a solid body which oscillates in a uniform stream. The effects of induced-mass and inertial forces appear naturally in the formulation and are separated from the contribution due to the surface vorticity and that due to the vorticity within the flow. Physical significance of the present analysis for vortical flows about a finite body is illustrated by examples, e.g. flow past a circular cylinder or an ellipsoid of revolution.

A global estimate for the gradient in a singular elliptic boundary value problem

Abstract: We investigate the singular problemwhere Ω is a bounded smooth domain, k a bounded, nonnegative measurable function and v Ω 0. For the solution u to this problem, which is shown to exist if k(x) > 0 on some subset of Ω with positive measure, a uniform bound for |∇u| in Ω is derived when k(x) ≧ ψ (dist (x, ∂Ω)) with ψ (s)/sv ∈ Lp(0, a) for some a > 0, p > 1.

Existence of Optimal Controls for Viscous Flow Problems

Abstract: A class of optimal control problems in viscous flow is studied. Main result is the existence theorem for optimal control. Three typical flow control problems are formulated within this general class.

On the Global Structure of Robinson-Trautman Space-Times

Abstract: The global structure of Robinson–Trautman space-times is studied. When the space-time topology is R + x R x S 2 it is shown that all Robinson–Trautman space-time can be C 117 extended (in the vacuum Robinson–Trautman class of metrics) beyond the r = 2 m 9Schwarzschild-like9 event horizon; evidence is given supporting the conjecture, that no smooth extensions beyond the r = 2 m event horizon exist unless the metric is the Schwarzschild one. When the space-time topology is R + x R x 2 M , with 2 M a higher genus surface, and the mass parameter m is negative, Schwarzchild-like event horizons are shown to occur. The Proofs of these results are based on the derivation of a detailed asymptotic expansion describing the long-time behaviour of the solutions of a nonlinear parabolic equation.

Asymptotic behaviour of solutions to the coagulation-fragmentation equations. I : The strong fragmentation case

Abstract: The discrete coagulation-fragmentation equations are a model for the time-evolution of cluster growth. The processes described by the model are the coagulation of clusters via binary interactions and the fragmentation of clusters. The assumptions made on the fragmentation coefficients in this paper have the physical interpretation that surface effects are not important, i.e. it is unlikely that a large cluster will fragment into two large pieces. Since solutions of the initial-value problem are not unique, we have to restrict the class of solutions. With this restriction, we prove that the fragmentation acts as a strong damping mechanism and we obtain results on the asymptotic behaviour of solutions. The main tool used is an estimate on the moments of admissible solutions.

Exact Blow-Up Solutions to the Cauchy Problem for the Davey-Stewartson Systems

Abstract: We present exact blow-up solutions to the Cauchy problem for the Davey-Stewartson systems. It is shown that for any prescribed blow-up time there is an exact solution whose mass density converges to the Dirac measure as time goes to the blow-up time and that the solution extends beyond the blow-up time and behaves like the free solution as time tends to infinity.

On the slow dynamics for the Cahn–Hilliard equation in one space dimension

Abstract: We study the limiting behaviour of the solution of the Cahn-Hilliard equation using \`energy-type methods'. We assume that the initial data has a \`transition layer structure', i.e. u$^{\epsilon}\approx \pm $ 1 except near finitely many transition points. We show that, in the limit as $\epsilon \rightarrow $ 0, the solution maintains its transition layer structure, and the transition layers move slower than any power of $\epsilon $.