Showing papers in "Proceedings of The Royal Society A: Mathematical, Physical and Engineering Sciences in 1995"

Universality in Quantum Computation

Abstract: We show that in quantum computation almost every gate that operates on two or more bits is a universal gate. We discuss various physical considerations bearing on the proper definition of universality for computational components such as logic gates.

Approximate controllability of the semilinear heat equation

Abstract: This article is concerned with the study of approximate controllability for the semilinear heat equation in a bounded domain Ω when the control acts on any open and nonempty subset of Ω or on a part of the boundary. In the case of both an internal and a boundary control, the approximate controllability in LP(Ω) for 1 ≦ p < + ∞ is proved when the nonlinearity is globally Lipschitz with a control in L∞. In the case of the interior control, we also prove approximate controllability in C0(Ω). The proof combines a variational approach to the controllability problem for linear equations and a fixed point method. We also prove that the control can be taken to be of “quasi bang-bang” form.

The Mechanical Properties of Natural Materials. I. Material Property Charts

Abstract: The mechanical properties of natural materials as diverse as wood, muscle, shell and bone are plotted on material-property charts which show the relationships between properties. Performance indices are used to identify the load-bearing applications in which each performs particularly well. By these criteria, many natural materials are superior to the man-made materials of engineering. A companion paper examines some of the origins of this superiority; an explanation is sought in an analysis of the way structure influences properties.

Incoherent imaging of crystals using thermally scattered electrons

Abstract: Thermal diffuse scattering of electrons through large angles by a simple low-index crystal projection is examined in the context of a phonon model, based on the Warren approximation of X-ray diffraction The scattering from an individual atomic column is visualized in terms of an assembly of independent `packets' of atoms Within a packet, the scattering is partially coherent, causing the columnar intensity to deviate from that calculated with an Einstein independent oscillator model For typical atomic spacings, this deviation is limited to within 20%

A statistical description of polarimetric and interferometric synthetic aperture radar data

Abstract: This paper provides a nearly complete analysis of the key distributions encountered in single- and multi-look polarimetric synthetic aperture radar data under the bivariate Gaussian and K -distribution models. It contains new analytic results on the moments of the amplitude and phase difference in single look data and on the moments of the amplitude in multi-look data. As yet no analytic results for the moments of multi-look phase difference have been found, except in limiting cases. The maximum likelihood estimators of the covariance matrix elements of two jointly Gaussian channels are derived, together with their asymptotic variances. The problems in extending this analysis to the bivariate K distribution are also discussed.

A Universal Two-Bit Gate for Quantum Computation

Abstract: We prove the existence of a class of two-input, two-output gates any one of which is universal for quantum computation. This is done by explicitly constructing the three-bit gate introduced by Deutsch (Proc. R. Soc. Lond. A 425, 73 (1989)) as a network consisting of replicas of a single two-bit gate.

The Mechanical Properties of Natural Materials. II. Microstructures for Mechanical Efficiency

Abstract: Many natural materials have exceptionally high values of the mechanical performance indices described in the previous, companion paper. For beams and plates of a given stiffness or strength, or for a column of a given buckling resistance, woods, palms and bamboo are among the most efficient materials available. Their mechanical efficiency arises from their combination of composite and cellular microstructures. In this paper we analyse the microstructures which give rise to exceptional performance, describe the fabrication and testing of model materials with those microstructures and discuss the implications for design of mechanically efficient engineering materials.

Invariant subspaces and new explicit solutions to evolution equations with quadratic nonlinearities

Abstract: We present new explicit solutions to some classes of quasilinear evolution equations arising in different applications, including equations of the Boussinesq type:and quasilinear heat equations:The method is based on construction of finite-dimensional linear functional subspaces which are invariant with respect to spatial operators having quadratic nonlinearities. The corresponding nonlinear evolution equations on invariant subspaces are shown to be equivalent to finite-dimensional dynamical systems. Examples of two-, three- and five- dimensional invariant subspaces are given. Some generalisations to N-dimensional quadratic operators are also considered.

The law of the wall in turbulent flow

Abstract: The `law of the wall' for the inner part of a turbulent shear flow over a solid surface is one of the cornerstones of fluid dynamics, and one of the very few pieces of turbulence theory whose results include a simple analytic function for the mean velocity distribution, the logarithmic law Various aspects of the law have recently been questioned, and this paper is a summary of the present position Although the law of the wall for velocity has apparently been confirmed by experiment well outside its original range, the law of the wall for temperature seems to apply only to very simple flows Since the two laws are derived by closely analogous arguments this throws suspicion on the law of the wall for velocity Analysis of simulation data, for all the Reynolds stresses including the shear stress, shows that law-of-the-wall scaling fails spectacularly in the viscous wall region, even when the logarithmic law is relatively well behaved Virtually all turbulence models are calibrated to reproduce the law of the wall in simple flows, and we discuss whether, in practice or in principle, their range of validity is larger than that of the law of the wall itself: the present answer is that it is not; so that when the law of the wall (or the mixing-length formula) fails, current Reynolds-averaged turbulence models are likely to fail too

A Unified Procedure for Construction of Theories of Deformable Media. I. Classical Continuum Physics

Abstract: A new unified procedure for constructing continuum theories of deformable media is presented and used in this and a companion paper. The procedure starts with a balance of energy and derives from it all the relevant balance laws that may also include those that are associated with thermal, electrical and magnetic effects; the basic energetic ingredients that are included in the balance of energy depend, of course, on the nature of the particular theory of material behaviour desired. The advantage of the new procedure becomes especially apparent when one considers formulation of a new theory of material behaviour for which additional balance laws (involving new kinetic quantities) are required to accompany any additional basic kinematic and thermal variables additional to those in the classical formulation. Indeed, in the formulation of such new theories, usually little or no previous information is available concerning properties of the new kinetic quantities in the additional balance laws; and, in this connection, the unified procedure of this paper provides a simple attractive setting for deriving the basic equations that are automatically consistent with the energy balance. In this paper, first the basic features of the new procedure are illustrated in the context of classical thermomechanics. Generalizations of this thermomechanical theory are then discussed in two cases: (1) in the presence of an additional kinematic variable and (2) in the presence of full electromagnetic effects. Both of these generalizations bring out some interesting novel features when new theories are being constructed.

Turbulent transport in flames

Abstract: A review is presented of turbulent transport phenomena in flames. Both large-scale turbulent transport and small-scale mixing processes are considered and various mechanisms of interaction between combustion and turbulent flow are identified. Flame-surface density descriptions of turbulent combustion at high Damkohler numbers are discussed in detail and some topics are identified which require further attention. Emphasis is placed on problems of premixed turbulent combustion.

A Generalized Spatial-Temporal Model of Rainfall Based on a Clustered Point Process

Abstract: The objective in this paper is to present and fit a relatively simple stochastic spatial-temporal model of rainfall in which the arrival times of rain cells occur in a clustered point process. In the x - y plane, rain cells are represented as discs; each disc having a random radius; the locations of the disc centres being given by a two-dimensional Poisson process. The intensity of each cell is a random variable that remains constant over the area of the disc and throughout the lifetime of the cell, the lifetime being an exponential random variable. The cells are randomly classified from 1 to n with different parameters for the different cell types, so that the random variables of an arbitrary cell, e. g. radius and intensity, are correlated. Multi-site second-order properties are derived and used to fit the model to hourly rainfall data taken from six sites in the Thames basin, UK.

Three-Dimensional Time-Harmonic Elastodynamic Green's Functions for Anisotropic Solids

Abstract: A method based on the Radon transform is presented to determine the displacement field in a general anisotropic solid due to the application of a time-harmonic point force. The Radon transform reduces the system of coupled partial differential equations for the displacement components to a system of coupled ordinary differential equations. This system is reduced to an uncoupled form by the use of properties of eigenvectors and eigenvalues. The resulting simplified system can be solved easily. A back transformation to the original coordinate system and a subsequent application of the inverse Radon transform yields the displacements as a summation of a regular elastodynamic term and a singular static term. Both terms are integrals over a unit sphere. For the regular dynamic term, the surface integration can be evaluated numerically without difficulty. For the singular static term, the surface integral has been reduced to a line integral over half a unit circle. Reductions to the cases of isotropy and transverse isotropy have been worked out in detail. Examples illustrate applications of the method.

On Light Propagation in Helicoidal Bianisotropic Mediums

Abstract: We have shown liquid crystals to be members of the more general class of helicoidal bianisotropic mediums (HBMs) and have discussed physical realizations of HBMs as liquid crystals, cross-linked polymer networks and solid thin films. Solutions of the time-harmonic Maxwell postulates for axial propagation in a HBM have been derived in terms of the eigenvalues and eigenvectors of a 4 x 4 matrix. We have given a procedure to obtain the response of a HBM slab to a normally incident plane wave and exemplified our analytical results by application to four mediums.

Determination of the striking distance of lightning to earthed structures

Abstract: A theoretical determination based upon a critical charge criterion and ionization processes is presented of the striking distance rs at which the upward leader is initiated from a struck object. The striking distance is shown to be leader-current dependent in the form rs ∝ i2/3. Comparisons are made with earlier, more numerically based models, which used an alternative equivalent-radius approach. The altitude dependence of the striking distance is also investigated. Since pressure, absolute humidity and temperature decrease with increasing altitude, the variation of streamer properties with altitude can be inferred from laboratory experiments. Calculations on this basis show that the striking distance increases significantly with increasing altitude, causing a corresponding increase in the risk of lightning strikes for given structures in mountain or aviation environments.

A unified procedure for construction of theories of deformable media. ii. generalized continua

Abstract: In contrast to classical theories discussed in the accompanying paper (part I), the present paper deals exclusively with generalized continua, a term that refers to a body embedded in a euclidean three-dimensional space with each of its material points endowed with additional kinematic structure. Such generalized continua are conveniently identified here by Cosserat (or directed ) continua in which the additional structure is represented by independent director fields and the material domain of the body manifold is classified according to the four categories: (A) a three-dimensional volume, (B) a two-dimensional surface, (C) a one-dimensional space curve and (D) a point. In the derivation of a thermomechanical theory for each of these categories, we use the same unified procedure as in part I but now the various energies that enter the balance of energy must be modified to include additional kinetic ingredients. New theories of this kind have increasingly provided, over the last three decades, effective means of formulating and studying new behaviour of materials (in both fluid and solid mechanics) that were not previously possible by the classical theories. Our attention is first focused in the development of the theory for the first category and this is discussed in two stages: first for a Cosserat continuum with a single director and then for N directors. Our derived results include for the first time a derivation from the balance of energy of balances of director momenta and balances of entropies (in the presence of more than one independent temperature field). The treatments of categories (B) to (D) are somewhat shorter since their developments are formally similar to that of category (A).

Breakup Phenomena in Coaxial Airblast Atomizers

Abstract: The breakup of a liquid jet with length-to-diameter ratio of 22 surrounded by a coaxial flow of air has been examined by a combination of high-speed photography and phase-Doppler velocimetry. The air-to-liquid momentum and kinetic energy ratios, the Reynolds number of the coaxial water and air jet flows and the exit-plane Weber number have been varied over extensive ranges and the results examined in terms of the breakup length, frequency, droplet size distributions and velocity characteristics. The photographs reveal the deterministic nature of the liquid flow at Reynolds numbers which are sufficient to guarantee turbulent flow, with the formation of a wave-like structure for a short distance followed by the formation of a liquid cluster and subsequent breakup into ligaments and droplets, with the entire process repeated in a periodic manner. Attempts are made to relate the breakup length and the frequency of the process to the air-to-liquid momentum and energy ratios, the exit Weber number and the slip velocity between the two streams at the nozzle exit. The results confirm that the ratio of the frequencies of the wave-like structures and breakup decreased with the slip velocity between the two streams and asymptotically approached a value of around one for values higher than 150 m s -1 . The photographs indicate that the droplet sizes in the sprays are due mainly to disintegration of liquid clusters produced after the initial breakup of the liquid jet and the phase Doppler measurements confirm that most of the liquid remained close to the centreline, where the mean diameter reached a maximum and the slip velocity between the droplets and the air flow was low. An atomization model based on the value of the local Weber number on the centreline of the sprays is used to explain the size characteristics of the sprays. The atomization process was affected by the air-to-liquid momentum ratio at the nozzle exit, the annular width of the coaxial atomizer, the liquid-to-air density ratio, the surface tension and the kinematic viscosity and density of the air. The rate of spread of the sprays close to the nozzle reduced with increase of the air and liquid flowrates and was affected by the initial breakup of the liquid jet and the amplitude of the wave-like structure of the liquid jet during breakup rather than by the air flow turbulence.

Sturmian comparison theorem for half-linear second-order differential equations

Abstract: Let φ:ℝ→ℝ be defined by φ(s) = |s|p−2s, with p > 1 a fixed number. We extend Sturm Comparison Theorem of the linear differential equationto the nonlinear differential equationby using the Wirtinger inequality. A Lyapunov inequality and some oscillation criteria of (E) are also given.

Variational estimates for the creep behaviour of polycrystals

Abstract: A variational procedure is developed for estimating the effective constitutive behaviour of polycrystalline materials undergoing high-temperature creep. The procedure is based on a new variational principle allowing the determination of the effective potential function of a given nonlinear polycrystal in terms of the corre­sponding potential for a linear comparison polycrystal with an identical geometric arrangements of its constituent single-crystal grains. As such, it constitutes an extension, to locally anisotropic behaviour, of the variational procedure devel­oped by Ponte Castaneda (1991) for nonlinear heterogeneous media with locally isotropic behaviour. By way of an example, the procedure is applied to the de­termination of bounds of the Hashin-Shtrikman type for the effective potentials of statistically isotropic nonlinear polycrystals. The bounds are computed for the special class of untextured FCC polycrystals with isotropic pure power-law viscous behaviour, first considered by Hutchinson (1976), in the context of a calculation of the self-consistent type. The new bounds are found to be more restrictive than the corresponding classical Taylor-Bishop-Hill bounds, and also more re­strictive, if only slightly so, than related bounds of the Hashin-Shtrikman type by Dendievel et al . (1991). The new procedure has the advantage over the self-consistent procedure of Hutchinson (1976) that it may be applied, without any essential complications, to aggregates of crystals with slip systems exhibiting dif­ferent creep rules - with, for example, different power exponents - and to general loading conditions. However, the distinctive feature of the new variational proce­dure is that it may be used in conjunction with other types of known bounds and estimates for linear polycrystals to generate corresponding bounds and estimates for nonlinear polycrystals.

A unified procedure for construction of theories of deformable media. iii. mixtures of interacting continua

Abstract: This paper is a continuation of parts I and II under the same title (Green & Naghdi 1995 a , b , Proc . R . Soc . Lond . A 448, 335, 357). In contrast to the earlier two papers which dealt with single phase continua, this paper is concerned with a new thermomechanical theory of multiphase interacting continua. We use the same unified procedure as in part I but now the various energies that enter the balance of energy must be modified to accommodate energetic contributions arising from N interacting finite constituents. Again, our derivation of the various basic balance laws is effected by an appeal to the form invariance of the balance of energy and leads to a system of basic equations of mixtures which are automatically consistent with the balance of energy or the first law of thermodynamics for mixtures of N constituents.

Polynomial Systems: A Lower Bound for the Hilbert Numbers

Abstract: Let H$\_{n}$ be the maximum possible number of limit cycles of systems $\dot{x}$ = P(x, y), $\dot{y}$ = Q(x, y), where P and Q are polynomials of degree at most n. We are concerned with the rate of growth of H$\_{n}$ as n increases: it is known that H$\_{n}\geq $ kn$^{2}$ for some constant k. In this paper we show that H$\_{n}$ grows at least as rapidly as n$^{2}$ log n.

On the dynamical theory of incompressible viscous fluids and the determination of the criterion

Abstract: 1. The equations of motion of viscous fluid (obtained by grafting on certain terms to the abstract equations of the Eulerian form, so as to adapt these equations to the case of fluids subject to stresses depending in some hypothetical manner on the rates of distortion, which equations Navier seems to have first introduced in 1822, and which were much studied by Cauchy and Poisson) were finally shown by St Venant and Sir Gabriel Stokes, in 1845, to involve no other assumption than that the stresses, other than that of pressure uniform in all directions, are linear functions of the rates of distortion, with a coefficient depending on the physical state of the fluid. By obtaining a singular solution of these equations as applied to the case of pendulums in steady periodic motion, Sir G. Stokes was able to compare the theoretical results with the numerous experiments that had been recorded, with the result that the theoretical calculations agreed so closely with the experimental determinations as seemingly to prove the truth of the assumption involved. This was also the result of comparing the flow of water through uniform tubes with the flow calculated from a singular solution of the equations, so long as the tubes were small and the velocities slow. On the other hand, these results, both theoretical and practical, were directly at variance with common experience as to the resistance encountered by larger bodies moving with higher velocities through water, or by water moving with greater velocities through larger tubes. This discrepancy Sir G. Stokes considered as probably resulting from eddies, which rendered the actual motion other than that to which the singular solution referred, and not as disproving the assumption.

Hydrodynamics of Quantized Shape Transitions of Lipid Domains

Abstract: The time-dependent distortions of nearly circular liquid domains of lipids at the air-water interface are studied analytically by accounting for line tension and electrostatic effects in the monolayer and viscous effects in both the monolayer and subphase fluid. One driving force for the shape changes arises from the elec­trostatic repulsions between molecular dipoles. This force is opposed by a line tension that favors circular shapes. The growth rate of small amplitude disturbances is determined as a function of the distortion (azimuthal) symmetry number n , the domain radius, the dipole density difference and line tension between the lipid domain and the surrounding lipid layer, and the viscosities of the monolayer and underlying liquid. For representative parameter values, the disturbance growth rate is independent of the monolayer viscosity and depends only on the subphase viscosity.

One-Point Modelling of Rapidly Deformed Homogeneous Turbulence

Abstract: One-point turbulence models are important tools for engineering analysis. A good model should have a viscoelastic character, predicting turbulent stresses proportional to the mean strain rate for slow deformations and stresses determined by the amount of strain for rapid distortions. Our goal is to build a one-point turbulence model with this character, and this requires a one-point model for rapid distortions. Here it is shown that the turbulent stresses introduced by Osborne Reynolds do not, by themselves, provide an adequate tensorial base for one-point modelling of rapidly distorted turbulence because they do not carry critical information about the turbulence structure. The deficiency is shown to be most pronounced in flows subjected to strong mean rotation. Additional one-point tensors that do carry the missing information are introduced, and the complexities of a model that would have an adequate tensorial base are assessed. A new type of one-point structure-based turbulence model that overcomes the basic deficiency of Reynolds-stress transport models, but without the excessive complexity of multiple tensor variables, is then described. The ideas behind the rapid distortion version of this new model are presented, along with results for some special cases.

Stochastic Backscatter of Turbulence Energy and Scalar Variance by Random Subgrid-Scale Fluxes

Abstract: For large-eddy simulation with a finite-difference scheme, a simple stochastic subgridscale (SGS) model is introduced which describes the effects of random SGS motions on the resolved (filtered) scales of incompressible turbulent motions. The model extends the Smagorinsky-Lilly model by adding realizable random stresses and fluxes which are constructed as quadratic expressions of Gaussian random velocity and temperature fields. The random components reduce the correlations between stresses and strain rates to in between 0.16 and 0.5, in agreement with observations. The random stresses (fluxes) also induce random accelerations (temperature changes) with a k 4 power spectrum. Such random sources backscatter energy (variance) from SGS motions to resolved scale motions when temporally correlated with finite timescales. The timescales are different for momentum and heat flux. The analysis of the model provides an upper estimate of the magnitude of backscatter which is close to previous predictions. The analysis identifies the influence of the quasi-normal assumption and of numerical filters and determines the variance of the pressure fluctuations induced by the random accelerations at grid scales. Backscatter increases the SGS turbulent Prandtl number to a degree depending strongly on the numerical filter. Tests of the model in large-eddy simulation of isotropic turbulence show energy decay rates in close agreement with expected rates when the stochastic SGS model is included. Backscatter cannot be simulated with reduced diffusivities or filter widths.

Counting stationary solutions of the Cahn–Hilliard equation by transversality arguments

Abstract: In this paper we use arguments based on Picard-Fuchs equations and transversality of intersections of level curves to obtain an exact count of the number of stationary solutions of the one-dimensional Cahn-Hilliard equation with a cubic nonlinearity.

Quantum Spacetime Fluctuations and Primary State Diffusion

Abstract: Non-differentiable fluctuations in spacetime on a Planck scale introduce stochastic terms into the equations for quantum states, resulting in a proposed new foundation for an existing alternative quantum theory: primary state diffusion (PSD). Planckscale stochastic spacetime structure results in quantum fluctuations, whilst larger-scale curvature is responsible for gravitational forces. The gravitational field and the quantum fluctuation field are the same, differing only in scale. The quantum mechanics of small systems, classical mechanics of large systems and the physics of quantum experiments are all derived dynamically, without any prior division into classical and quantum domains, and without any measurement hypothesis. Unlike the earlier derivation of PSD, the new derivation, based on a stochastic spacetime differential geometry, has essentially no free parameters. However, many features of this structure remain to be determined. The theory is falsifiable in the laboratory, and critical matter interferometry experiments, to distinguish it from ordinary quantum mechanics, might be feasible within the next decade.

Weak solutions to a hydrodynamic model for semiconductors: the Cauchy problem

Abstract: We investigate the Cauchy problem for a hydrodynamic model for semiconductors. An existence theorem of global weak solutions with large initial data is obtained by using the fractional step Lax—Friedrichs scheme and Godounov scheme.

The random packing of fibres in three dimensions

Abstract: An investigation has been carried out of the limiting packing density of an array of long straight rigid fibres distributed randomly in space as a function of the length of the fibre. We derive an approximate relationship between the limiting volume fraction V f and the slenderness λ of the fibres defined as length divided by diameter. The formula agrees well with our experimental results and those found in the literature.

A theoretical justification for the application of the Arrhenius equation to kinetics of solid state reactions (mainly ionic crystals)

Abstract: Although the Arrhenius equation has been widely and successfully applied to innumerable solid state reactions, this use lacks a theoretical justification because the energy distribution amongst the immobilized constituents of a crystalline reactant is not represented by the Maxwell-Boltzmann equation. The present analysis focuses attention on the role of the reactant-product interface, the active zone within which chemical changes preferentially proceed in many solid state rate processes. We identify interface energy levels, that are the precursors to the bond redistribution step, as extensions to the band structure of the solid into the structurally less-regular reaction zone. These interface energy levels are analogous to impurity levels. Electron reorganization requires a locally high energy so that interface levels are appreciably above the Fermi level of the crystalline reactant (and product). Occupancy is determined by energy distribution functions based on Fermi-Dirac statistics for electrons and Bose-Einstein statistics for phonons. For the highest energies, necessary for reaction, both distributions approximate to the exponential energy term, thereby providing a theoretical justification for the application of the Arrhenius equation to reactions of solids. The treatment given here has been largely developed from the theory applicable to ionic solids and the conclusions are most directly relevant to reactions of this class of substance. It is intended, however, that the approach should be of value in extending theoretical understanding of all rate processes involving solids which require the preinvestment of energy in an electron reorganization step.