Showing papers in "Proceedings of The Royal Society A: Mathematical, Physical and Engineering Sciences in 1989"

Quantum computational networks

Abstract: The theory of quantum computational networks is the quantum generalization of the theory of logic circuits used in classical computing machines. Quantum gates are the generalization of classical logic gates. A single type of gate, the universal quantum gate, together with quantum ‘unit wires’, is adequate for constructing networks with any possible quantum computational property.

A logic of authentication

Abstract: Questions of belief are essential in analysing protocols for the authentication of principals in distributed computing systems. In this paper we motivate, set out, and exemplify a logic specifically designed for this analysis: we show how various protocols differ subtly with respect to the required initial assumptions of the participants and their final beliefs. Our formalism has enabled us to isolate and express these differences with a precision that was not previously possible. It has drawn attention to features of protocols of which we and their authors were previously unaware, and allowed us to suggest improvements to the protocols. The reasoning about some protocols has been mechanically verified. This paper starts with an informal account of the problem, goes on to explain the formalism to be used, and gives examples of its application to protocols from the literature, both with shared-key cryptography and with public-key cryptography. Some of the examples are chosen because of their practical importance, whereas others serve to illustrate subtle points of the logic and to explain how we use it. We discuss extensions of the logic motivated by actual practice; for example, to account for the use of hash functions in signatures. The final sections contain a formal semantics of the logic and some conclusions.

Front migration in the nonlinear Cahn-Hilliard equation

Abstract: The method of matched asymptotic expansions is used to describe solutions of the nonlinear Cahn-Hilliard equation for phase separation in N > 1 space dimensions The expansion is formally valid when the thickness of internal transition layers is small compared with the distance separating layers and with their radii of curvature On the dominant (slowest) timescale the interface velocity is determined by the mean curvature of the interface, by a non-local relation which is identical to that in a well-known quasi-static model of solidification, which exhibits a shape instability discovered by Mullins & Sekerka (J appl Phys 34, 323-329 (1963)) On a faster timescale, the Cahn-Hilliard equation regularizes a classic two-phase Stefan problem Similarity solutions of the two-phase Stefan problem should describe boundary layers Existence and uniqueness of such similarity solutions which admit metastable states is proved rigorously in an appendix

A theoretical study of the Brinell hardness test

Abstract: Brinell tests have long been the preferred method of assaying the hardness of metals during forming operations. The general significance of the test has been codified in empirical laws, especially those of Meyer, O'Neill and Tabor. On the other hand, the indentation of elastoplastic media by a ball has never been thoroughly analysed in the context of modern mechanics of continua; this is the objective here. The actual boundary-value problem is non-steady but can be made steady in terms of reduced variables when the material response is suitably modelled. Namely, the strain should be infinitesimal and expressible as the tensor gradient of a potential function of the stress deviator; the function must be homogeneous of degree n + 1 (?> 2), but is otherwise arbitrary. Meyer's law is then derivable rigorously ahead of a detailed solution. Moreover the predicted index is (2n + 1)/n, substantiating O'Neill's rule for materials whose strain under uniaxial tension varies as some nth power of the stress. It is predicted also that the piling-up or sinking-in around the indenter is correlated with n in the manner observed. These immediate implications of the model amount to a priori evidence of its overall ability to simulate elastoplastic response of the kind induced in Brinell tests. Evidence a posteriori was supplied by finite element computations for a standard potential whose level surfaces are of Mises type. Mixed nine-node quadrilateral elements were adopted; these are known to promote optimal convergence and are well suited to handling incompressibility. A carefully graded mesh provided about 24000 degrees of freedom. Computations were performed for n = 1, 2, 4 and 10, covering the practical range. The results include (i) distributions of the contact pressure- and the radial and circumferential in-surface stresses; (ii) profiles of the deformed surface; and (iii) contours of representative strain in the main body of material. Excellent agreement was obtained with Tabor's experimental findings that the representative strain at the contact perimeter is y0.4 a/D for any n (a is the contact radius and D the ball diameter), while the average pressure is 2.8 times the flow stress at strain y in a tension test. Finally, strain paths at fixed stations were sampled to check the simulation of elastoplastic response locally as well as overall.

The perturbed test function method for viscosity solutions of nonlinear PDE

Abstract: The method of viscosity solutions for nonlinear partial differential equations (PDEs) justifies passages to limits by in effect using the maximum principle to convert to the corresponding limit problem for smooth test functions. We describe in this paper a “perturbed test function” device, which entails various modifications of the test functions by lower order correctors. Applications include homogenisation for quasilinear elliptic PDEs and approximation of quasilinear parabolic PDEs by systems of Hamilton-Jacobi equations.

Uniform Asymptotic Smoothing of Stokes's Discontinuities

Abstract: Across a Stokes line, where one exponential in an asymptotic expansion maximally dominates another, the multiplier of the small exponential changes rapidly. If the expansion is truncated near its least term the change is not discontinuous but smooth and moreover universal in form. In terms of the singulant F - the difference between the larger and smaller exponents, and real on the Stokes line - the change in the multiplier is the error function $\pi^{-\frac{1}{2}}\int^\sigma_{-\infty}dt \exp (-t^2) \text{where} \sigma = ImF/(2ReF)^{\frac{1}{2}}.$ The derivation requires control of exponentially small terms in the dominant series; this is achieved with Dingle's method of Borel summation of late terms, starting with the least term. In numerical illustrations the multiplier is extracted from Dawson's integral (erfi) and the Airy function of the second kind (Bi): the small exponential emerges in the predicted universal manner from the dominant one, which can be 10$^{10}$ times larger.

Quantum scars of classical closed orbits in phase space

Abstract: The way in which quantum eigenstates are influenced by the closed orbits of a chaotic classical system is analysed in phase space x ═ ( q, p ) through the spectral Wigner function W ( x ; E, ∊ ). This is a sum over Wigner functions of eigenstates within a range ∊ of energy E . In the classical limit, W is concentrated on the energy surface and smoothly distributed over it. Closed orbits provide oscillatory corrections (scars) for which explicit semiclassical formulae are calculated. Each scar is a fringe pattern decorating the orbit. As x moves off the energy surface the fringes form an Airy pattern with spacing of order h ⅔ . As x moves off the closed orbit the fringes form a complex gaussian with spacing h ½ .

Local minimisers and singular perturbations

Abstract: We construct local minimisers to certain variational problems. The method is quite general and relies on the theory of Γ-convergence. The approach is demonstrated through the model problemIt is shown that in certain nonconvex domains Ω ⊂ ℝn and for e small, there exist nonconstant local minimisers ue satisfying ue ≈ ± 1 except in a thin transition layer. The location of the layer is determined through the requirement that in the limit ue →u0, the hypersurface separating the states u0 = 1 and u0 = −1 locally minimises surface area. Generalisations are discussed with, for example, vector-valued u and “anisotropic” perturbations replacing |∇u|2.

Time-dependent kinematic dynamos with stationary flows

Abstract: Numerical solutions to the magnetic induction equation in a sphere have been obtained for a number of stationary velocity models. By searching for non-steady magnetic fields and in some circumstances showing that all magnetic field modes decay, the inability of several earlier researchers to find convergent steady solutions is explained. Results of previous authors are generally confirmed, but also extended to cover non-steady fields, different values of magnetic Reynolds number and other parameters, and higher truncation limits. Some non-decaying fields are found where only decaying or non-convergent results have previously been reported. Two flows $\epsilon s^0\_2 + t^0\_2$ and $\epsilon s^0\_2 + t^0\_1$, each consisting of two very simple axisymmetric rolls are seen to sustain growing fields provided that (i) the magnetic Reynolds number R and the poloidal to toroidal flow ratio $\epsilon$ are of appropriate magnitudes, and (ii) the meridional s$^0\_2$ flow is directed inwards along the equatorial plane and out towards the poles. An even simpler axisymmetric single roll flow $\epsilon s^0\_1 + t^0_1$ is also seen to support growing fields for appropriate $\epsilon$ and R. These simple flows dispel the somewhat prevalent belief that dynamo maintenance relies on the supporting flow being complex, and having length scale significantly less than that of the conducting fluid volume.

Diffusive logistic equations with indefinite weights: population models in disrupted environments

Abstract: The dynamics of a population inhabiting a strongly heterogeneous environment are modelledby diffusive logistic equations of the form ut = d Δu + [m(x) — cu]u in Ω × (0, ∞), where u represents the population density, c, d > 0 are constants describing the limiting effects of crowding and the diffusion rate of the population, respectively, and m(x) describes the local growth rate of the population. If the environment ∞ is bounded and is surrounded by uninhabitable regions, then u = 0 on ∂∞× (0, ∞). The growth rate m(x) is positive on favourablehabitats and negative on unfavourable ones. The object of the analysis is to determine how the spatial arrangement of favourable and unfavourable habitats affects the population being modelled. The models are shown to possess a unique, stable, positive steady state (implying persistence for the population) provided l/d> where is the principle positive eigenvalue for the problem — Δϕ=λm(x)ϕ in Χ,ϕ=0 on ∂Ω. Analysis of how depends on m indicates that environments with favourable and unfavourable habitats closely intermingled are worse for the population than those containing large regions of uniformly favourable habitat. In the limit as the diffusion rate d ↓ 0, the solutions tend toward the positive part of m(x)/c, and if m is discontinuous develop interior transition layers. The analysis uses bifurcation and continuation methods, the variational characterisation of eigenvalues, upper and lower solution techniques, and singular perturbation theory.

A Weighted Average Flux Method for Hyperbolic Conservation Laws

Abstract: A numerical technique, called a 'weighted average flux' (WAF) method, for the solution of initial-value problems for hyperbolic conservation laws is presented. The intercell fluxes are defined by a weighted average through the complete structure of the solution of the relevant Riemann problem. The aim in this definition is the achievement of higher accuracy without the need for solving 'generalized' Riemann problems or adding an anti-diffusive term to a given first-order upwind method. Second-order accuracy is proved for a model equation in one space dimension; for nonlinear systems second-order accuracy is supported by numerical evidence. An oscillation-free formulation of the method is easily constructed for a model equation. Applications of the modified technique to scalar equations and nonlinear systems gives results of a quality comparable with those obtained by existing good high resolution methods. An advantage of the present method is its simplicity. It also has the potential for efficiency, because it is well suited to the use of approximations in the solution of the associated Riemann problem. Application of WAF to multidimensional problems is illustrated by the treatment using dimensional splitting of a simple model problem in two dimensions.

Fractal geometry: what is it, and what does it do?

Abstract: Fractal geometry is a workable geometric middle ground between the excessive geometric order of Euclid and the geometric chaos of general mathematics. It is based on a form of symmetry that had previously been underused, namely invariance under contraction or dilation. Fractal geometry is conveniently viewed as a language that has proven its value by its uses. Its uses in art and pure mathematics, being without ‘practical’ application, can be said to be poetic. Its uses in various areas of the study of materials and of other areas of engineering are examples of practical prose. Its uses in physical theory, especially in conjunction with the basic equations of mathematical physics, combine poetry and high prose. Several of the problems that fractal geometry tackles involve old mysteries, some of them already known to primitive man, others mentioned in the Bible, and others familiar to every landscape artist.

The gradient theory of phase transitions for systems with two potential wells

Abstract: In this paper we generalise the gradient theory of phase transitions to the vector valued case. We consider the family of perturbationsof the nonconvex functionalwhere W:RN→R supports two phases and N ≧1. We obtain the Γ(L1(Ω))-limit of the sequenceMoreover, we improve a compactness result ensuring the existence of a subsequence of minimisers of Ee(·) converging in L1(Ω) to a minimiser of E0(·) with minimal interfacial area.

Velocity and particle-flux characteristics of turbulent particle-laden jets

Abstract: The velocity and flux of spherical glass beads with nominal diameters of 200, 80 and 40 $\mu m$ have been obtained by phase-Doppler anemometry in a round unconfined air jet over the first 28 diameters. The jet diameter was 15 mm and the exit velocity was 13 m s$^{-1}$ giving a Reynolds number of 13 000 and a timescale of 1.15 ms, which increased quadratically with axial distance: the bead inertial time constants were 298, 48 and 12 ms. The purposes of the experiments were to quantify the velocity and flux characteristics of the beads and of the gas phase in the presence of the beads as a function of bead diameter and of the mass loading in the jet nozzle. Due to the large inertia of the 200 $\mu m$ beads, the mean bead velocity downstream of the exit of the jet was constant and independent of mass loading up to 0.37 and the axial root mean square (r.m.s.) bead velocity decayed by about one-fifth: at the exit of the jet, the axial r.m.s. bead velocity was higher than that of the corresponding clean jet. The mean centreline velocity of the 80 $\mu m$ beads decayed to about one-half of the bead exit velocity by 28 diameters downstream and was independent of mass loading up to 0.86. The decay rate of the mean gas centreline velocity in the presence of the beads reduced as the loading increased because of momentum transfer from the discrete to the gaseous phase. The axial r.m.s. velocity of the beads was comparable to that of the gas phase and both decreased with increasing loading and the rate of spread of the half width of the jet increased with increasing loading. For the 40 $\mu m$ beads, the decay rate of the mean centreline velocity of the beads decreased with increasing loading and, in contrast to the 80 $\mu m$ beads, the rate of spread decreased with increasing loading up to 0.80. The axial r.m.s. velocity of the beads became largest at a position downstream of the nozzle exit, which moved downstream with increasing loading and was larger than the axial r.m.s. velocity of the clean jet, although the beads were not expected to be responsive to the frequencies of the energy-containing eddies. The bead axial r.m.s. velocity was more than twice as large as the radial r.m.s. velocity and the correlation coefficient of the cross correlation was larger than that of the clean jet. The large bead turbulence, anisotropy and strong correlation coefficient are explained by the superposition of bead trajectories from regions of different bead mean velocity and are not because of acquisition of axial turbulent motion from the gaseous phase.

New Theoretical Model of Stress Transfer Between Fibre and Matrix in a Uniaxially Fibre-Reinforced Composite

Abstract: A new analytical method has been developed that can predict the stress transfer between fibre and matrix in a uniaxially fibre-reinforced composite associated with either a single matrix crack or a fibre break. Account is taken of thermal residual stresses arising from a mismatch in thermal expansion coefficients between the fibre and matrix. In addition Poisson ratio mismatches are also taken into account. The theoretical approach retains all relevant stress and displacement components, and satisfies exactly the equilibrium equations, the interface conditions and other boundary conditions involving stresses. Two of the four stress-strain-temperature relations are satisfied exactly, whereas the remaining two are satisfied in an average sense. The required non-interface displacement boundary conditions are also satisfied in an average sense. The general representation is used to solve three types of stress transfer problem. A matrix crack and a broken fibre are analysed for the case when there is perfect bonding between fibre and matrix. The third type of problem takes account of frictional slip at the interface governed by the Coulomb friction law. The approximate analytic approach described in this paper, and the preliminary numerical predictions presented, indicate that the stress transfer between fibres and matrix in a unidirectional fibre-reinforced composite, loaded in tension, can now be investigated theoretically in more detail than before. The paper includes some discussion of singularities in the stress fields, which are smoothed by the averaging techniques employed in the analysis. The analytical approach has enabled the development of a micro-mechanical model of frictional slip at the fibre-matrix interface based on the Coulomb friction law, which is more realistic than assuming that the interfacial shear stress is a constant. Predictions are presented of the stress distributions along the fibre-matrix interface and, in particular, it is shown how the length of the frictional slip zone is related to applied strain, friction coefficient, fibre volume fraction and the difference between the test and ‘manufacturing’ temperatures. An indication is given of many other areas of composite modelling where the new theory will be applied.

On Boundary Integral Equations for Crack Problems

Abstract: A ubiquitous linear boundary-value problem in mathematical physics involves solving a partial differential equation exterior to a thin obstacle. One typical example is the scattering of scalar waves by a curved crack or rigid strip (Neumann boundary condition) in two dimensions. This problem can be reduced to an integrodifferential equation, which is often regularized. We adopt a more direct approach, and prove that the problem can be reduced to a hypersingular boundary integral equation. (Similar reductions will obtain in more complicated situations.) Computational schemes for solving this equation are described, with special emphasis on smoothness requirements. Extensions to three-dimensional problems involving an arbitrary smooth bounded crack in an elastic solid are discussed.

The Brittle-Ductile Transition in Silicon. I. Experiments

Abstract: The experiments described in this paper investigate the fundamental processes involved in the brittle–ductile transition (BDT) in silicon, and form the basis of a new theoretical model (see following paper). The fracture (or bending) stresses of four-point bend specimens of silicon containing semicircular surface cracks, introduced by surface indentation, were determined over a range of temperatures and strain rates. A sharp transition, characterized by a rapid increase in fracture stress with temperature, occurs at a temperature ( T c ) that depends on the strain rate and the doping of the material used; these data are used to derive activation energies, which are found to be equal to those for dislocation glide. At temperatures above the sharp transition region, the specimens deform by macroscopic plastic yielding. Etch pitting experiments show that below T c no significant dislocation activity occurs; the sharp brittle–ductile transition is associated with a sudden growth of well-defined dislocation arrays from certain points on the precursor flaw, before fracture occurs. These only appear in a dynamic test at T ≽ T c ; at T = T c , they form only when the applied stress intensity factor K is of the same order as that for brittle failure ( K Ic ) at T T c . These experiments suggest that at T ≈ T c , a 9nucleation9 event precedes the generation of avalanches of dislocations when K ≈ K Ic . Static tests show that dislocations can be made to move from crack tips of K value as low as ca . 0.3 K Ic . Above the transition region general plasticity occurs, with slip being concentrated particularly around and spreading from the precursor flow. A 9warm-prestressing9 effect is observed, whereby the low-temperature fracture stress is increased by prestressing above T c .

Structural localization phenomena and the dynamical phase-space analogy

Abstract: The localization of buckle patterns in elastic structures is reviewed from three complementary viewpoints: (a) from a modal perspective, (b) from a formulation which allows the amplitude to modulate in an asymptotically defined 'slow' space and (c) from a dynamical analogy in phase space suggested by the form of the underlying differential equation. A simple strut on an (asymmetric) nonlinear foundation provides a typical illustrative example. The three approaches emphasize different features of the localization phenomenon. The modal view illustrates the distinctive effects of boundary conditions, the modulated approach generates a convenient second-order differential equation in the amplitude function and the dynamical phase-space analogy suggests a useful interpretation of localization as a homoclinic connection. Comparisons are also made with nonlinear numerical solutions. As the strut length approaches infinity it is shown that the fully localized solution represents the unstable post-buckled state with the lowest energy, allowing evaluation of the minimum energy barrier relevant to dynamical impact studies. Attention is drawn to the possibility of spatial chaos, becoming manifest as a randomly spaced sequence of localizations caused by a regular sinusoidal spatial imperfection.

Evaluating the fractal dimension of surfaces

Abstract: Fractal objects derive from many interface phenomena, as they arise in, for example, materials science, chemistry and geology. Hence the problem of estimating fractal dimension becomes of both theoretical and practical importance. Existing algorithms implement the standard definitions of fractal dimension directly, but, as we show, often give unreliable results when applied to digitized and quantized data. We present a new algorithm for estimating the fractal dimension of surfaces - the variation method - that is more reliable and robust than the standard ones. It is based on a new definition of fractal dimension particularly suited for graphs of functions. The variation method is validated with both fractional brownian surfaces and Takagi surfaces, two classes of mathematical objects with known fractal dimension, and is shown to give more accurate results than the classical algorithms. Finally, our new algorithm is applied to data from sand-blasted metal surfaces.

On the dynamics of rigid block motion under harmonic forcing

Abstract: In this paper the simplest and most widely used model of a rigid block undergoing harmonic forcing is analysed in detail. The block is shown to possess extremely complicated dynamics, with many different types of response being revealed. Symmetric single-impact subharmonic orbits of all orders are found and regions of parameter space in which they occur are given. In particular, period-doubling cascades of asymmetric orbits are found, which ultimately produce an apparently non-periodic or chaotic response. Sensitivity to initial conditions is illustrated, which leads to uncertainty in the prediction of the asymptotic dynamics. Nevertheless, the transient response may be the most important in connection with real earthquakes. To this end, the concept of the domain of maximum transients is introduced. In this light the response is shown to be quite ordered and predictable, despite the chaotic nature of the asymptotic domain of attraction. It is shown that safety issues cannot be satisfactorily resolved until an agreed set of initial conditions is established. It appears that blocks may survive under very high accelerations and topple at very low accelerations provided the initial conditions are correct. Consideration is also given to the use of actual earthquake recordings in attempting to reproduce responses in given structures. If the present conclusions carry over to general excitations, then small errors in recordings may produce large differences in response. The present methods include orbital stability techniques together with detailed numerical computations. These results are backed up by encouraging qualitative agreement from an electronic analogue circuit.

Mixing, entrainment and fractal dimensions of surfaces in turbulent flows

Abstract: Some basic thoughts are set down on the relation between the fractal dimension of various surfaces in turbulent flows, and the practically important processes of mixing between two streams (reacting or otherwise) separated by a convoluted surface, as well as of entrainment of irrotational flow by a turbulent stream. An expression based on heuristic arguments is derived for the flux of transportable properties (such as mass, momentum, and energy) across surfaces, and a prediction made on this basis for the fractal dimension of surfaces in fully turbulent flows is shown to be in essential agreement with measurements. It is further shown that this prediction remains robust when corrected for the non-uniform effects along the surface. A related prediction concerning the dependence of mixing on the Reynolds number and the fractal dimension of the surface is substantiated, in the developing as well as the fully developed states, by independent measurements of both the fractal dimension and the amount of mixing between reactants in a temporally evolving countercurrent shear flow.

Universality of fractal aggregates as probed by light scattering

Abstract: Fractal colloid aggregates are studied with both static and dynamic light scattering. The dynamic light scattering data are scaled onto a single master curve, whose shape is sensitive to the structure of the aggregates and their mass distribution. By using the structure factor determined from computer-simulated aggregates, and including the effects of rotational diffusion, we predict the shape of the master curve for different cluster distributions. Excellent agreement is found between our predictions and the data for the two limiting regimes, diffusion-limited and reaction-limited colloid aggregation. Furthermore, using data from several completely differient colloids, we find that the shapes of the master curves are identical for each regime. In addition, the cluster fractal dimensions and the aggregation kinetics are identical in each regime. This provides convincing experimental evidence of the universality of these two regimes of colloid aggregation.

The brittle-ductile transition in silicon. II. Interpretation

Abstract: A dynamic crack tip shielding model has been developed to describe the brittle-ductile transition (BDT) of precracked crystals in constant strain-rate tests. Dislocations are emitted from a discrete number of sources at or near the crack tip. At the BDT the dislocations are emitted and move sufficiently rapidly to shield the most vulnerable parts of the crack, furthest away from the sources, such that the local stress intensity factor remains below K Ic for values of the applied stress intensity factor K above K Ic . Computer simulations of the dynamics of dislocation generation from the crack tip sources, assuming mode III loading, suggest that a sharp transition as observed in silicon is predicted only if generation starts at K ≡ K 0 ≈ K Ic , but then continues at K ≡ K N ≪ K Ic . Dislocation etch pit studies reported by Samuels & Roberts ( Proc. R. Soc. Lond. A 421, 1─23 (1989)) (hereafter called I) confirm that generation begins at K 0 ≈ K Ic . It is suggested that K 0 corresponds to the value of K at which a crack tip source is nucleated by movement of an existing dislocation in the crystal to the crack tip. The model accounts quantitatively for the strain-rate dependence of the transition temperature T c reported in I, and predicts a dependence of T c on dislocation density, in qualitative agreement with (unpublished) experiments. Calcluations of the strees field around the crack tip of a semicircular precrack, suggest that the ends of the half loops emitted by crack tip sources undergo multiple cross slip to follow the crack profile. The predicted dislocation configurations agree with etch pit observations reported in I.

Single-Point Diamond Machining of Glasses

Abstract: A machine tool of very high stiffness has been constructed and used for single-point diamond grooving of blanks of soda-lime glass and optical glassy quartz. Results show that below a critical depth of cut predicted in order of magnitude by a fracture mechanics analysis, material is removed by the action of plastic flow, leaving crack-free surfaces. Subsequent observations by scanning electron microscopy indicate that a crucial part in the detachment of ribbons of swarf is played by the operation of residual stresses after the passage of the tool, particularly in the case of the amorphous ceramic.

Evolution equations with dynamic boundary conditions

Abstract: In this paper, we study boundary problems with dynamic boundary conditions, that is, with boundary operators containing time derivatives. The equations under consideration are transformed into abstract Cauchy problems x – Cx = f and x(0) = x0. Abstract theoretical results concerning the operators C are obtained by the study of a naturally arising pseudodifferential operator. For existence and uniqueness theorems concerning solutions of parabolic and hyperbolic equations, we then apply the theory of semigroups in Banach spaces. Some examples of semilinear and quasilinear problems, to which our results apply, are given.

Fractal bet and FHH theories of adsorption: a comparative study

Abstract: Two theories of multilayer adsorption of gases, namely the BrunauerEmmett-Teller (bet) theory and the Frenkel-Halsey-Hill (fhh) theory, have recently been extended to the case of fractal substrates in a number of different ways. We present a critical evaluation of the various predictions. The principal results are the following. At high coverage, the fractal bet and fhh isotherms apply to mass and surface fractals, respectively. Both give characteristic power laws with D -dependent exponents ( D = fractal dimension of the substrate). The bet isotherm additionally depends on the topological dimension D top of the substrate. For fractal aggregates ( D top = 1) with D

On subordinacy and analysis of the spectrum of Schrödinger operators with two singular endpoints

Abstract: The theory of subordinacy is extended to all one-dimensional Schrodinger operatorsfor which the corresponding differential expression L = – d2/(dr2) + V(r) is in the limit point case at both ends of an interval (a, b), with V(r) locally integrable. This enables a detailed classification of the absolutely continuous and singular spectra to be established in terms of the relative asymptotic behaviour of solutions of Lu = xu, x eℝ, as r→a and r→b. The result provides a rigorous but straightforward method of direct spectral analysis which has very general application, and somefurther properties of the spectrum are deduced from the underlying theory.

Flow through porous media: limits of fractal patterns

Abstract: By using experiments on micromodels and computer simulations, we have demonstrated the existence of three types of basic displacements when a non-wetting fluid invades a two-dimensional porous medium: capillary fingering when capillary forces are very strong compared to viscous forces, viscous fingering when a less viscous fluid is displacing a more viscous one, and stable displacement in the opposite case. These displacements are described by statistical models: invasion percolation, diffusion-limited aggregation (DLA) and anti-DLA. The domains of validity of the basic displacements are mapped onto the plane with axes Ca (capillary number) and M (viscosity ratio). The boundaries of these domains are calculated either by using theoretical laws describing transport properties of fractal patterns or by the interpretation of physical mechanisms at the pore scale. In addition, the prefactors that are not available from scaling theories are obtained by computer simulations on a network of capillaries, in which the flow equations are solved at each node.

The Identified $\Lambda\Lambda$-Hypernuclei and the Predicted H-Particle

Abstract: The existence of the H particle, the dihyperon predicted by Jaffe, would bring into question the existence of double hypernuclei. We review the two double hypernucleus events published in the literature. We include an independent report, hitherto unpublished, which was made on the $\Lambda\Lambda^{10}$Be event in 1963 and clarifies the salient features of the event; this report reaffirms its published interpretation. We have made a simple calculation of the energy spectrum for $\Xi^-$ hyperons produced with K$^-$ beams in past emulsion experiments, with a result which accounts adequately for the paucity of reported double hypernucleus events. We outline a hybrid emulsion experiment that would locate $\Xi^-$ hyperon interactions efficiently and could thereby greatly improve our knowledge of double hypernuclei.

Wave Propagation in Transversely Isotropic Elastic Media. I. Homogeneous Plane Waves

Abstract: In relation to transversely isotropic media, this paper presents a detailed study of those aspects of the propagation of homogeneous plane elastic waves which are essential to a basic understanding of the behaviour of surface waves. It is first shown how the ordering of the speeds of plane waves provides, directly and simply, a means of classifying the chosen materials, with the class label specifying the broad structure of the slowness surface and the location of its singular points. An examination of the shape of the outer sheet of the slowness surface follows, providing inter alia a complete account of the incidence of the various types of transonic states. The discussion turns next to exceptional waves, that is homogeneous plane waves which leave free of traction some family of parallel planes. The subset of the plane waves possessing this property is determined, after which the subset of the exceptional waves serving as limiting waves for an exceptional transonic state is picked out. Exceptional transonic states occur only when the axis of material symmetry lies either in the reference plane or at right angles to the reference vector and these orientations of the axis are referred to as $\alpha$ and $\beta$ configurations respectively. The exceptional states are arranged in a threefold classification, one class consisting of a continuous set of $\alpha$ configurations and the others discrete $\beta$ configurations. The paper ends with calculations of the limiting speed of the transonic state for the totality of $\alpha$ and $\beta$ configurations.