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Showing papers in "Proceedings of The Royal Society A: Mathematical, Physical and Engineering Sciences in 2013"


Journal ArticleDOI
TL;DR: In this paper, a nonlinear elliptic problem with fractional powers of the Laplacian operator together with a concave concvex term is studied and the range of parameters for which solutions of the problem exist is characterized.
Abstract: We study a nonlinear elliptic problem defined in a bounded domain involving fractional powers of the Laplacian operator together with a concave—convex term. We completely characterize the range of parameters for which solutions of the problem exist and prove a multiplicity result. We also prove an associated trace inequality and some Liouville-type results.

460 citations


Journal ArticleDOI
TL;DR: In this article, a thermodynamic inspired formalization of bounded rational decision-making where information processing is modelled as state changes in thermodynamic systems that can be quantified by differences in free energy is proposed.
Abstract: Perfectly rational decision-makers maximize expected utility, but crucially ignore the resource costs incurred when determining optimal actions. Here, we propose a thermodynamically inspired formalization of bounded rational decision-making where information processing is modelled as state changes in thermodynamic systems that can be quantified by differences in free energy. By optimizing a free energy, bounded rational decision-makers trade off expected utility gains and information-processing costs measured by the relative entropy. As a result, the bounded rational decision-making problem can be rephrased in terms of well-known variational principles from statistical physics. In the limit when computational costs are ignored, the maximum expected utility principle is recovered. We discuss links to existing decision-making frameworks and applications to human decision-making experiments that are at odds with expected utility theory. Since most of the mathematical machinery can be borrowed from statistical physics, the main contribution is to re-interpret the formalism of thermodynamic free-energy differences in terms of bounded rational decision-making and to discuss its relationship to human decision-making experiments.

247 citations


Journal ArticleDOI
TL;DR: In this article, the authors used code UG under grant no. NNX09AV99G for support of this research under the Microgravity Fluid Sciences (Code UG) program.
Abstract: United States. National Aeronautics and Space Administration (Microgravity Fluid Sciences (Code UG) for support of this research under grant no. NNX09AV99G)

225 citations


Journal ArticleDOI
TL;DR: A parallel quantum search algorithm is presented that can be used by algorithm designers without worrying whether the underlying architecture supports the connectivity of the circuit and improves the time–space trade-off for the element distinctness and collision finding problems.
Abstract: We provide algorithms for efficiently moving and addressing quantum memory in parallel. These imply that the standard circuit model can be simulated with a low overhead by a more realistic model of a distributed quantum computer. As a result, the circuit model can be used by algorithm designers without worrying whether the underlying architecture supports the connectivity of the circuit. In addition, we apply our results to existing memory-intensive quantum algorithms. We present a parallel quantum search algorithm and improve the time–space trade-off for the element distinctness and collision finding problems.

209 citations


Journal ArticleDOI
TL;DR: A new bound ω < 2.37369 is given for the exponent of complexity of matrix multiplication, giving a small improvement on the previous bound obtained by Coppersmith and Winograd.
Abstract: We give a new bound ω < 2.37369 for the exponent of complexity of matrix multiplication, giving a small improvement on the previous bound obtained by Coppersmith and Winograd. The proof involves an extension of the method used by these authors. We have attempted to make the exposition self-contained.

152 citations


Journal ArticleDOI
TL;DR: The state of the art of modelling by means of Boussinesq-type models (BTMs) is described, with special attention to the interplay between the physics to be described, the chosen model equations and the numerics in use.
Abstract: This paper, which is largely the fruit of an invited talk on the topic at the latest International Conference on Coastal Engineering, describes the state of the art of modelling by means of Boussinesq-type models (BTMs). Motivations for using BTMs as well as their fundamentals are illustrated, with special attention to the interplay between the physics to be described, the chosen model equations and the numerics in use. The perspective of the analysis is that of a physicist/engineer rather than of an applied mathematician. The chronological progress of the currently available BTMs from the pioneering models of the late 1960s is given. The main applications of BTMs are illustrated, with reference to specific models and methods. The evolution in time of the numerical methods used to solve BTMs (e.g. finite differences, finite elements, finite volumes) is described, with specific focus on finite volumes. Finally, an overview of the most important BTMs currently available is presented, as well as some indications on improvements required and fields of applications that call for attention.

151 citations


Journal ArticleDOI
TL;DR: In this article, an upper bound for the power resource of the Pentland Firth was derived using a depth-averaged numerical model of the tidal dynamics in the region and validated against a series of test cases.
Abstract: This paper assesses an upper bound for the tidal stream power resource of the Pentland Firth. A depth-averaged numerical model of the tidal dynamics in the region is set-up and validated against fi...

140 citations


Journal ArticleDOI
Nicolas Noiray1, Bruno Schuermans1
TL;DR: In this article, a detailed analysis of the spatial Fourier amplitudes extracted from these data reveals that the acoustic modes are continuously switching between standing, clockwise and counter-clockwise travelling waves.
Abstract: This paper deals with the dynamics of standing and rotating azimuthal thermoacoustic modes in annular combustion chambers. Simultaneous acoustic measurements have been made at multiple circumferential positions in an annular gas turbine combustion chamber. A detailed statistical analysis of the spatial Fourier amplitudes extracted from these data reveals that the acoustic modes are continuously switching between standing, clockwise and counter-clockwise travelling waves. A theoretical framework from which the modal dynamics can be explained is proposed and supported by real gas turbine data. The stochastic differential equations that govern these systems have been derived and used as a basis for system identification of the measured engine data. The model describes the probabilities of the two azimuthal wave components as a function of the random source intensity, the asymmetry in the system and the strength of the thermoacoustic interaction. The solution of the simplified system is in good agreement with experimental observations on a gas turbine combustion chamber.

125 citations


Journal ArticleDOI
TL;DR: In this article, the authors describe flow-relative and flow-aided navigation of a biomimetic underwater vehicle using an artificial lateral line for flow sensing and demonstrate navigation with respect to the flow in periodic turbulence and show that controlling the position of the robot in the reduced flow zone in the wake of an object reduces a vehicle's energy consumption.
Abstract: This paper describes flow-relative and flow-aided navigation of a biomimetic underwater vehicle using an artificial lateral line for flow sensing. Most of the aquatic animals have flow sensing organs, but there are no man-made analogues to those sensors currently in use on underwater vehicles. Here, we show that artificial lateral line sensing can be used for detecting hydrodynamic regimens and for controlling the robot’s motion with respect to the flow. We implement station holding of an underwater vehicle in a steady stream and in the wake of a bluff object. We show that lateral line sensing can provide a speed estimate of an underwater robot thus functioning as a short-term odometry for robot navigation. We also demonstrate navigation with respect to the flow in periodic turbulence and show that controlling the position of the robot in the reduced flow zone in the wake of an object reduces a vehicle’s energy consumption.

114 citations


Journal ArticleDOI
TL;DR: In this paper, the authors investigated the effect of the acoustic azimuthal velocity, u, on the flames, and proposed a phenomenological model based on experiments performed on transversely forced flames to determine the type of thermoacoustic oscillations found in annular combustors.
Abstract: This theoretical study investigates spinning and standing modes in azimuthally symmetric annular combustion chambers. Both modes are observed in experiments and simulations, and an existing model predicts that spinning modes are the only stable state of the system. We extend this model to take into account the effect that the acoustic azimuthal velocity, u, has on the flames, and propose a phenomenological model based on experiments performed on transversely forced flames. This model contains a parameter, δ, that quantifies the influence that the transversal excitation has on the fluctuating heat release. For small values of δ, spinning modes are the only stable state of the system. In an intermediate range of δ, both spinning and standing modes are stable states. For large values of δ, standing modes are the only stable state. This study shows that a flame’s response to azimuthal velocity fluctuations plays an important role in determining the type of thermoacoustic oscillations found in annular combustors.

103 citations


Journal ArticleDOI
TL;DR: In this article, a dynamical system explanation of the metastability of an explicit family of solutions, referred to as bar states, of the two-dimensional incompressible Navier-Stokes equation on the torus was proposed.
Abstract: Quasi-stationary, or metastable, states play an important role in two-dimensional turbulent fluid flows, where they often emerge on timescales much shorter than the viscous timescale, and then dominate the dynamics for very long time intervals. In this paper we propose a dynamical systems explanation of the metastability of an explicit family of solutions, referred to as bar states, of the two-dimensional incompressible Navier–Stokes equation on the torus. These states are physically relevant because they are associated with certain maximum entropy solutions of the Euler equations, and they have been observed as one type of metastable state in numerical studies of two-dimensional turbulence. For small viscosity (high Reynolds number), these states are quasi-stationary in the sense that they decay on the slow, viscous timescale. Linearization about these states leads to a time-dependent operator. We show that if we approximate this operator by dropping a higher-order, non-local term, it produces a decay rate much faster than the viscous decay rate. We also provide numerical evidence that the same result holds for the full linear operator, and that our theoretical results give the optimal decay rate in this setting.

Journal ArticleDOI
TL;DR: In this paper, the collective ability of organisms to move coherently in space and time is ubiquitous in any group of autonomous agents that can move and sense each other and the environment.
Abstract: The collective ability of organisms to move coherently in space and time is ubiquitous in any group of autonomous agents that can move and sense each other and the environment. Here, we investigate...

Journal ArticleDOI
Jian Xu1, Engui Fan1
TL;DR: The unified transform method is implemented to the initial-boundary value (IBV) problem of the Sasa–Satsuma equation on the half line and the associated general Dirichlet to Neumann map is analysed using the so-called global relation.
Abstract: We implement the unified transform method to the initial-boundary value (IBV) problem of the Sasa–Satsuma equation on the half line. In addition to presenting the basic Riemann–Hilbert formalism, which linearizes this IBV problem, we also analyse the associated general Dirichlet to Neumann map using the so-called global relation.

Journal ArticleDOI
TL;DR: The orthodox emission hypothesis is a set of physical and mathematical assumptions that permit well-specified analysis of measured currentvoltage data relating to field electron emission (FE) as mentioned in this paper, and it is based on the assumption that the electron emission is caused by a single electron.
Abstract: The orthodox emission hypothesis is a set of physical and mathematical assumptions that permit well-specified analysis of measured currentvoltage data relating to field electron emission (FE). If t...

Journal ArticleDOI
TL;DR: In this paper, it is proved that the integrals are of Stratonovich type for continuous time and neither additive noise nor one-dimensional multiplicative noise is present in the continuous time integrals.
Abstract: A recent paper of Melbourne & Stuart, A note on diffusion limits of chaotic skew product flows, Nonlinearity 24 (2011) 1361–1367, gives a rigorous proof of convergence of a fast-slow deterministic system to a stochastic differential equation with additive noise. In contrast to other approaches, the assumptions on the fast flow are very mild. In this paper, we extend this result from continuous time to discrete time. Moreover we show how to deal with one-dimensional multiplicative noise. This raises the issue of how to interpret certain stochastic integrals; it is proved that the integrals are of Stratonovich type for continuous time and neither

Journal ArticleDOI
TL;DR: In this article, it was found that transgranular fracture is promoted by the segregation of hydrogen to mechanical twin interfaces and to any ϵ-martensite induced during deformation.
Abstract: It is known empirically that the addition of aluminium as a solute in high-Mn austenitic steels dramatically improves their resistance to hydrogen-induced embrittlement. A variety of experimental techniques, including the characterization of trapping sites and high-resolution observation of fracture facets, have been used to reveal the mechanism by which aluminium induces this effect. It is found that transgranular fracture is promoted by the segregation of hydrogen to mechanical twin interfaces and to any ϵ-martensite that is induced during deformation. Because aluminium increases the stacking fault energy of austenite, the tendency for mechanical twinning is reduced, and the formation of deformation-induced martensite eliminated. These two effects contribute to the resistance of the aluminium-alloyed steel to hydrogen embrittlement.

Journal ArticleDOI
TL;DR: Assessment of rupture risk could be improved by future experiments that delineate and quantify different aspects of patient-specific collagen turnover and that such understanding could lead to new targeted therapeutics.
Abstract: Abdominal aortic aneurysms (AAAs) are characterized by significant changes in the architecture of the aortic wall, notably, loss of functional elastin and smooth muscle. Because collagen is the principal remaining load-bearing constituent of the aneurysmal wall, its turnover must play a fundamental role in the natural history of the lesion. Nevertheless, detailed investigations of the effects of different aspects of collagen turnover on AAA development are lacking. A finite-element membrane model of the growth and remodelling of idealized AAAs was thus used to investigate parametrically four of the primary aspects of collagen turnover: rates of production, half-life, deposition stretch (prestretch) and material stiffness. The predicted rates of aneurysmal expansion and spatio-temporal changes in wall thickness, biaxial stresses and maximum collagen fibre stretch at the apex of the lesion depended strongly on all four factors, as did the predicted clinical endpoints (i.e. arrest, progressive expansion or rupture). Collagen turnover also affected the axial expansion, largely due to mechanical changes within the shoulder region of the lesion. We submit, therefore, that assessment of rupture risk could be improved by future experiments that delineate and quantify different aspects of patient-specific collagen turnover and that such understanding could lead to new targeted therapeutics.

Journal ArticleDOI
TL;DR: Partially ionized fluids can gain net momentum under an electric field, as charged particles undergo momentum transfer collisions with neutral molecules in a phenomenon termed an ionic wind as discussed by the authors, which is referred to as a wind.
Abstract: Partially ionized fluids can gain net momentum under an electric field, as charged particles undergo momentum-transfer collisions with neutral molecules in a phenomenon termed an ionic wind. Electr...

Journal ArticleDOI
Mazen Diab1, Teng Zhang1, Ruike Zhao1, Huajian Gao1, Kyung-Suk Kim1 
TL;DR: In this article, the surface creasing is set off when the compressive strain is large enough, and the longest-admissible perturbation wavelength relative to the decay length of the elastic modulus is shorter than a critical value.
Abstract: We present mechanics of surface creasing caused by lateral compression of a nonlinear neo-Hookean solid surface, with its elastic stiffness decaying exponentially with depth. Nonlinear bifurcation stability analysis reveals that neo-Hookean solid surfaces can develop instantaneous surface creasing under compressive strains greater than 0.272 but less than 0.456. It is found that instantaneous creasing is set off when the compressive strain is large enough, and the longest-admissible perturbation wavelength relative to the decay length of the elastic modulus is shorter than a critical value. A compressive strain smaller than 0.272 can only trigger bifurcation of a stable wrinkle that can prompt a setback crease upon further compression. The minimum compressive strain required to develop setback creasing is found to be 0.174. If the relative longest-admissible perturbation wavelength is long enough, then the wrinkle can fold before it creases, and the specimen can be compressed further beyond the Biot critical strain limit of 0.456. Various bifurcation branches on a plane of normalized longest-admissible wavelength versus compressive strain delineate different phases of corrugated surface configurations to form a ruga phase diagram. The phase diagram will be useful for understating surface crease, as well as for controlling ruga structures for various applications, such as designing stretchable electronics.

Journal ArticleDOI
TL;DR: This work considers methods for evaluating the probability mass function of the total number of infections over the course of a stochastic epidemic, with a focus on homogeneous finite populations, but also considering heterogeneous and large populations.
Abstract: Epidemic models have become a routinely used tool to inform policy on infectious disease A particular interest at the moment is the use of computationally intensive inference to parametrize these models In this context, numerical efficiency is critically important We consider methods for evaluating the probability mass function of the total number of infections over the course of a stochastic epidemic, with a focus on homogeneous finite populations, but also considering heterogeneous and large populations Relevant methods are reviewed critically, with existing and novel extensions also presented We provide code in MATLAB and a systematic comparison of numerical efficiency

Journal ArticleDOI
TL;DR: In this article, a bending-dominated model material (i.e., transversely loaded hexagonal honeycomb) was explored for iterative structural refinement, which is susceptible to improvement by simple iterative refinement that replaces each three-edge structural node with a smaller hexagon.
Abstract: Hierarchical structures are observed in nature, and can be shown to offer superior efficiency. However, the potential advantages of structural hierarchy are not well understood. We extensively explored a bending-dominated model material (i.e. transversely loaded hexagonal honeycomb) which is susceptible to improvement by simple iterative refinement that replaces each three-edge structural node with a smaller hexagon. Using a blend of analytical and numerical techniques, both elastic and plastic properties were explored over a range of loadings and iteration parameters. A wide variety of specific stiffness and specific strengths (up to fourfold increase) were achieved. The results offer insights into the potential value of iterative structural refinement for creating low-density materials with desired properties and function.

Journal ArticleDOI
TL;DR: A theoretical model is constructed to predict the mechanical energy associated with the compression of TMP bellows, which is compared with the experimentally measured energy, resulting in the gap between the mechanical work by the compression force and the bending energy distributed along all the crease lines.
Abstract: In this paper, we examine the folding behaviour of Tachi–Miura polyhedron (TMP) bellows made of paper, which is known as a rigid-foldable structure, and construct a theoretical model to predict the mechanical energy associated with the compression of TMP bellows, which is compared with the experimentally measured energy, resulting in the gap between the mechanical work by the compression force and the bending energy distributed along all the crease lines. The extended Hamilton's principle is applied to explain the gap which is considered to be energy dissipation in the mechanical behaviour of TMP bellows.

Journal ArticleDOI
TL;DR: In this paper, the second law of classical thermodynamics in terms of the entropy principle was presented, and an empirically accessible axiomatic derivati cation for the second Law was provided.
Abstract: In earlier work, we presented a foundation for the second law of classical thermodynamics in terms of the entropy principle. More precisely, we provided an empirically accessible axiomatic derivati...

Journal ArticleDOI
TL;DR: An absolute chronology for Early Egypt is produced by combining radiocarbon and archaeological evidence within a Bayesian paradigm and indicates that the process occurred more rapidly than previously thought.
Abstract: The Egyptian state was formed prior to the existence of verifiable historical records. Conventional dates for its formation are based on the relative ordering of artefacts. This approach is no longer considered sufficient for cogent historical analysis. Here, we produce an absolute chronology for Early Egypt by combining radiocarbon and archaeological evidence within a Bayesian paradigm. Our data cover the full trajectory of Egyptian state formation and indicate that the process occurred more rapidly than previously thought. We provide a timeline for the First Dynasty of Egypt of generational-scale resolution that concurs with prevailing archaeological analysis and produce a chronometric date for the foundation of Egypt that distinguishes between historical estimates.

Journal ArticleDOI
TL;DR: In this article, the authors revisited the classical problem of horizontal and vertical point loads suddenly applied onto the surface of a homogeneous, elastic half-space, and provided a complete set of exact, explicit formulae which are cast in the most compact format and with the simplest possible structure.
Abstract: This article revisits the classical problem of horizontal and vertical point loads suddenly applied onto the surface of a homogeneous, elastic half-space, and provides a complete set of exact, explicit formulae which are cast in the most compact format and with the simplest possible structure The formulae given are valid for the full range of Poisson9s ratios from 0 to 05, and they treat real and complex poles alike, as a result of which a single set of formulae suffices and also exact formulae for dipoles can be given

Journal ArticleDOI
TL;DR: In this article, a multi-scale analysis based on a sharp interface limit for the dynamics of bilayer structures of the functionalized Cahn-Hilliard equation is presented, which yields a quenched mean-curvature-driven normal velocity at O( e −1) whereas on the longer O(e −2) time scale, it leads to a total surface area preserving Willmore flow.
Abstract: We use a multi-scale analysis to derive a sharp interface limit for the dynamics of bilayer structures of the functionalized Cahn–Hilliard equation. In contrast to analysis based on single-layer interfaces, we show that the Stefan and Mullins–Sekerka problems derived for the evolution of single-layer interfaces for the Cahn–Hilliard equation are trivial in this context, and the sharp interface limit yields a quenched mean-curvature-driven normal velocity at O ( e −1), whereas on the longer O ( e −2) time scale, it leads to a total surface area preserving Willmore flow. In particular, for space dimension n =2, the constrained Willmore flow drives collections of spherically symmetric vesicles to a common radius, whereas for n =3, the radii are constant, and for n ≥4 the largest vesicle dominates.

Journal ArticleDOI
TL;DR: A geometric framework to calculate the residual stress fields and deformations of nonlinear solids with inclusions and eigenstrains is introduced and it is shown how singularities in the stress distribution emerge as a consequence of a mismatch between radial and circumferential eigen Strains at the centre of a sphere or the axis of a cylinder.
Abstract: We introduce a geometric framework to calculate the residual stress fields and deformations of nonlinear solids with inclusions and eigenstrains. Inclusions are regions in a body with different reference configurations from the body itself and can be described by distributed eigenstrains. Geometrically, the eigenstrains define a Riemannian 3-manifold in which the body is stress-free by construction. The problem of residual stress calculation is then reduced to finding a mapping from the Riemannian material manifold to the ambient Euclidean space. Using this construction, we find the residual stress fields of three model systems with spherical and cylindrical symmetries in both incompressible and compressible isotropic elastic solids. In particular, we consider a finite spherical ball with a spherical inclusion with uniform pure dilatational eigenstrain and we show that the stress in the inclusion is uniform and hydrostatic. We also show how singularities in the stress distribution emerge as a consequence of a mismatch between radial and circumferential eigenstrains at the centre of a sphere or the axis of a cylinder.

Journal ArticleDOI
TL;DR: In this paper, a new set of laboratory observations are presented and the results compared with the commonly applied statistical distributions, taken as a whole, the data confirm that the crest-height distributions are critically dependent upon the directionality of the sea state.
Abstract: This paper concerns the crest height statistics arising in sea states that are broad banded in both frequency and direction. A new set of laboratory observations are presented and the results compared with the commonly applied statistical distributions. Taken as a whole, the data confirm that the crest-height distributions are critically dependent upon the directionality of the sea state. Although nonlinear effects arising at third order and above are most pronounced in uni-directional seas, the present data show that they are also important in directionally spread seas, provided the seas are sufficiently steep and not too short crested. The data also highlight the limiting effects of wave breaking. With individual breaking events dependent upon the local wave steepness, the directionality of the sea state again plays a significant role. Indeed, the present observations confirm that the two competing processes of nonlinear amplification and wave breaking can have a profound influence on the crest-height distributions leading to significant departures from established theory. In such cases, the key parameters are the sea state steepness and directional spread; the latter acting to counter the former in terms of nonlinear changes in the crest-height distributions.

Journal ArticleDOI
TL;DR: In this paper, the authors propose a rigorous test for the detection of structural sensitivity in a system with respect to the local stability of equilibria, the main idea being to project infinite dimensional function space onto a finite dimensional space by considering the local properties of the model functions.
Abstract: It is well recognized that models in the life sciences can be sensitive to small variations in their model functions, a phenomenon known as ‘structural sensitivity’. Conventionally, modellers test for sensitivity by varying parameters for a specific formulation of the model functions, but models can show structural sensitivity to the choice of functional representations used: a particularly concerning problem when system processes are too complex, or insufficiently understood, to theoretically justify specific parameterizations. Here we propose a rigorous test for the detection of structural sensitivity in a system with respect to the local stability of equilibria, the main idea being to project infinite dimensional function space onto a finite dimensional space by considering the local properties of the model functions. As an illustrative example, we use our test to demonstrate structural sensitivity in the seminal Rosenzweig–MacArthur predator–prey model, and show that the conventional parameter-based approach can fail to do so. We also consider some implications that structural sensitivity has for ecological modelling: we argue that when the model exhibits structural sensitivity but experimental results remain consistent it may indicate that there is a problem with the model construction, and that in some cases trying to find an ‘optimal’ parameterization of a model function may simply be impossible when the model exhibits structural sensitivity. Finally, we suggest that the phenomenon of structural sensitivity in biological models may help explain the irregular oscillations often observed in real ecosystems.

Journal ArticleDOI
TL;DR: It is demonstrated how one replaces metamaterial and photonic crystal structures asymptotically by a continuum, and therefore by a set of equations, that captures the behaviour of potentially high-frequency waves propagating through a periodic medium.
Abstract: Metamaterial and photonic crystal structures are central to modern optics and are typically created from multiple elementary repeating cells. We demonstrate how one replaces such structures asymptotically by a continuum, and therefore by a set of equations, that captures the behaviour of potentially high-frequency waves propagating through a periodic medium. The high-frequency homogenization that we use recovers the classical homogenization coefficients in the low-frequency long-wavelength limit. The theory is specifically developed in electromagnetics for two-dimensional square lattices where every cell contains an arbitrary hole with Neumann boundary conditions at its surface and implemented numerically for cylinders and split-ring resonators. Illustrative numerical examples include lensing via all-angle negative refraction, as well as omni-directive antenna, endoscope and cloaking effects. We also highlight the importance of choosing the correct Brillouin zone and the potential of missing interesting physical effects depending upon the path chosen.