# Showing papers in "Proceedings of The Royal Society A: Mathematical, Physical and Engineering Sciences in 2009"

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TL;DR: The mother protocol described here is easily transformed into the so-called ‘father’ protocol whose children provide the quantum capacity and the entanglement-assisted capacity of a quantum channel, demonstrating that the division of single-sender/single-receiver protocols into two families was unnecessary.

Abstract: We give a simple, direct proof of the ‘mother’ protocol of quantum information theory. In this new formulation, it is easy to see that the mother, or rather her generalization to the fully quantum Slepian–Wolf protocol, simultaneously accomplishes two goals: quantum communication-assisted entanglement distillation and state transfer from the sender to the receiver. As a result, in addition to her other ‘children’, the mother protocol generates the state-merging primitive of Horodecki, Oppenheim and Winter, a fully quantum reverse Shannon theorem, and a new class of distributed compression protocols for correlated quantum sources which are optimal for sources described by separable density operators. Moreover, the mother protocol described here is easily transformed into the so-called ‘father’ protocol whose children provide the quantum capacity and the entanglement-assisted capacity of a quantum channel, demonstrating that the division of single-sender/single-receiver protocols into two families was unnecessary: all protocols in the family are children of the mother.

287 citations

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TL;DR: In this paper, the most general objective stored elastic energy for a second gradient material is deduced using a literature result of Fortune ´ & Vallee, proving that these materials are characterized by seven elastic moduli and generalizing previous studies by Toupin, Mindlin and Sokolowski.

Abstract: In the spirit of Germain the most general objective stored elastic energy for a second gradient material is deduced using a literature result of Fortune ´ & Vallee. Linear isotropic constitutive relations for stress and hyperstress in terms of strain and strain- gradient are then obtained proving that these materials are characterized by seven elastic moduli and generalizing previous studies by Toupin, Mindlin and Sokolowski. Using a suitable decomposition of the strain-gradient, it is found a necessary and sufficient condition, to be verified by the elastic moduli, assuring positive definiteness of the stored elastic energy. The problem of warping in linear torsion of a prismatic second gradient cylinder is formulated, thus obtaining a possible measurement procedure for one of the second gradient elastic moduli.

234 citations

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TL;DR: Starting from microscopic interactions among individuals, this work arrives at a macroscopic description of the opinion formation process that is characterized by a system of Fokker–Planck-type equations.

Abstract: We propose a mathematical model for opinion formation in a society that is built of two groups, one group of `ordinary? people and one group of `strong opinion leaders?. Our approach is based on an opinion formation model introduced in Toscani (Toscani 2006 Commun. Math. Sci.4, 481?496) and borrows ideas from the kinetic theory of mixtures of rarefied gases. Starting from microscopic interactions among individuals, we arrive at a macroscopic description of the opinion formation process that is characterized by a system of Fokker?Planck-type equations. We discuss the steady states of this system, extend it to incorporate emergence and decline of opinion leaders and present numerical results.

217 citations

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TL;DR: In this paper, a new phase-field model for strongly anisotropic crystal and epitaxial growth using regularized Cahn-Hilliard-type equations is presented, where the square of the mean curvature is added to the energy to remove the ill-posedness.

Abstract: We present a new phase-field model for strongly anisotropic crystal and epitaxial growth using regularized, anisotropic Cahn–Hilliard-type equations. Such problems arise during the growth and coarsening of thin films. When the anisotropic surface energy is sufficiently strong, sharp corners form and unregularized anisotropic Cahn–Hilliard equations become ill-posed. Our models contain a high-order Willmore regularization, where the square of the mean curvature is added to the energy, to remove the ill-posedness. The regularized equations are sixth order in space. A key feature of our approach is the development of a new formulation in which the interface thickness is independent of crystallographic orientation. Using the method of matched asymptotic expansions, we show the convergence of our phase-field model to the general sharp-interface model. We present two- and three-dimensional numerical results using an adaptive, nonlinear multigrid finite-difference method. We find excellent agreement between the dynamics of the new phase-field model and the sharp-interface model. The computed equilibrium shapes using the new model also match a recently developed analytical sharp-interface theory that describes the rounding of the sharp corners by the Willmore regularization.

190 citations

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TL;DR: Using the theory of binary matroids, it is argued that the paradigm is rich enough to enable sampling from probability distributions that cannot, classically, be sampled efficiently and accurately.

Abstract: We examine theoretic architectures and an abstract model for a restricted class of quantum computation, called here temporally unstructured (instantaneous) quantum computation because it allows for...

187 citations

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TL;DR: Reduced Bloch mode expansion (RBME) as mentioned in this paper employs a natural basis composed of a selected reduced set of Bloch eigenfunctions, which is selected within the irreducible Brillouin zone at high symmetry points determined by the medium's crystal structure and group theory.

Abstract: Reduced Bloch mode expansion (RBME) is presented for fast periodic media band structure calculations. The expansion employs a natural basis composed of a selected reduced set of Bloch eigenfunctions. The reduced basis is selected within the irreducible Brillouin zone at high symmetry points determined by the medium’s crystal structure and group theory (and possibly at additional related points). At each of the reciprocal lattice selection points, a number of Bloch eigenfunctions are selected up to the frequency/energy range of interest for the band structure calculations. As it is common to initially discretize the periodic unit cell and solution field using some choice of basis, RBME is practically a secondary expansion that uses a selected set of Bloch eigenvectors. Such expansion therefore keeps, and builds on, any favourable attributes a primary expansion approach might exhibit. Being in line with the well-known concept of modal analysis, the proposed approach maintains accuracy while reducing the computation time by up to two orders of magnitudes or more depending on the size and extent of the calculations. Results are presented for phononic, photonic and electronic band structures.

186 citations

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TL;DR: This work bound the trade-off between AB's and AC's violation of the Clauser–Horne–Shimony–Holt inequality and demonstrates that forcing B and C to be classically correlated prevents A and B from violating certain Bell inequalities, relevant for interactive proof systems and cryptography.

Abstract: We describe a new technique for obtaining Tsirelson bounds, which are upper bounds on the quantum value of a Bell inequality. Since quantum correlations do not allow signalling, we obtain a Tsirelson bound by maximizing over all no-signalling probability distributions. This maximization can be cast as a linear programme. In a setting where three parties, A, B and C, share an entangled quantum state of arbitrary dimension, we (i) bound the trade-off between AB's and AC's violation of the Clauser-Horne-Shimony Holt inequality and (ii) demonstrate that forcing B and C to be classically correlated prevents A and B from violating certain Bell inequalities, relevant for interactive proof systems and cryptography.

168 citations

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TL;DR: On its opening day, the London Millennium Bridge (LMB) experienced unexpected large amplitude lateral vibrations due to crowd loading as mentioned in this paper, and this form of pedestrianstructure interaction has since been id...

Abstract: On its opening day, the London Millennium Bridge (LMB) experienced unexpected large amplitude lateral vibrations due to crowd loading. This form of pedestrianstructure interaction has since been id...

162 citations

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TL;DR: A new numerical scheme for the time discretization of the finite-dimensional Galerkin stochastic differential equations is introduced, which is called the exponential Euler scheme, and it is proved that it converges faster than the classical numerical schemes for this equation with the general noise.

Abstract: We consider the numerical approximation of parabolic stochastic partial differential equations driven by additive space–time white noise. We introduce a new numerical scheme for the time discretization of the finite-dimensional Galerkin stochastic differential equations, which we call the exponential Euler scheme, and show that it converges (in the strong sense) faster than the classical numerical schemes, such as the linear-implicit Euler scheme or the Crank–Nicholson scheme, for this equation with the general noise. In particular, we prove that our scheme applied to a semilinear stochastic heat equation converges with an overall computational order 1/3 which exceeds the barrier order 1/6 for numerical schemes using only basic increments of the noise process reported previously. By contrast, our scheme takes advantage of the smoothening effect of the Laplace operator and of a linear functional of the noise and, therefore overcomes this order barrier.

161 citations

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TL;DR: In this article, the relationship between the structure and the activity of a glass composition in a biological environment has not been studied in detail, which negatively affects further progress, for instance, to improve the chemical durability and tailor the biodegradability of these materials for specific applications.

Abstract: The bioactive mechanism, by which living tissues attach to and integrate with an artificial implant through stable chemical bonds, is at the core of many current medical applications of biomaterials, as well as of novel promising applications in tissue engineering. Having been employed in these applications for almost 40 years, soda-lime phosphosilicate glasses such as 45S5 represent today the paradigm of bioactive materials. Despite their strategical importance in the field, the relationship between the structure and the activity of a glass composition in a biological environment has not been studied in detail. This fundamental gap negatively affects further progress, for instance, to improve the chemical durability and tailor the biodegradability of these materials for specific applications. This paper reviews recent advances in computer modelling of bioactive glasses based on molecular dynamics simulations, which are starting to unveil key structural features of these materials, thus contributing to improve our fundamental understanding of how bioactive materials work.

142 citations

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TL;DR: In this paper, the authors derived the first known numerical shallow water model on the sphere using radial basis function (RBF) spatial discretization, a novel numerical methodology that does not require any grid or mesh.

Abstract: The paper derives the first known numerical shallow water model on the sphere using radial basis function (RBF) spatial discretization, a novel numerical methodology that does not require any grid or mesh. In order to perform a study with regard to its spatial and temporal errors, two nonlinear test cases with known analytical solutions are considered. The first is a global steady-state flow with a compactly supported velocity field, while the second is an unsteady flow where features in the flow must be kept intact without dispersion. This behaviour is achieved by introducing forcing terms in the shallow water equations. Error and time stability studies are performed, both as the number of nodes are uniformly increased and the shape parameter of the RBF is varied, especially in the flat basis function limit. Results show that the RBF method is spectral, giving exceptionally high accuracy for low number of basis functions while being able to take unusually large time steps. In order to put it in the context of other commonly used global spectral methods on a sphere, comparisons are given with respect to spherical harmonics, double Fourier series and spectral element methods.

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TL;DR: This paper derives asymmetric codes using a combination of Bose–Chaudhuri–Hocquenghem (BCH) and finite geometry low-density parity-check (LDPC) codes and shows that the asymmetric quantum codes offer two advantages, namely to allow a higher rate without sacrificing performance when compared with symmetric codes and vice versa to allowed a higher performance whenCompared with symmetrical codes of comparable rates.

Abstract: Recently, quantum error-correcting codes have been proposed that capitalize on the fact that many physical error models lead to a significant asymmetry between the probabilities for bit- and phase-flip errors. An example for a channel that exhibits such asymmetry is the combined amplitude damping and dephasing channel, where the probabilities of bit and phase flips can be related to relaxation and dephasing time, respectively. We study asymmetric quantum codes that are obtained from the Calderbank–Shor–Steane (CSS) construction. For such codes, we derive upper bounds on the code parameters using linear programming. A central result of this paper is the explicit construction of some new families of asymmetric quantum stabilizer codes from pairs of nested classical codes. For instance, we derive asymmetric codes using a combination of Bose–Chaudhuri–Hocquenghem (BCH) and finite geometry low-density parity-check (LDPC) codes. We show that the asymmetric quantum codes offer two advantages, namely to allow a higher rate without sacrificing performance when compared with symmetric codes and vice versa to allow a higher performance when compared with symmetric codes of comparable rates. Our approach is based on a CSS construction that combines BCH and finite geometry LDPC codes.

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TL;DR: In this article, the authors developed fully Eulerian, implicit constitutive equations for the mechanical response of a class of materials that do not dissipate mechanical work in any process, and obtained a form for the Helmholtz potential as a function of the current mass density, the Cauchy stress and certain other parameters that capture anisotropic response.

Abstract: The purpose of this brief note is to develop fully Eulerian, implicit constitutive equations for the mechanical response of a class of materials that do not dissipate mechanical work in any process. We show that such materials can be modelled by obtaining a form for the Helmholtz potential as a function of the current mass density, the Cauchy stress and certain other parameters that capture anisotropic response. The resulting constitutive equations are of the form ![Graphic][1] , where ![Graphic][2] and ![Graphic][3] are functions of the state variables of the system. The class of materials that can be obtained from such a constitutive relation is considerably more general than conventional Green-elastic hyperelastic materials. Such response functions may be suitable for the modelling of biological tissue where, due to the constant remodelling that takes place, there may be no physical meaning to a ‘reference configuration’.
[1]: /embed/inline-graphic-1.gif
[2]: /embed/inline-graphic-2.gif
[3]: /embed/inline-graphic-3.gif

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TL;DR: In this article, first-principles calculations relevant to the adsorption of aromatic molecules on metal surfaces are reviewed, highlighting an area where the predictive power of theory is likely to prove decisive in the future.

Abstract: We review first-principles calculations relevant to the adsorption of aromatic molecules on metal surfaces. Benzene has been intensively studied on a variety of substrates, providing an opportunity to comment upon trends from one metal to another. Meanwhile, calculations elucidating the adsorption of polycyclic aromatic molecules are more sparse, but nevertheless yield important insights into the role of non-covalent interactions. Heterocyclic and substituted aromatic compounds introduce the complicating possibility of electronic and steric effects, whose relative importance can thus far only be gauged on a case-by-case basis. Finally, the coadsorption and/or reaction of aromatic molecules is discussed, highlighting an area where the predictive power of theory is likely to prove decisive in the future.

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TL;DR: In this article, the authors proposed an extension of continuum thermomechanics to fractal porous media that are specified by a mass fractal dimension D, a surface fractal dimensions d and a resolution length scale R. This procedure allows a specification of a geometry configuration of continua by fractal metric coefficients, on which the continuum mechanics is subsequently constructed.

Abstract: This paper builds on the recently begun extension of continuum thermomechanics to fractal porous media that are specified by a mass (or spatial) fractal dimension D, a surface fractal dimension d and a resolution length scale R. The focus is on pre-fractal media (i.e. those with lower and upper cut-offs) through a theory based on a dimensional regularization, in which D is also the order of fractional integrals employed to state global balance laws. In effect, the governing equations are cast in forms involving conventional (integer order) integrals, while the local forms are expressed through partial differential equations with derivatives of integer order but containing coefficients involving D, d and R. This procedure allows a specification of a geometry configuration of continua by ‘fractal metric’ coefficients, on which the continuum mechanics is subsequently constructed. While all the derived relations depend explicitly on D, d and R, upon setting D = 3 and d = 2, they reduce to conventional forms of governing equations for continuous media with Euclidean geometries. Whereas the original formulation was based on a Riesz measure—and thus more suited to isotropic media—the new model is based on a product measure, making it capable of grasping local fractal anisotropy. Finally, the one-, two- and three-dimensional wave equations are developed, showing that the continuum mechanics approach is consistent with that obtained via variational energy principles.

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TL;DR: In this paper, a model for elasticity, plasticity and twinning in anisotropic single crystals subjected to large deformations is developed, with shearing rates on discrete glide and deformation twinning modelled explicitly.

Abstract: A model is developed for elasticity, plasticity and twinning in anisotropic single crystals subjected to large deformations. Dislocation glide and deformation twinning are dissipative, while energy storage mechanisms associated with dislocation lines and twin boundaries are described via scalar internal state variables. Concepts from continuum crystal plasticity are invoked, with shearing rates on discrete glide and twinning systems modelled explicitly. The model describes aspects of thermomechanical behaviour of single crystals of alumina over a range of loading conditions. Resolved shear stresses necessary for glide or twin nucleation at low to moderate temperatures are estimated from nonlinear elastic calculations, theoretical considerations of Peierls barriers and stacking fault energies and observations from shock physics experiments. These estimates are combined with the existing data from high-temperature experiments to provide initial yield conditions spanning a wide range of temperatures. The model reflects hardening of glide and twin systems from dislocations accumulated during basal slip. Residual elastic volume changes, predicted from nonlinear elastic considerations and approximated dislocation line energies, are positive and proportional to the dislocation line density. While the model suggests that generation of very large dislocation densities could influence the pressure–volume response, volume increases from defects are predicted to be small in crystals deformed via basal glide on a single system.

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TL;DR: In this article, closed timelike curves (CTC) are studied and their consequences have led to non-trivial insights into general relativity, quantum information and other areas.

Abstract: While closed timelike curves (CTCs) are not known to exist, studying their consequences has led to non-trivial insights into general relativity, quantum information and other areas. In this paper, ...

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TL;DR: In this article, a boundary element implementation of the regularized Stokeslet method of Cortez is applied to cilia and flagella-driven flows in biology, and it is found that reducing cilia spacing reduces transport, and increasing cilia number increases the transport, up to a plateau at 9×9 cilia.

Abstract: A boundary element implementation of the regularized Stokeslet method of Cortez is applied to cilia and flagella-driven flows in biology. Previously published approaches implicitly combine the force discretization and the numerical quadrature used to evaluate boundary integrals. By contrast, a boundary element method can be implemented by discretizing the force using basis functions, and calculating integrals using accurate numerical or analytic integration. This substantially weakens the coupling of the mesh size for the force and the regularization parameter, and greatly reduces the number of degrees of freedom required. When modelling a cilium or flagellum as a one-dimensional filament, the regularization parameter can be considered a proxy for the body radius, as opposed to being a parameter used to minimize numerical errors. Modelling a patch of cilia, it is found that: (i) for a fixed number of cilia, reducing cilia spacing reduces transport, (ii) for fixed patch dimension, increasing cilia number increases the transport, up to a plateau at 9×9 cilia. Modelling a choanoflagellate cell, it is found that the presence of a lorica structure significantly affects transport and flow outside the lorica, but does not significantly alter the force experienced by the flagellum.

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TL;DR: In this paper, a two-level method is presented to determine the feasible regions of lamination parameters where potential ply orientations are a predefined finite set, and a nonlinear algebraic identity is used to relate the in-plane, coupling and out-of-plane laminar parameters to each other.

Abstract: The stiffness tensors of a laminated composite may be expressed as a linear function of material invariants and lamination parameters. Owing to the nature of orienting unidirectional laminae ply by ply, lamination parameters, which are trigonometric functions of the ply orientation, are interrelated. In optimization studies, lamination parameters are often treated as independent design variables constrained by inequality relationships to feasible regions that depend on their values. The relationships between parameters enclose a convex feasible region of lamination parameters which is generally unknown. The convexity properties allow the efficient optimization of laminated composite structures where lamination parameters are used as design variables. Herein, a two-level method is presented to determine the feasible regions of lamination parameters where potential ply orientations are a predefined finite set. At the first level, the feasible region of the in-plane, coupling and out-of-plane lamination parameters is determined separately using convex hulls. At the second level, a nonlinear algebraic identity is used to relate the in-plane, coupling and out-of-plane lamination parameters to each other. This general approach yields all constraints on the feasible regions of lamination parameters for a predefined set of ply orientations.

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TL;DR: In this article, a single-order time-fractional diffusion-wave equation is generalized by introducing a time distributed-order fractional derivative and forcing term, while a Laplacian is replaced by a general linear multi-dimensional spatial differential operator.

Abstract: A single-order time-fractional diffusion-wave equation is generalized by introducing a time distributed-order fractional derivative and forcing term, while a Laplacian is replaced by a general linear multi-dimensional spatial differential operator. The obtained equation is (in the case of the Laplacian) called a time distributed-order diffusion-wave equation. We analyse a Cauchy problem for such an equation by means of the theory of an abstract Volterra equation. The weight distribution, occurring in the distributed-order fractional derivative, is specified as the sum of the Dirac distributions and the existence and uniqueness of solutions to the Cauchy problem, and the corresponding Volterra-type equation were proven for a general linear spatial differential operator, as well as in the special case when the operator is Laplacian.

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TL;DR: In this paper, the authors studied the existence of travelling waves for a class of epidemic models structured in space and with respect to the age of infection, and derived a necessary and sufficient condition for the presence of traveling waves for such models.

Abstract: In this article, we study the existence of travelling waves for a class of epidemic models structured in space and with respect to the age of infection. We obtain a necessary and sufficient condition for the existence of travelling waves for such a class of problems. As a consequence of our main result, we also derive the existence of travelling waves of a class of functional partial derivative equations.

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TL;DR: In this paper, the authors used a power law exponent of 1 4 to estimate the ages of fire-clay artefacts from Roman to modern dates, consistent with the theory of fractional (anomalous) single file diffusion.

Abstract: Fired-clay materials such as brick, tile and ceramic artefacts are found widely in archaeological deposits. The slow progressive chemical recombination of ceramics with environmental moisture (rehydroxylation) provides the basis for archaeological dating. Rehydroxylation rates are described by a (time)1/4 power law. A ceramic sample may be dated by first heating it to determine its lifetime water mass gain, and then exposing it to water vapour to measure its mass gain rate and hence its individual rehydroxylation kinetic constant. The kinetic constant depends on temperature. Mean lifetime temperatures are estimated from historical meteorological data. Calculated ages of samples of established provenance from Roman to modern dates agree excellently with assigned (known) ages. This agreement shows that the power law holds precisely on millennial time scales. The power law exponent is accurately 1 4, consistent with the theory of fractional (anomalous) single-file diffusion.

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TL;DR: In this article, a Cauchy problem for a time distributed-order multi-dimensional diffusion-wave equation containing a forcing term is reinterpreted in the space of tempered distributions, and a distributional diffusionwave equation is obtained.

Abstract: A Cauchy problem for a time distributed-order multi-dimensional diffusion-wave equation containing a forcing term is reinterpreted in the space of tempered distributions, and a distributional diffusion-wave equation is obtained. The distributional equation is solved in the general case of weight function (or distribution). Solutions are given in terms of solution kernels (Green’s functions), which are studied separately for two cases. The first case is when the order of the fractional derivative is in the interval [0, 1], while, in the second case, the order of the fractional derivative is in the interval [0, 2]. Solutions of fractional diffusionwave and fractional telegraph equations are obtained as special cases. Numerical experiments are also performed. An analogue of the maximum principle is also presented.

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TL;DR: In this paper, a two-dimensional semi-analytical finite element (SAFE) method is applied to provide a modal study of the elastic waves that are guided by the welded joint in a plate.

Abstract: The inspection of large areas of complex structures is a growing interest for industry. An experimental observation on a large welded plate found that the weld can concentrate and guide the energy of a guided wave travelling along the direction of the weld. This is attractive for non-destructive evaluation (NDE) since it offers the potential to quickly inspect for defects such as cracking or corrosion along long lengths of welds. In this paper, a two-dimensional semi-analytical finite-element (SAFE) method is applied to provide a modal study of the elastic waves that are guided by the welded joint in a plate. This brings understanding to the compression wave that was previously observed in the experiment. However, during the study, a shear weld-guided mode, which is non-leaky and almost non-dispersive, has also been discovered. Its characteristics are particularly attractive for NDE, so this is a significant new finding. The properties for both the compression and the shear mode are discussed and compared, and the physical reason for the energy trapping phenomena is then explained. Experiments have been undertaken to validate the existence of the shear weld-guided mode and the accuracy of the FE model, showing very good results.

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TL;DR: In this paper, a model is proposed to predict rebound kinematics of the spheres during oblique impacts, including the tangential coefficient of restitution, the rebound velocity at the contact patch and the rebound rotational speed of the sphere during impact.

Abstract: Results of finite-element analysis (FEA) of oblique impacts of elastic and elastic, perfectly plastic spheres with an elastic flat substrate are presented. The FEA results are in excellent agreement with published data available in the literature. A simple model is proposed to predict rebound kinematics of the spheres during oblique impacts. In this model, the oblique impacts are classified into two regimes: (i) persistent sliding impact, in which sliding occurs throughout the impact, the effect of tangential (elastic or plastic) deformation is insignificant and the model reproduces the well-established theoretical solutions based on rigid body dynamics for predicting the rebound kinematics and (ii) non-persistent sliding impact, in which sliding does not occur throughout the impact duration and the rebound kinematics depends upon both Poisson's ratio and the normal coefficient of restitution (i.e. the yield stress of the materials). For non-persistent sliding impacts, the variation of impulse ratio with impact angle is approximated using an empirical equation with four parameters. These parameters are sensitive to the values of Poisson's ratio and the normal coefficient of restitution, but can be obtained by fitting numerical data. Consequently, a complete set of solutions is obtained for the rebound kinematics, including the tangential coefficient of restitution, the rebound velocity at the contact patch and the rebound rotational speed of the sphere during oblique impacts. The accuracy and robustness of this model is demonstrated by comparisons with FEA results and data published in the literature. The model is capable of predicting complete rebound behaviour of spheres for both elastic and elastoplastic oblique impacts.

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TL;DR: In this article, the authors show that the efficient excitation of different modes would require different numbers of structural periods of the chiral STF, while all other modes exist only above some minimum value of the structural period, the minimum value being different for each mode.

Abstract: The solution of a dispersion equation indicates the theoretical existence of multiple modes of surface plasmon polariton wave propagation at the planar interface of a metal and a chiral sculptured thin film (STF). One mode appears to occur over a wide range of the structural period of the chiral STF, while all other modes exist only above some minimum value of the structural period, the minimum value being different for each mode. In order to excite the different modes, the interface can be incorporated in the commonplace Kretschmann configuration, for which our calculations show that the efficient excitation of different modes would require different numbers of structural periods of the chiral STF.

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TL;DR: In this paper, the second law of thermodynamics is used to explain the evolution of space-time as a causal chain of events in a non-Euclidean energy landscape.

Abstract: The concept of time is examined using the second law of thermodynamics that was recently formulated as an equation of motion. According to the statistical notion of increasing entropy, flows of energy diminish differences between energy densities that form space. The flow of energy is identified with the flow of time. The non-Euclidean energy landscape, i.e. the curved space–time, is in evolution when energy is flowing down along gradients and levelling the density differences. The flows along the steepest descents, i.e. geodesics are obtained from the principle of least action for mechanics, electrodynamics and quantum mechanics. The arrow of time, associated with the expansion of the Universe, identifies with grand dispersal of energy when high-energy densities transform by various mechanisms to lower densities in energy and eventually to ever-diluting electromagnetic radiation. Likewise, time in a quantum system takes an increment forwards in the detection-associated dissipative transformation when the stationary-state system begins to evolve pictured as the wave function collapse. The energy dispersal is understood to underlie causality so that an energy gradient is a cause and the resulting energy flow is an effect. The account on causality by the concepts of physics does not imply determinism; on the contrary, evolution of space–time as a causal chain of events is non-deterministic.

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TL;DR: This work constructs near-optimal quadratures for the sphere that are invariant under the icosahedral rotation group and discretizes the reproducing kernel on a rotationally invariant subspace to construct an analogue of Lagrange interpolation on the sphere, providing a much better localization than spherical harmonic expansions.

Abstract: We construct near-optimal quadratures for the sphere that are invariant under the icosahedral rotation group. These quadratures integrate all ( N +1) 2 linearly independent functions in a rotationally invariant subspace of maximal order and degree N . The nodes of these quadratures are nearly uniformly distributed, and the number of nodes is only marginally more than the optimal ( N +1) 2 /3 nodes. Using these quadratures, we discretize the reproducing kernel on a rotationally invariant subspace to construct an analogue of Lagrange interpolation on the sphere. This representation uses function values at the quadrature nodes. In addition, the representation yields an expansion that uses a single function centred and mostly concentrated at nodes of the quadrature, thus providing a much better localization than spherical harmonic expansions. We show that this representation may be localized even further. We also describe two algorithms of complexity for using these grids and representations. Finally, we note that our approach is also applicable to other discrete rotation groups.

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TL;DR: In this article, the spiral growth mechanism of Burton, Cabrera and Frank is used to predict the steady-state shape of organic molecular crystals grown from solution, and the predicted crystal shapes are in good agreement with experiment.

Abstract: We present a systematic modelling methodology using the spiral growth mechanism of Burton, Cabrera and Frank to predict the steady-state shape of organic molecular crystals grown from solution. This methodology has been developed to eliminate the need for special modifications for each new crystal system studied. Therefore, the mechanisms and choices for spiral shapes, edges and evolution are mathematically determined as governed by the underlying solid-state chemistry and physics. The power of the approach is demonstrated for several crystal systems: naphthalene grown from both ethanol and cyclohexane; anthracene grown from 2-propanol; and glycine grown from water. The predicted crystal shapes are in good agreement with experiment.

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TL;DR: In this article, a differential inequality technique is used to determine a lower bound on the blowup time for solutions to the heat equation subject to a nonlinear boundary condition when blow-up of the solution does occur.

Abstract: A differential inequality technique is used to determine a lower bound on the blow-up time for solutions to the heat equation subject to a nonlinear boundary condition when blow-up of the solution does occur. In addition, a sufficient condition which implies that blow-up does occur is determined.