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Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences RSS feed -- current issue1471-2946June, 2017Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences1364-5021<![CDATA[Is a time symmetric interpretation of quantum theory possible without retrocausality?]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2202/20160607?rss=1
Huw Price has proposed an argument that suggests a time symmetric ontology for quantum theory must necessarily be retrocausal, i.e. it must involve influences that travel backwards in time. One of Price's assumptions is that the quantum state is a state of reality. However, one of the reasons for exploring retrocausality is that it offers the potential for evading the consequences of no-go theorems, including recent proofs of the reality of the quantum state. Here, we show that this assumption can be replaced by a different assumption, called -mediation, that plausibly holds independently of the status of the quantum state. We also reformulate the other assumptions behind the argument to place them in a more general framework and pin down the notion of time symmetry involved more precisely. We show that our assumptions imply a timelike analogue of Bell's local causality criterion and, in doing so, give a new interpretation of timelike violations of Bell inequalities. Namely, they show the impossibility of a (non-retrocausal) time symmetric ontology.
]]>2017-06-21T00:08:19-07:00info:doi/10.1098/rspa.2016.0607hwp:master-id:royprsa;rspa.2016.06072017-06-21Research articles47322022016060720160607<![CDATA[Liquid toroidal drop under uniform electric field]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2202/20160633?rss=1
The problem of a stationary liquid toroidal drop freely suspended in another fluid and subjected to an electric field uniform at infinity is addressed analytically. Taylor’s discriminating function implies that, when the phases have equal viscosities and are assumed to be slightly conducting (leaky dielectrics), a spherical drop is stationary when Q=(2R^{2}+3R+2)/(7R^{2}), where R and Q are ratios of the phases’ electric conductivities and dielectric constants, respectively. This condition holds for any electric capillary number, Ca_{E}, that defines the ratio of electric stress to surface tension. Pairam and Fernández-Nieves showed experimentally that, in the absence of external forces (Ca_{E}=0), a toroidal drop shrinks towards its centre, and, consequently, the drop can be stationary only for some Ca_{E}>0. This work finds Q and Ca_{E} such that, under the presence of an electric field and with equal viscosities of the phases, a toroidal drop having major radius and volume 4/3 is qualitatively stationary—the normal velocity of the drop’s interface is minute and the interface coincides visually with a streamline. The found Q and Ca_{E} depend on R and , and for large , e.g. ≥3, they have simple approximations: Q~(R^{2}+R+1)/(3R^{2}) and CaE~33/2 (6 ln +2 ln[96]–9)/(12 ln +4 ln[96]–17) (R+1)2/(R–1)2.
]]>2017-06-21T03:05:36-07:00info:doi/10.1098/rspa.2016.0633hwp:master-id:royprsa;rspa.2016.06332017-06-21Research articles47322022016063320160633<![CDATA[Nonlinear model identification and spectral submanifolds for multi-degree-of-freedom mechanical vibrations]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2202/20160759?rss=1
In a nonlinear oscillatory system, spectral submanifolds (SSMs) are the smoothest invariant manifolds tangent to linear modal subspaces of an equilibrium. Amplitude–frequency plots of the dynamics on SSMs provide the classic backbone curves sought in experimental nonlinear model identification. We develop here, a methodology to compute analytically both the shape of SSMs and their corresponding backbone curves from a data-assimilating model fitted to experimental vibration signals. This model identification utilizes Taken’s delay-embedding theorem, as well as a least square fit to the Taylor expansion of the sampling map associated with that embedding. The SSMs are then constructed for the sampling map using the parametrization method for invariant manifolds, which assumes that the manifold is an embedding of, rather than a graph over, a spectral subspace. Using examples of both synthetic and real experimental data, we demonstrate that this approach reproduces backbone curves with high accuracy.
]]>2017-06-14T00:07:50-07:00info:doi/10.1098/rspa.2016.0759hwp:master-id:royprsa;rspa.2016.07592017-06-14Research articles47322022016075920160759<![CDATA[On the regularization of impact without collision: the Painleve paradox and compliance]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2202/20160773?rss=1
We consider the problem of a rigid body, subject to a unilateral constraint, in the presence of Coulomb friction. We regularize the problem by assuming compliance (with both stiffness and damping) at the point of contact, for a general class of normal reaction forces. Using a rigorous mathematical approach, we recover impact without collision (IWC) in both the inconsistent and the indeterminate Painlevé paradoxes, in the latter case giving an exact formula for conditions that separate IWC and lift-off. We solve the problem for arbitrary values of the compliance damping and give explicit asymptotic expressions in the limiting cases of small and large damping, all for a large class of rigid bodies.
]]>2017-06-14T00:07:50-07:00info:doi/10.1098/rspa.2016.0773hwp:master-id:royprsa;rspa.2016.07732017-06-14Research articles47322022016077320160773<![CDATA[Energy-based analysis of biomolecular pathways]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2202/20160825?rss=1
Decomposition of biomolecular reaction networks into pathways is a powerful approach to the analysis of metabolic and signalling networks. Current approaches based on analysis of the stoichiometric matrix reveal information about steady-state mass flows (reaction rates) through the network. In this work, we show how pathway analysis of biomolecular networks can be extended using an energy-based approach to provide information about energy flows through the network. This energy-based approach is developed using the engineering-inspired bond graph methodology to represent biomolecular reaction networks. The approach is introduced using glycolysis as an exemplar; and is then applied to analyse the efficiency of free energy transduction in a biomolecular cycle model of a transporter protein [sodium-glucose transport protein 1 (SGLT1)]. The overall aim of our work is to present a framework for modelling and analysis of biomolecular reactions and processes which considers energy flows and losses as well as mass transport.
]]>2017-06-21T00:08:19-07:00info:doi/10.1098/rspa.2016.0825hwp:master-id:royprsa;rspa.2016.08252017-06-21Research articles47322022016082520160825<![CDATA[Knotted fields and explicit fibrations for lemniscate knots]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2202/20160829?rss=1
We give an explicit construction of complex maps whose nodal lines have the form of lemniscate knots. We review the properties of lemniscate knots, defined as closures of braids where all strands follow the same transverse (1, ) Lissajous figure, and are therefore a subfamily of spiral knots generalizing the torus knots. We then prove that such maps exist and are in fact fibrations with appropriate choices of parameters. We describe how this may be useful in physics for creating knotted fields, in quantum mechanics, optics and generalizing to rational maps with application to the Skyrme–Faddeev model. We also prove how this construction extends to maps with weakly isolated singularities.
]]>2017-06-07T01:27:08-07:00info:doi/10.1098/rspa.2016.0829hwp:master-id:royprsa;rspa.2016.08292017-06-07Research articles47322022016082920160829<![CDATA[The cutting of metals via plastic buckling]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2202/20160863?rss=1
The cutting of metals has long been described as occurring by laminar plastic flow. Here we show that for metals with large strain-hardening capacity, laminar flow mode is unstable and cutting instead occurs by plastic buckling of a thin surface layer. High speed in situ imaging confirms that the buckling results in a small bump on the surface which then evolves into a fold of large amplitude by rotation and stretching. The repeated occurrence of buckling and folding manifests itself at the mesoscopic scale as a new flow mode with significant vortex-like components—sinuous flow. The buckling model is validated by phenomenological observations of flow at the continuum level and microstructural characteristics of grain deformation and measurements of the folding. In addition to predicting the conditions for surface buckling, the model suggests various geometric flow control strategies that can be effectively implemented to promote laminar flow, and suppress sinuous flow in cutting, with implications for industrial manufacturing processes. The observations impinge on the foundations of metal cutting by pointing to the key role of stability of laminar flow in determining the mechanism of material removal, and the need to re-examine long-held notions of large strain deformation at surfaces.
]]>2017-06-07T00:07:57-07:00info:doi/10.1098/rspa.2016.0863hwp:master-id:royprsa;rspa.2016.08632017-06-07Research articles47322022016086320160863<![CDATA[Regularly configured structures with polygonal prisms for three-dimensional auxetic behaviour]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2202/20160926?rss=1
We report here structures, constructed with regular polygonal prisms, that exhibit negative Poisson’s ratios. In particular, we show how we can construct such a structure with regular n-gonal prism-shaped unit cells that are again built with regular n-gonal component prisms. First, we show that the only three possible values for n are 3, 4 and 6 and then discuss how we construct the unit cell again with regular n-gonal component prisms. Then, we derive Poisson’s ratio formula for each of the three structures and show, by analysis and numerical verification, that the structures possess negative Poisson’s ratio under certain geometric conditions.
]]>2017-06-14T00:07:51-07:00info:doi/10.1098/rspa.2016.0926hwp:master-id:royprsa;rspa.2016.09262017-06-14Research articles47322022016092620160926<![CDATA[Nonlinear evolution of a thin anodic film]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2202/20160930?rss=1
The formation of pores in anodic aluminium oxide films is treated with a model equation. The model treats the oxide layer as a thin viscous liquid in two dimensions. Surface tension on the top boundary, electrostriction due to the external electric field and mass flow through the bottom boundary due to oxide formation are all included. Viscous flow is treated with the creeping flow assumption. The model equation is solved numerically using a Fourier spectral method in space and Adams–Bashforth/Adams–Moulton methods in time. Initial conditions include sinusoidal shapes as well as random shapes. The results show that pores form at the trough of the initial sinusoidal shape. Random shapes get smoothed before forming pore structures with spacing different than predicted by linear theory.
]]>2017-06-07T00:07:57-07:00info:doi/10.1098/rspa.2016.0930hwp:master-id:royprsa;rspa.2016.09302017-06-07Research articles47322022016093020160930<![CDATA[Wave reflection and transmission in multiply stented blood vessels]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2202/20170015?rss=1
Closed circulatory systems display an exquisite balance between vascular elasticity and viscous fluid effects, to induce pulse-smoothing and avoid resonance during the cardiac cycle. Stents in the arterial tree alter this balance through stiffening and because a periodic structure is introduced, capable of interacting with the fluid in a complex way. While the former feature has been investigated, the latter received no attention so far. But periodic structures are the building blocks of metamaterials, known for their ‘non-natural’ behaviour. Thus, the investigation of a stent's periodic microstructure dynamical interactions is crucial to assess possible pathological responses. A one-dimensional fluid–structure interaction model, simple enough to allow an analytical solution for situations of interest involving one or two interacting stents, is introduced. It is determined: (i) whether or not frequency bands exist in which reflected blood pulses are highly increased and (ii) if these bands are close to the characteristic frequencies of arteries and finally, (iii) if the internal structure of the stent can sensibly affect arterial blood dynamics. It is shown that, while the periodic structure of an isolated stent can induce anomalous reflection only in pathological conditions, the presence of two interacting stents is more critical, and high reflection can occur at frequencies not far from the physiological values.
]]>2017-06-07T01:27:08-07:00info:doi/10.1098/rspa.2017.0015hwp:master-id:royprsa;rspa.2017.00152017-06-07Research articles47322022017001520170015<![CDATA[Ultrasonic defect characterization using parametric-manifold mapping]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2202/20170056?rss=1
The aim of ultrasonic non-destructive evaluation includes the detection and characterization of defects, and an understanding of the nature of defects is essential for the assessment of structural integrity in safety critical systems. In general, the defect characterization challenge involves an estimation of defect parameters from measured data. In this paper, we explore the extent to which defects can be characterized by their ultrasonic scattering behaviour. Given a number of ultrasonic measurements, we show that characterization information can be extracted by projecting the measurement onto a parametric manifold in principal component space. We show that this manifold represents the entirety of the characterization information available from far-field harmonic ultrasound. We seek to understand the nature of this information and hence provide definitive statements on the defect characterization performance that is, in principle, extractable from typical measurement scenarios. In experiments, the characterization problem of surface-breaking cracks and the more general problem of elliptical voids are studied, and a good agreement is achieved between the actual parameter values and the characterization results. The nature of the parametric manifold enables us to explain and quantify why some defects are relatively easy to characterize, whereas others are inherently challenging.
]]>2017-06-07T01:27:08-07:00info:doi/10.1098/rspa.2017.0056hwp:master-id:royprsa;rspa.2017.00562017-06-07Research articles47322022017005620170056<![CDATA[The rolling suitcase instability: a coupling between translation and rotation]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2202/20170076?rss=1
A two-wheel suitcase or trolley can exhibit undamped rocking oscillations from one wheel to the other when pulled fast enough. We study this instability both experimentally—with a toy model of a suitcase rolling on a treadmill—and theoretically. The suitcase oscillates only if a finite perturbation is applied. This is because intrinsic dissipation occurs when the supporting wheel switches. When unstable, the suitcase either increasingly rocks until overturning or reaches a stable limit cycle. The friction force at the rolling wheels constrains wheels to roll without slipping. This constraint imposes a coupling between the translational motion and the three-dimensional rotational motion of the suitcase that drives the rocking instability. The same behaviours are observed in the experiments and in the simulations. The asymptotic scaling laws we observe in the simulations are explained by means of a simplified model where the coupling force is explicit.
]]>2017-06-21T00:08:19-07:00info:doi/10.1098/rspa.2017.0076hwp:master-id:royprsa;rspa.2017.00762017-06-21Research articles47322022017007620170076<![CDATA[Addressing the discrepancy of finding the equilibrium melting point of silicon using molecular dynamics simulations]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2202/20170084?rss=1
We performed molecular dynamics simulations to study the equilibrium melting point of silicon using (i) the solid–liquid coexistence method and (ii) the Gibbs free energy technique, and compared our novel results with the previously published results obtained from the Monte Carlo (MC) void-nucleated melting method based on the Tersoff-ARK interatomic potential (Agrawal et al. Phys. Rev. B72, 125206. (doi:10.1103/PhysRevB.72.125206)). Considerable discrepancy was observed (approx. 20%) between the former two methods and the MC void-nucleated melting result, leading us to question the applicability of the empirical MC void-nucleated melting method to study a wide range of atomic and molecular systems. A wider impact of the study is that it highlights the bottleneck of the Tersoff-ARK potential in correctly estimating the melting point of silicon.
]]>2017-06-07T00:07:57-07:00info:doi/10.1098/rspa.2017.0084hwp:master-id:royprsa;rspa.2017.00842017-06-07Research articles47322022017008420170084<![CDATA[Analytical solutions for two-dimensional Stokes flow singularities in a no-slip wedge of arbitrary angle]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2202/20170134?rss=1
An analytical method to find the flow generated by the basic singularities of Stokes flow in a wedge of arbitrary angle is presented. Specifically, we solve a biharmonic equation for the stream function of the flow generated by a point stresslet singularity and satisfying no-slip boundary conditions on the two walls of the wedge. The method, which is readily adapted to any other singularity type, takes full account of any transcendental singularities arising at the corner of the wedge. The approach is also applicable to problems of plane strain/stress of an elastic solid where the biharmonic equation also governs the Airy stress function.
]]>2017-06-07T00:07:57-07:00info:doi/10.1098/rspa.2017.0134hwp:master-id:royprsa;rspa.2017.01342017-06-07Research articles47322022017013420170134<![CDATA[Extensional channel flow revisited: a dynamical systems perspective]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2202/20170151?rss=1
Extensional self-similar flows in a channel are explored numerically for arbitrary stretching–shrinking rates of the confining parallel walls. The present analysis embraces time integrations, and continuations of steady and periodic solutions unfolded in the parameter space. Previous studies focused on the analysis of branches of steady solutions for particular stretching–shrinking rates, although recent studies focused also on the dynamical aspects of the problems. We have adopted a dynamical systems perspective, analysing the instabilities and bifurcations the base state undergoes when increasing the Reynolds number. It has been found that the base state becomes unstable for small Reynolds numbers, and a transitional region including complex dynamics takes place at intermediate Reynolds numbers, depending on the wall acceleration values. The base flow instabilities are constitutive parts of different codimension-two bifurcations that control the dynamics in parameter space. For large Reynolds numbers, the restriction to self-similarity results in simple flows with no realistic behaviour, but the flows obtained in the transition region can be a valuable tool for the understanding of the dynamics of realistic Navier–Stokes solutions.
]]>2017-06-07T01:27:08-07:00info:doi/10.1098/rspa.2017.0151hwp:master-id:royprsa;rspa.2017.01512017-06-07Research articles47322022017015120170151<![CDATA[A free-boundary model of diffusive valley growth: theory and observation]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2202/20170159?rss=1
Valleys that form around a stream head often develop characteristic finger-like elevation contours. We study the processes involved in the formation of these valleys and introduce a theoretical model that indicates how shape may inform the underlying processes. We consider valley growth as the advance of a moving boundary travelling forward purely through linearly diffusive erosion, and we obtain a solution for the valley shape in three dimensions. Our solution compares well to the shape of slowly growing groundwater-fed valleys found in Bristol, Florida. Our results identify a new feature in the formation of groundwater-fed valleys: a spatially variable diffusivity that can be modelled by a fixed-height moving boundary.
]]>2017-06-21T00:08:19-07:00info:doi/10.1098/rspa.2017.0159hwp:master-id:royprsa;rspa.2017.01592017-06-21Research articles47322022017015920170159<![CDATA[Windowed Green function method for the Helmholtz equation in the presence of multiply layered media]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2202/20170161?rss=1
This paper presents a new methodology for the solution of problems of two- and three-dimensional acoustic scattering (and, in particular, two-dimensional electromagnetic scattering) by obstacles and defects in the presence of an arbitrary number of penetrable layers. Relying on the use of certain slow-rise windowing functions, the proposed windowed Green function approach efficiently evaluates oscillatory integrals over unbounded domains, with high accuracy, without recourse to the highly expensive Sommerfeld integrals that have typically been used to account for the effect of underlying planar multilayer structures. The proposed methodology, whose theoretical basis was presented in the recent contribution (Bruno et al. 2016 SIAM J. Appl. Math.76, 1871–1898. (doi:10.1137/15M1033782)), is fast, accurate, flexible and easy to implement. Our numerical experiments demonstrate that the numerical errors resulting from the proposed approach decrease faster than any negative power of the window size. In a number of examples considered in this paper, the proposed method is up to thousands of times faster, for a given accuracy, than corresponding methods based on the use of Sommerfeld integrals.
]]>2017-06-14T00:07:51-07:00info:doi/10.1098/rspa.2017.0161hwp:master-id:royprsa;rspa.2017.01612017-06-14Research articles47322022017016120170161<![CDATA[New conformal mapping for adaptive resolving of the complex singularities of Stokes wave]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2202/20170198?rss=1
A new highly efficient method is developed for computation of travelling periodic waves (Stokes waves) on the free surface of deep water. A convergence of numerical approximation is determined by the complex singularities above the free surface for the analytical continuation of the travelling wave into the complex plane. An auxiliary conformal mapping is introduced which moves singularities away from the free surface thus dramatically speeding up numerical convergence by adapting the numerical grid for resolving singularities while being consistent with the fluid dynamics. The efficiency of that conformal mapping is demonstrated for the Stokes wave approaching the limiting Stokes wave (the wave of the greatest height) which significantly expands the family of numerically accessible solutions. It allows us to provide a detailed study of the oscillatory approach of these solutions to the limiting wave. Generalizations of the conformal mapping to resolve multiple singularities are also introduced.
]]>2017-06-21T00:08:19-07:00info:doi/10.1098/rspa.2017.0198hwp:master-id:royprsa;rspa.2017.01982017-06-21Research articles47322022017019820170198<![CDATA[Correction to 'Homogenized boundary conditions and resonance effects in Faraday cages]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2202/20170331?rss=1
2017-06-14T00:07:51-07:00info:doi/10.1098/rspa.2017.0331hwp:master-id:royprsa;rspa.2017.03312017-06-14Corrections47322022017033120170331