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Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences RSS feed -- current issue1471-2946January, 2017Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences1364-5021<![CDATA[Why do students quit school? Implications from a dynamical modelling study]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2197/20160204?rss=1
In 2012, more than three million students dropped out from high school. At this pace, we will have more than 30 million Americans without a high school degree by 2022 and relatively high dropout rates among Hispanic and African American students. We have developed and analysed a data-driven mathematical model that includes multiple interacting mechanisms and estimates of parameters using data from a specifically designed survey applied to a certain group of students of a high school in Chicago to understand the dynamics of dropouts. Our analysis suggests students' academic achievement is directly related to the level of parental involvement more than any other factors in our study. However, if the negative peer influence (leading to lower academic grades) increases beyond a critical value, the effect of parental involvement on the dynamics of dropouts becomes negligible.
]]>2017-01-18T00:05:26-08:00info:doi/10.1098/rspa.2016.0204hwp:master-id:royprsa;rspa.2016.02042017-01-18Research articles47321972016020420160204<![CDATA[Learning partial differential equations via data discovery and sparse optimization]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2197/20160446?rss=1
We investigate the problem of learning an evolution equation directly from some given data. This work develops a learning algorithm to identify the terms in the underlying partial differential equations and to approximate the coefficients of the terms only using data. The algorithm uses sparse optimization in order to perform feature selection and parameter estimation. The features are data driven in the sense that they are constructed using nonlinear algebraic equations on the spatial derivatives of the data. Several numerical experiments show the proposed method's robustness to data noise and size, its ability to capture the true features of the data, and its capability of performing additional analytics. Examples include shock equations, pattern formation, fluid flow and turbulence, and oscillatory convection.
]]>2017-01-18T03:28:12-08:00info:doi/10.1098/rspa.2016.0446hwp:master-id:royprsa;rspa.2016.04462017-01-18Research articles47321972016044620160446<![CDATA[Integrability of systems of two second-order ordinary differential equations admitting four-dimensional Lie algebras]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2197/20160461?rss=1
We suggest an algorithm for integrating systems of two second-order ordinary differential equations with four symmetries. In particular, if the admitted transformation group has two second-order differential invariants, the corresponding system can be integrated by quadratures using invariant representation and the operator of invariant differentiation. Otherwise, the systems reduce to partially uncoupled forms and can also be integrated by quadratures.
]]>2017-01-11T00:05:17-08:00info:doi/10.1098/rspa.2016.0461hwp:master-id:royprsa;rspa.2016.04612017-01-11Research articles47321972016046120160461<![CDATA[On compression and damage evolution in two thermoplastics]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2197/20160495?rss=1
The well-known Taylor cylinder impact test, which follows the impact of a flat-ended cylindrical rod onto a rigid stationary anvil, is conducted over a range of impact speeds for two polymers, polytetrafluoroethylene (PTFE) and polyetheretherketone (PEEK). In previous work, experiments and a model were developed to capture the deformation behaviour of the cylinder after impact. These works showed a region in which spatial and temporal variation of both longitudinal and radial deformation provided evidence of changes in phase within the material. In this further series of experiments, this region is imaged in a range of impacted targets at the Diamond synchrotron. Further techniques were fielded to resolve compressed regions within the recovered polymer cylinders that showed a fracture zone in the impact region. The combination of macroscopic high-speed photography and three-dimensional X-ray imaging has identified the development of failure with these polymers and shown that there is no abrupt transition in behaviours but rather a continuous range of responses to competing operating mechanisms. The behaviours noted in PEEK in these polymers show critical gaps in understanding of polymer high strain-rate response.
]]>2017-01-18T00:05:27-08:00info:doi/10.1098/rspa.2016.0495hwp:master-id:royprsa;rspa.2016.04952017-01-18Research articles47321972016049520160495<![CDATA[Identification of internal properties of fibres and micro-swimmers]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2197/20160517?rss=1
In this paper, we address the identifiability of constitutive parameters of passive or active micro-swimmers. We first present a general framework for describing fibres or micro-swimmers using a bead-model description. Using a kinematic constraint formulation to describe fibres, flagellum or cilia, we find explicit linear relationship between elastic constitutive parameters and generalized velocities from computing contact forces. This linear formulation then permits one to address explicitly identifiability conditions and solve for parameter identification. We show that both active forcing and passive parameters are both identifiable independently but not simultaneously. We also provide unbiased estimators for generalized elastic parameters in the presence of Langevin-like forcing with Gaussian noise using a Bayesian approach. These theoretical results are illustrated in various configurations showing the efficiency of the proposed approach for direct parameter identification. The convergence of the proposed estimators is successfully tested numerically.
]]>2017-01-18T03:28:12-08:00info:doi/10.1098/rspa.2016.0517hwp:master-id:royprsa;rspa.2016.05172017-01-18Research articles47321972016051720160517<![CDATA[Beyond linear fields: the Lie-Taylor expansion]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2197/20160525?rss=1
The work extends the linear fields’ solution of compressible nonlinear magnetohydrodynamics (MHD) to the case where the magnetic field depends on superlinear powers of position vector, usually, but not always, expressed in Cartesian components. Implications of the resulting Lie–Taylor series expansion for physical applicability of the Dolzhansky–Kirchhoff (D–K) equations are found to be positive. It is demonstrated how resistivity may be included in the D–K model. Arguments are put forward that the D–K equations may be regarded as illustrating properties of nonlinear MHD in the same sense that the Lorenz equations inform about the onset of convective turbulence. It is suggested that the Lie–Taylor series approach may lead to valuable insights into other fluid models.
]]>2017-01-04T00:05:20-08:00info:doi/10.1098/rspa.2016.0525hwp:master-id:royprsa;rspa.2016.05252017-01-04Research articles47321972016052520160525<![CDATA[The problem of missing terms in term by term integration involving divergent integrals]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2197/20160567?rss=1
Term by term integration may lead to divergent integrals, and naive evaluation of them by means of, say, analytic continuation or by regularization or by the finite part integral may lead to missing terms. Here, under certain analyticity conditions, the problem of missing terms for the incomplete Stieltjes transform, 0af(x)(+x)–1 dx, and the Stieltjes transform itself, 0f(x)(+x)–1 dx, is resolved by lifting the integration in the complex plane. It is shown that the missing terms arise from the singularities of the complex-valued function f(z)(+z)^{–1}, with the divergent integrals arising from term by term integration interpreted as finite part integrals.
]]>2017-01-18T03:28:12-08:00info:doi/10.1098/rspa.2016.0567hwp:master-id:royprsa;rspa.2016.05672017-01-18Research articles47321972016056720160567<![CDATA[The onset of thermalization in finite-dimensional equations of hydrodynamics: insights from the Burgers equation]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2197/20160585?rss=1
Solutions to finite-dimensional (all spatial Fourier modes set to zero beyond a finite wavenumber KG), inviscid equations of hydrodynamics at long times are known to be at variance with those obtained for the original infinite dimensional partial differential equations or their viscous counterparts. Surprisingly, the solutions to such Galerkin-truncated equations develop sharp localized structures, called tygers (Ray et al. 2011 Phys. Rev. E84, 016301 (doi:10.1103/PhysRevE.84.016301)), which eventually lead to completely thermalized states associated with an equipartition energy spectrum. We now obtain, by using the analytically tractable Burgers equation, precise estimates, theoretically and via direct numerical simulations, of the time c at which thermalization is triggered and show that c~KG, with =–49. Our results have several implications, including for the analyticity strip method, to numerically obtain evidence for or against blow-ups of the three-dimensional incompressible Euler equations.
]]>2017-01-18T00:05:27-08:00info:doi/10.1098/rspa.2016.0585hwp:master-id:royprsa;rspa.2016.05852017-01-18Research articles47321972016058520160585<![CDATA[An investigation into inflection-point instability in the entrance region of a pulsating pipe flow]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2197/20160590?rss=1
This paper investigates the inflection-point instability that governs the flow disturbance initiated in the entrance region of a pulsating pipe flow. Under such a flow condition, the flow instability grows within a certain phase region in a pulsating cycle, during which the inflection point in the unsteady mean flow lifts away from the viscous effect-dominated region known as the Stokes layer. The characteristic frequency of the instability is found to be in agreement with that predicted by the mixing-layer model. In comparison with those cases not falling in this category, it is further verified that the flow phenomenon will take place only if the inflection point lifts away sufficiently from the Stokes layer.
]]>2017-01-11T00:05:17-08:00info:doi/10.1098/rspa.2016.0590hwp:master-id:royprsa;rspa.2016.05902017-01-11Research articles47321972016059020160590<![CDATA[A study on the temperature field of a bronze-bonded diamond wheel dressed using a laser/ultrasonic vibration combined method]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2197/20160597?rss=1
To solve problems in dressings for metal-bonded super-hard abrasive wheels, such as low efficiency and rapid wear, this article introduces a laser-assisted ultrasonic vibration dressing technique. Firstly, finite-element simulations were conducted on the dressing process of a bronze-bonded diamond wheel, and the wheel's temperature field distributions under different laser parameters were simulated. By analysing the simulation results of temperature fields and the melting point of the bronze bond, the laser parameters for laser-assisted ultrasonic vibration dressing tests were optimized, and then actual tests were carried out on the bronze-bonded diamond wheel. Results showed that, with appropriate technology parameters, the laser-assisted ultrasonic vibration dressing technique achieved desirable dressing results; specifically, the dressing force was low, the abrasive particles had high protrusions and the wheel had a large chip space and favourable surface topography.
]]>2017-01-18T00:05:26-08:00info:doi/10.1098/rspa.2016.0597hwp:master-id:royprsa;rspa.2016.05972017-01-18Research articles47321972016059720160597<![CDATA[Entropy in sound and vibration: towards a new paradigm]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2197/20160602?rss=1
This paper describes a discussion on the method and the status of a statistical theory of sound and vibration, called statistical energy analysis (SEA). SEA is a simple theory of sound and vibration in elastic structures that applies when the vibrational energy is diffusely distributed. We show that SEA is a thermodynamical theory of sound and vibration, based on a law of exchange of energy analogous to the Clausius principle. We further investigate the notion of entropy in this context and discuss its meaning. We show that entropy is a measure of information lost in the passage from the classical theory of sound and vibration and SEA, its thermodynamical counterpart.
]]>2017-01-11T00:05:17-08:00info:doi/10.1098/rspa.2016.0602hwp:master-id:royprsa;rspa.2016.06022017-01-11Research articles47321972016060220160602<![CDATA[Reconfiguration of a smart surface using heteroclinic connections]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2197/20160614?rss=1
A reconfigurable smart surface with multiple equilibria is presented, modelled using discrete point masses and linear springs with geometric nonlinearity. An energy-efficient reconfiguration scheme is then investigated to connect equal-energy unstable (but actively controlled) equilibria. In principle, zero net energy input is required to transition the surface between these unstable states, compared to transitions between stable equilibria across a potential barrier. These transitions between equal-energy unstable states, therefore, form heteroclinic connections in the phase space of the problem. Moreover, the smart surface model developed can be considered as a unit module for a range of applications, including modules which can aggregate together to form larger distributed smart surface systems.
]]>2017-01-11T00:05:17-08:00info:doi/10.1098/rspa.2016.0614hwp:master-id:royprsa;rspa.2016.06142017-01-11Research articles47321972016061420160614<![CDATA[Domain structure of ultrathin ferromagnetic elements in the presence of Dzyaloshinskii-Moriya interaction]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2197/20160666?rss=1
Recent advances in nanofabrication make it possible to produce multilayer nanostructures composed of ultrathin film materials with thickness down to a few monolayers of atoms and lateral extent of several tens of nanometers. At these scales, ferromagnetic materials begin to exhibit unusual properties, such as perpendicular magnetocrystalline anisotropy and antisymmetric exchange, also referred to as Dzyaloshinskii–Moriya interaction (DMI), because of the increased importance of interfacial effects. The presence of surface DMI has been demonstrated to fundamentally alter the structure of domain walls. Here we use the micromagnetic modelling framework to analyse the existence and structure of chiral domain walls, viewed as minimizers of a suitable micromagnetic energy functional. We explicitly construct the minimizers in the one-dimensional setting, both for the interior and edge walls, for a broad range of parameters. We then use the methods of -convergence to analyse the asymptotics of the two-dimensional magnetization patterns in samples of large spatial extent in the presence of weak applied magnetic fields.
]]>2017-01-11T00:05:17-08:00info:doi/10.1098/rspa.2016.0666hwp:master-id:royprsa;rspa.2016.06662017-01-11Research articles47321972016066620160666<![CDATA[The theory on 'gravity-driven horizontal locomotion is flawed; a commentary on 'Gravity-driven horizontal locomotion: theory and experiment by Kanstad & Kononoff (2015)]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2197/20160683?rss=1
In a recent paper, Kanstad & Kononoff (Proc. R. Soc. A471, 20150287. (doi:10.1098/rspa.2015.0287)) presented a theoretical analysis of the mechanical energetics of a particular style of human walking and running. According to their analysis, the force of gravity provides energy when this style of horizontal walking/running is adopted. Furthermore, Kanstad & Kononoff suggested that uphill walking at zero energy cost is possible when the suggested style of walking is adopted. In this commentary, we argue that these claims violate the basic laws of thermodynamics, and are based on erroneous application of the basic laws of classical mechanics.
]]>2017-01-18T00:05:27-08:00info:doi/10.1098/rspa.2016.0683hwp:master-id:royprsa;rspa.2016.06832017-01-18Comments and invited replies47321972016068320160683<![CDATA['Full fusion is not ineluctable during vesicular exocytosis of neurotransmitters by endocrine cells]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2197/20160684?rss=1
Vesicular exocytosis is an essential and ubiquitous process in neurons and endocrine cells by which neurotransmitters are released in synaptic clefts or extracellular fluids. It involves the fusion of a vesicle loaded with chemical messengers with the cell membrane through a nanometric fusion pore. In endocrine cells, unless it closes after some flickering (‘Kiss-and-Run’ events), this initial pore is supposed to expand exponentially, leading to a full integration of the vesicle membrane into the cell membrane—a stage called ‘full fusion’. We report here a compact analytical formulation that allows precise measurements of the fusion pore expansion extent and rate to be extracted from individual amperometric spike time courses. These data definitively establish that, during release of catecholamines, fusion pores enlarge at most to approximately one-fifth of the radius of their parent vesicle, hence ruling out the ineluctability of ‘full fusion’.
]]>2017-01-11T00:05:17-08:00info:doi/10.1098/rspa.2016.0684hwp:master-id:royprsa;rspa.2016.06842017-01-11Special feature47321972016068420160684<![CDATA[Whitham modulation theory for the Ostrovsky equation]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2197/20160709?rss=1
This paper derives the Whitham modulation equations for the Ostrovsky equation. The equations are then used to analyse localized cnoidal wavepacket solutions of the Ostrovsky equation in the weak rotation limit. The analysis is split into two main parameter regimes: the Ostrovsky equation with normal dispersion relevant to typical oceanic parameters and the Ostrovsky equation with anomalous dispersion relevant to strongly sheared oceanic flows and other physical systems. For anomalous dispersion a new steady, symmetric cnoidal wavepacket solution is presented. The new wavepacket can be represented as a solution of the modulation equations and an analytical solution for the outer solution of the wavepacket is given. For normal dispersion the modulation equations are used to describe the unsteady finite-amplitude wavepacket solutions produced from the rotation-induced decay of a Korteweg–de Vries solitary wave. Again, an analytical solution for the outer solution can be given. The centre of the wavepacket closely approximates a train of solitary waves with the results suggesting that the unsteady wavepacket is a localized, modulated cnoidal wavetrain. The formation of wavepackets from solitary wave initial conditions is considered, contrasting the rapid formation of the packets in anomalous dispersion with the slower formation of unsteady packets under normal dispersion.
]]>2017-01-18T03:28:12-08:00info:doi/10.1098/rspa.2016.0709hwp:master-id:royprsa;rspa.2016.07092017-01-18Research articles47321972016070920160709<![CDATA[Observation-based correction of dynamical models using thermostats]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2197/20160730?rss=1
Models used in simulation may give accurate short-term trajectories but distort long-term (statistical) properties. In this work, we augment a given approximate model with a control law (a ‘thermostat’) that gently perturbs the dynamical system to target a thermodynamic state consistent with a set of prescribed (possibly evolving) observations. As proof of concept, we provide an example involving a point vortex fluid model on the sphere, for which we show convergence of equilibrium quantities (in the stationary case) and the ability of the thermostat to dynamically track a transient state.
]]>2017-01-18T03:28:12-08:00info:doi/10.1098/rspa.2016.0730hwp:master-id:royprsa;rspa.2016.07302017-01-18Research articles47321972016073020160730<![CDATA[Finite-element modelling of elastic wave propagation and scattering within heterogeneous media]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2197/20160738?rss=1
The scattering treated here arises when elastic waves propagate within a heterogeneous medium defined by random spatial fluctuation of its elastic properties. Whereas classical analytical studies are based on lower-order scattering assumptions, numerical methods conversely present no such limitations by inherently incorporating multiple scattering. Until now, studies have typically been limited to two or one dimension, however, owing to computational constraints. This article seizes recent advances to realize a finite-element formulation that solves the three-dimensional elastodynamic scattering problem. The developed methodology enables the fundamental behaviour of scattering in terms of attenuation and dispersion to be studied. In particular, the example of elastic waves propagating within polycrystalline materials is adopted, using Voronoi tessellations to randomly generate representative models. The numerically observed scattering is compared against entirely independent but well-established analytical scattering theory. The quantitative agreement is found to be excellent across previously unvisited scattering regimes; it is believed that this is the first quantitative validation of its kind which provides significant support towards the existence of the transitional scattering regime and facilitates future deployment of numerical methods for these problems.
]]>2017-01-04T00:05:20-08:00info:doi/10.1098/rspa.2016.0738hwp:master-id:royprsa;rspa.2016.07382017-01-04Research articles47321972016073820160738<![CDATA[Topological origin and not purely antisymmetric wave functions of many-body states in the lowest Landau level]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2197/20160758?rss=1
In this paper, we recall the topological approach to quantum Hall effects. We note that, in the presence of a magnetic field, trajectories representing elements of the system’s braid group are of cyclotron orbit type. In two-dimensional spaces, this leads to the restriction of the full braid group, _{1}()—loopless generators (exchanges of M^{N} coordinates or classical particles) are unenforceable. As a result, the identification of a possible Hall-like state comes down to the identification of a possible subgroup of _{1}(). The latter follows from the connection between the one-dimensional unitary representation of the system’s braid group and particle statistics (unavoidable for any correlated state). In this work, we implement the topological approach to derive the lowest Landau-level pyramid of fillings. We point out that it contains all mysterious odd-denominator filling factors—like 411, 413 or 617—not trivial to explain within the standard picture. We also introduce, explicitly, cyclotron subgroup generators for all derived fractions. Preliminary results on wave functions, supported by several Monte Carlo calculations, are presented. It is worth emphasizing that not all proposed many-body functions are purely antisymmetric—they, however, transform in agreement with the scalar representations of the system’s braid group. The latter is enforced by standard quantization methods.
]]>2017-01-18T03:28:12-08:00info:doi/10.1098/rspa.2016.0758hwp:master-id:royprsa;rspa.2016.07582017-01-18Research articles47321972016075820160758<![CDATA[Recycling of waste gasket rubber granules by bulk CuCl2 and nano CuCl2: removal of Hg(II) ions by recycled rubber granules]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2197/20160771?rss=1
Environmental problems arise due to the millions of tons of waste rubber that are thrown away in the natural environment. Management of this waste rubber is a big environmental challenge. So, a new, simple and cost-effective recycling method for obtaining recycled waste rubber should be developed. In this study, we found that waste gasket rubber can be desulfurized by means of bulk and nano-sized transition metal halides in the presence of solvents. The recycled product of desulfurized waste gasket rubber granules that is obtained can be used as the cheapest adsorbent in the removal of mercury(II) ions from aqueous solution. Comparative batch studies have been conducted to elucidate the adsorption efficiency of desulfurized rubber using bulk copper chloride and also using nano-sized copper chloride under optimum conditions with commercial activated carbon.
]]>2017-01-18T00:05:26-08:00info:doi/10.1098/rspa.2016.0771hwp:master-id:royprsa;rspa.2016.07712017-01-18Research articles47321972016077120160771<![CDATA[Real wave propagation in the isotropic-relaxed micromorphic model]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2197/20160790?rss=1
For the recently introduced isotropic-relaxed micromorphic generalized continuum model, we show that, under the assumption of positive-definite energy, planar harmonic waves have real velocity. We also obtain a necessary and sufficient condition for real wave velocity which is weaker than the positive definiteness of the energy. Connections to isotropic linear elasticity and micropolar elasticity are established. Notably, we show that strong ellipticity does not imply real wave velocity in micropolar elasticity, whereas it does in isotropic linear elasticity.
]]>2017-01-11T00:28:48-08:00info:doi/10.1098/rspa.2016.0790hwp:master-id:royprsa;rspa.2016.07902017-01-11Research articles47321972016079020160790<![CDATA[Correction to 'On the energy partition in oscillations and waves]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2197/20160839?rss=1
2017-01-04T00:05:20-08:00info:doi/10.1098/rspa.2016.0839hwp:master-id:royprsa;rspa.2016.08392017-01-04Erratum47321972016083920160839<![CDATA[Editorial January 2017]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2197/20160897?rss=1
2017-01-11T00:05:17-08:00info:doi/10.1098/rspa.2016.0897hwp:master-id:royprsa;rspa.2016.08972017-01-11Editorial47321972016089720160897