Showing papers in "Proceedings of The Royal Society A: Mathematical, Physical and Engineering Sciences in 2010"

Multivariate empirical mode decomposition

Abstract: Despite empirical mode decomposition (EMD) becoming a de facto standard for time-frequency analysis of nonlinear and non-stationary signals, its multivariate extensions are only emerging; yet, they are a prerequisite for direct multichannel data analysis. An important step in this direction is the computation of the local mean, as the concept of local extrema is not well defined for multivariate signals. To this end, we propose to use real-valued projections along multiple directions on hyperspheres ( n -spheres) in order to calculate the envelopes and the local mean of multivariate signals, leading to multivariate extension of EMD. To generate a suitable set of direction vectors, unit hyperspheres ( n -spheres) are sampled based on both uniform angular sampling methods and quasi-Monte Carlo-based low-discrepancy sequences. The potential of the proposed algorithm to find common oscillatory modes within multivariate data is demonstrated by simulations performed on both hexavariate synthetic and real-world human motion signals.

Contagion in financial networks

Abstract: This paper develops an analytical model of contagion in financial networks with arbitrary structure. We explore how the probability and potential impact of contagion is influenced by aggregate and ...

The rheology of suspensions of solid particles

Abstract: We present data for the rheology of suspensions of monodisperse particles of varying aspect ratio, from oblate to prolate, and covering particle volume fractions from dilute to highly concentrated....

Micro-architectured materials: past, present and future

Abstract: Micro-architectured materials offer the opportunity of obtaining unique combinations of material properties. First, a historical perspective is given to the expansion of material property space by the introduction of new alloys and new microstructures. Principles of design of micro-architecture are then given and the role of nodal connectivity is emphasized for monoscale and multi-scale microstructures. The stiffness, strength and damage tolerance of lattice materials are reviewed and compared with those of fully dense solids. It is demonstrated that micro-architectured materials are able to occupy regions of material property space (such as high stiffness, strength and fracture toughness at low density) that were hitherto empty. Some challenges for the development of future materials are highlighted.

Constitutive modelling of arteries

Abstract: This review article is concerned with the mathematical modelling of the mechanical properties of the soft biological tissues that constitute the walls of arteries. Many important aspects of the mechanical behaviour of arterial tissue can be treated on the basis of elasticity theory, and the focus of the article is therefore on the constitutive modelling of the anisotropic and highly nonlinear elastic properties of the artery wall. The discussion focuses primarily on developments over the last decade based on the theory of deformation invariants, in particular invariants that in part capture structural aspects of the tissue, specifically the orientation of collagen fibres, the dispersion in the orientation, and the associated anisotropy of the material properties. The main features of the relevant theory are summarized briefly and particular forms of the elastic strain-energy function are discussed and then applied to an artery considered as a thick-walled circular cylindrical tube in order to illustrate its extension–inflation behaviour. The wide range of applications of the constitutive modelling framework to artery walls in both health and disease and to the other fibrous soft tissues is discussed in detail. Since the main modelling effort in the literature has been on the passive response of arteries, this is also the concern of the major part of this article. A section is nevertheless devoted to reviewing the limited literature within the continuum mechanics framework on the active response of artery walls, i.e. the mechanical behaviour associated with the activation of smooth muscle, a very important but also very challenging topic that requires substantial further development. A final section provides a brief summary of the current state of arterial wall mechanical modelling and points to key areas that need further modelling effort in order to improve understanding of the biomechanics and mechanobiology of arteries and other soft tissues, from the molecular, to the cellular, tissue and organ levels.

High-frequency homogenization for periodic media

Abstract: An asymptotic procedure based upon a two-scale approach is developed for wave propagation in a doubly periodic inhomogeneous medium with a characteristic length scale of microstructure far less than that of the macrostructure. In periodic media, there are frequencies for which standing waves, periodic with the period or double period of the cell, on the microscale emerge. These frequencies do not belong to the low-frequency range of validity covered by the classical homogenization theory, which motivates our use of the term ‘high-frequency homogenization’ when perturbing about these standing waves. The resulting long-wave equations are deduced only explicitly dependent upon the macroscale, with the microscale represented by integral quantities. These equations accurately reproduce the behaviour of the Bloch mode spectrum near the edges of the Brillouin zone, hence yielding an explicit way for homogenizing periodic media in the vicinity of ‘cell resonances’. The similarity of such model equations to high-frequency long wavelength asymptotics, for homogeneous acoustic and elastic waveguides, valid in the vicinities of thickness resonances is emphasized. Several illustrative examples are considered and show the efficacy of the developed techniques.

A glance beyond the quantum model

Abstract: One of the most important problems in physics is to reconcile quantum mechanics with general relativity, and some authors have suggested that this may be realized at the expense of having to drop the quantum formalism in favour of a more general theory. Here, we propose a mechanism to make general claims on the microscopic structure of the Universe by postulating that any post-quantum theory should recover classical physics in the macroscopic limit. We use this mechanism to bound the strength of correlations between distant observers in any physical theory. Although several quantum limits are recovered, such as the set of two-point quantum correlators, our results suggest that there exist plausible microscopic theories of Nature that predict correlations impossible to reproduce in any quantum mechanical system.

On the thermodynamic entropy of fatigue fracture

Abstract: Entropy production during the fatigue process can serve as a measure of degradation. We postulate that the thermodynamic entropy of metals undergoing repeated cyclic load reaching the point of fracture is a constant, independent of geometry, load and frequency. That is, the necessary and sufficient condition for the final fracture of a metal undergoing fatigue load corresponds to a constant irreversible entropy gain. To examine validity, we present the results of an extensive set of both experimental tests and analytical predictions that involve bending, torsion and tension-compression of aluminium 6061-T6 and stainless steel 304 specimens. The concept of tallying up the entropy generation has application in determining the fatigue life of components undergoing cyclic bending, torsion and tension-compression.

Heterofunctionalization catalysis with organometallic complexes of calcium, strontium and barium

Abstract: Despite the routine employment of Grignard reagents and Hauser bases as stoichiometric carbanion reagents in organic and inorganic synthesis, a defined reaction chemistry encompassing the heavier e...

Cosmic strings and superstrings

Abstract: Cosmic strings are predicted by many field-theory models, and may have been formed at a symmetry-breaking transition early in the history of the universe, such as that associated with grand unification. They could have important cosmological effects. Scenarios suggested by fundamental string theory or M-theory, in particular the popular idea of brane inflation, also strongly suggest the appearance of similar structures. Here we review the reasons for postulating the existence of cosmic strings or superstrings, the various possible ways in which they might be detected observationally and the special features that might discriminate between ordinary cosmic strings and superstrings.

The effect of crystal orientation on the indentation response of commercially pure titanium: experiments and simulations

Abstract: This study combines nanoindentation, electron backscatter diffraction (EBSD) and crystal plasticity finite element analysis to examine the anisotropy in the indentation behaviour of individual grains within an α-Ti polycrystal. Nanoindentation is utilized to mechanically probe small volumes of material within grains for which orientations are known from prior EBSD mapping. Both indentation modulus and hardness decrease significantly as the indentation axis is inclined further from the c -axis; the plastic response showing the more marked anisotropy. Recently developed high angular resolution EBSD has been utilized to examine selected indents, providing maps of elastic strain variations and lattice rotations. From such maps lower bound solutions for the density of geometrically necessary dislocations (GNDs) have been established. Crystal plasticity modelling showed promise in capturing correctly the orientation dependence of load–displacement response and in lattice rotations local to the indenter, particularly for indentation into a basal plane which generated threefold rotational symmetry about an axis parallel with the indentation direction which was also observed in experiments.

A probabilistic twin nucleation model for HCP polycrystalline metals

Abstract: This article presents a basic probabilistic theory for the nucleation of deformation twins in hexagonal close packed (HCP) metals. Twin nucleation is assumed to rely on the dissociation of grain bo...

The fourth element: characteristics, modelling and electromagnetic theory of the memristor

Abstract: In 2008, researchers at the HewlettPackard (HP) laboratories published a paper in Nature reporting the development of a new basic circuit element that completes the missing link between charge and ...

Fast-forward of adiabatic dynamics in quantum mechanics

Abstract: We propose a method to accelerate adiabatic dynamics of wave functions (WFs) in quantum mechanics to obtain a final adiabatic state except for the spatially uniform phase in any desired short time. In our previous work, acceleration of the dynamics of WFs was shown to obtain the final state in any short time by applying driving potential. We develop the previous theory of fast-forward to derive a driving potential for the fast-forward of adiabatic dynamics. A typical example is the fast-forward of adiabatic transport of a WF, which is the ideal transport in the sense that a stationary WF is transported to an aimed position in any desired short time without leaving any disturbance at the final time of the fast-forward. As other important examples, we show accelerated manipulations of WFs, such as their splitting and squeezing. The theory is also applicable to macroscopic quantum mechanics described by the nonlinear Schrodinger equation.

Modelling bacterial behaviour close to a no-slip plane boundary: the influence of bacterial geometry

Abstract: We describe a boundary-element method used to model the hydrodynamics of a bacterium propelled by a single helical flagellum. Using this model, we optimize the power efficiency of swimming with respect to cell body and flagellum geometrical parameters, and find that optima for swimming in unbounded fluid and near a no-slip plane boundary are nearly indistinguishable. We also consider the novel optimization objective of torque efficiency and find a very different optimal shape. Excluding effects such as Brownian motion and electrostatic interactions, it is demonstrated that hydrodynamic forces may trap the bacterium in a stable, circular orbit near the boundary, leading to the empirically observable surface accumulation of bacteria. Furthermore, the details and even the existence of this stable orbit depend on geometrical parameters of the bacterium, as described in this article. These results shed some light on the phenomenon of surface accumulation of micro-organisms and offer hydrodynamic explanations as to why some bacteria may accumulate more readily than others based on morphology.

Solar change and climate: an update in the light of the current exceptional solar minimum

Abstract: Solar outputs during the current solar minimum are setting record low values for the space age. Evidence is here reviewed that this is part of a decline in solar activity from a grand solar maximum and that the Sun has returned to a state that last prevailed in 1924. Recent research into what this means, and does not mean, for climate change is reviewed.

Semi-classical limit for Schrödinger equations with magnetic field and Hartree-type nonlinearities

Abstract: The semi-classical regime of standing wave solutions of a Schrodinger equation in the presence of non-constant electric and magnetic potentials is studied in the case of non-local nonlinearities of Hartree type. It is shown that there exists a family of solutions having multiple concentration regions which are located around the minimum points of the electric potential.

Normal typicality and von Neumann's quantum ergodic theorem

Abstract: We discuss the content and significance of John von Neumann’s quantum ergodic theorem (QET) of 1929, a strong result arising from the mere mathematical structure of quantum mechanics. The QET is a precise formulation of what we call normal typicality, i.e. the statement that, for typical large systems, every initial wave function ψ 0 from an energy shell is ‘normal’: it evolves in such a way that | ψ t 〉〈 ψ t | is, for most t , macroscopically equivalent to the micro-canonical density matrix. The QET has been mostly forgotten after it was criticized as a dynamically vacuous statement in several papers in the 1950s. However, we point out that this criticism does not apply to the actual QET, a correct statement of which does not appear in these papers, but to a different (indeed weaker) statement. Furthermore, we formulate a stronger statement of normal typicality, based on the observation that the bound on the deviations from the average specified by von Neumann is unnecessarily coarse and a much tighter (and more relevant) bound actually follows from his proof.

Travelling waves of a diffusive Kermack-McKendrick epidemic model with non-local delayed transmission

Abstract: We obtain full information about the existence and non-existence of travelling wave solutions for a general class of diffusive Kermack–McKendrick SIR models with non-local and delayed disease transmission. We show that this information is determined by the basic reproduction number of the corresponding ordinary differential model, and the minimal wave speed is explicitly determined by the delay (such as the latent period) and non-locality in disease transmission, and the spatial movement pattern of the infected individuals. The difficulty is the lack of order-preserving property of the general system, and we obtain the threshold dynamics for spatial spread of the disease by constructing an invariant cone and applying Schauder’s fixed point theorem.

Thin films for solar control applications

Abstract: The excessive use of heating systems in cold climates and air conditioning systems in hotter climates is resulting in the extensive use of electricity in order to maintain such systems. This in turn leads to the greater use of fossil fuels and higher emissions of carbon dioxide and other pollutant gases. The growing amount of carbon dioxide emissions is contributing to the problem of global warming, hence increasing the need for alternative technologies to heating and air conditioning systems. One such alternative is the production of thin films which can be used as window glazing coatings to construct ‘smart windows’. These windows have the greatest use within constant climates. In cold climates, windows with high solar transmittance and low thermal emittance are needed; this allows sunlight into the building to brighten the room but stops heat from escaping thus warming the room. In constantly hot climates, materials that are transparent in the visible region but reflective in the infrared, such as thin metallic coatings, can be used to ensure that the inside of the building remains cool. These solar control coatings, however, pose a problem in varying climates such as in northern and central Europe. For these cases, materials that have altering properties owing to external surroundings could be the solution. These ‘chromatic’ materials include several categories, such as photochromic glasses and polymers, thermochromic metal oxides and electrochromic materials. This review will begin with a discussion as to the chemistry behind the solar control coatings and chromic materials including ambient radiation and the ideal of a black-body object. It will then look in depth at each class of these chromic materials considering experimental results and theoretical insight as well as production techniques and applications.

Dropouts of the outer electron radiation belt in response to solar wind stream interfaces: global positioning system observations

Abstract: We present a statistical study of relativistic electron counts in the electron radiation belt across a range of drift shells (L*>4) combining data from nine combined X-ray dosimeters (CXD) on the global positioning system (GPS) constellation. The response of the electron counts as functions of time, energy and drift shell are examined statistically for 67 solar wind stream interfaces (SIs); two-dimensional superposed epoch analysis is performed with the CXD data. For these epochs we study the radiation belt dropouts and concurrent variations in key geophysical parameters. At higher L* we observe a tendency for a gradual drop in the electron counts over the day preceding the SI, consistent with outward diffusion and magnetopause shadowing. At all L*, dropouts occur with a median time scale of ≃7 h and median counts fall by 0.4–1.8 orders of magnitude. The central tendencies of radiation belt dropout and recovery depend on both L* and energy. For ≃70 per cent of epochs Sym-H more than −30 nT, yet only three of 67 SIs did not have an associated dropout in the electron data. Statistical maps of electron precipitation suggest that chorus-driven relativistic electron microbursts might be major contributors to radiation belt losses under high-speed stream driving.

Periodic frameworks and flexibility

Abstract: We formulate a concise deformation theory for periodic bar-and-joint frameworks in R d and illustrate our algebraic–geometric approach on frameworks related to various crystalline structures. Particular attention is given to periodic frameworks modelled on silica, zeolites and perovskites. For frameworks akin to tectosilicates, which are made of one-skeleta of d -dimensional simplices, with each vertex common to exactly two simplices, we prove the existence of a space of periodicity-preserving infinitesimal flexes of dimension at least ![Graphic][1] . However, these infinitesimal flexes need not come from genuine flexibility, as shown by rigid examples. The changes implicated in passing from a given lattice of periods to a sublattice of periods are illustrated with frameworks modelled on perovskites. [1]: /embed/inline-graphic-1.gif

Modelling rigid origami with quaternions and dual quaternions

Abstract: This paper examines the mathematical modelling of rigid origami, a type of origami where all the panels are rigid and can only rotate about crease lines. The rotating vector model is proposed, which establishes the loop-closure conditions among a group of characteristic vectors. By building up an explicit relationship between the single-vertex origami and the spherical linkage mechanism, the rotating vector model can conveniently and directly describe arbitrary three-dimensional configurations and can detect some self-intersection. Quaternion and dual quaternion are then employed to represent the origami model, based on which two numerical methods have been developed. Through examples, it has been shown that the first method can effectively track the entire rigid-folding procedure of an initially flat or a non-flat pattern with a single vertex or multiple vertices, and thereby provide judgment for its rigid foldability and flat foldability. Furthermore, its ability to rule out some self-intersecting configurations during folding is illustrated in detail, leading to its ability of checking rigid foldability in a more or less sufficient way. The second method is especially for analysing the multi-vertex origami. It can also effectively track the trajectories of multiple vertices during folding.

Evolving graphs: dynamical models, inverse problems and propagation

Abstract: Applications such as neuroscience, telecommunication, online social networking, transport and retail trading give rise to connectivity patterns that change over time. In this work, we address the resulting need for network models and computational algorithms that deal with dynamic links. We introduce a new class of evolving range-dependent random graphs that gives a tractable framework for modelling and simulation. We develop a spectral algorithm for calibrating a set of edge ranges from a sequence of network snapshots and give a proof of principle illustration on some neuroscience data. We also show how the model can be used computationally and analytically to investigate the scenario where an evolutionary process, such as an epidemic, takes place on an evolving network. This allows us to study the cumulative effect of two distinct types of dynamics.

Mode-mixity-dependent adhesive contact of a sphere on a plane surface

Abstract: Tangential loading in the presence of adhesion is highly relevant to biological locomotion, but mixed-mode contact of biological materials or similar soft elastomers remains to be well understood. To better capture the effects of dissipation in such contact problems owing to viscoelasticity or irreversible interfacial adhesive processes, a model is developed for the combined adhesive and tangential loading of a rigid sphere on a flat half-space which incorporates a phenomenological model of energy dissipation in the form of increased effective work of adhesion with increasing degree of mode mixity. To verify the model, contact experiments are performed on polydimethylsiloxane (PDMS) samples using a custom-built microtribometer. Measurements of contact area during mixed normal/tangential loading indicate that the strong dependence of the effective work of adhesion upon mode mixity can be captured effectively by the phenomenological model in the regime where the contact area stayed circular and the slip was negligible. Rate effects were seen to be described by a power-law dependence upon the crack front velocity, similar to observations of rate-dependent contact seen for pure normal loading.

Curvature in nematic elastica responding to light and heat

Abstract: Nematic elastic bodies can develop a gradient of response to heat, light and other stimuli They then bend and develop curvature in a complex manner depending on director field distributions, on whether they are monodomain or polydomain structures and on linear or nonlinear light absorptive processes In each case, we derive the general weak response where bend in each direction is treated independently of that in others In a subsequent paper, we address the reverse phenomenon, that is of strong spontaneous distortion leading to curvature suppression

Geological pattern formation by growth and dissolution in aqueous systems

Abstract: Progress towards the development of a better understanding of the formation of geological patterns in wet systems due to precipitation and dissolution is reviewed. Emphasis is placed on the formation of terraces, stalactites, stalagmites and other carbonate patterns due to precipitation from flowing supersaturated solutions and the formation of scallops by dissolution in undersaturated turbulent fluids. In addition, the formation of spherulites, dendrites and very large, essentially euhedral, crystals is discussed. In most cases, the formation of very similar patterns as a result of the freezing/melting of ice and the precipitation/dissolution of minerals strongly suggests that complexity associated with aqueous chemistry, interfacial chemistry and biological processes has only a secondary effect on these pattern formation processes.

Could light harvesting complexes exhibit non-classical effects at room temperature?

Abstract: Mounting experimental and theoretical evidence suggest that coherent quantum effects play a role in the efficient transfer of an excitation from a chlorosome antenna to a reaction centre in the Fenna–Matthews–Olson protein complex. However, it is conceivable that a satisfying alternate interpretation of the results is possible in terms of a classical theory. To address this possibility, we consider a class of classical theories satisfying the minimal postulates of macrorealism and frame Leggett–Garg-type tests that could rule them out. Our numerical simulations indicate that even in the presence of decoherence, several tests could exhibit the required violations of the Leggett–Garg inequality. Remarkably, some violations persist even at room temperature for our decoherence model.

Stability of ice-sheet grounding lines

Abstract: Recent observations of the West Antarctic Ice Sheet document rapid changes in the mass balance of its component glaciers. These observations raise the question of whether changing climatic conditions have triggered a dynamical instability in the ice-sheet–ice-shelf system. The dynamics of marine ice sheets are sensitive to grounding-line position and variation, characteristics that are poorly captured by most current models. We present a theory for grounding-line dynamics in three spatial dimensions and time. Our theory is based on a balance of forces across the grounding line; it is expressed as a differential equation that is analogous to the canonical Stefan condition. We apply this theory to the question of grounding-line stability under conditions of retrograde bed slope in a suite of calculations with different basal topography. A subset of these have basal topography inspired by the Pine Island glacier, where basal depth varies in both the along-flow and across-flow directions. Our results indicate that unstable retreat of the grounding line over retrograde beds is a robust feature of models that evolve based on force balance at the grounding line. We conclude, based on our simplified model, that unstable grounding-line recession may already be occurring at the Pine Island glacier.

The 8 year thicket of triazine dendrimers: strategies, targets and applications

Abstract: This manuscript focuses on the routes, methods and reagents used to synthesize triazine-based dendrimers. Our pursuit of macromolecular architectures for drug delivery—dendrimers based on triazines—has been an ongoing effort for 8 years. To date, we have produced complex dendrimers with diverse peripheries as proof-of-concept, less complex molecules tailored for specific applications including DNA and RNA delivery and drug-decorated dendrimers for potential therapeutic applications including infectious disease and cancer. These syntheses have been executed at scales that range from high milligrams to over a kilogram. The routes, reagents and diversity displayed by a target anchors it in time. Early targets derive from convergent synthetic routes while later targets are prepared using divergent syntheses. The core of early dendrimers was a simple diamine, including piperazine, yielding the so-called bow-tie structures, middle period targets boast either a trispiperazinyltriazine core or a ‘super-core’ with six piperazine groups. Later targets return to the trispiperazinyltriazine core. The choice of linking diamine has also changed. Over time, p -aminobenzylamine was replaced by piperazine and then by aminomethylpiperidine with more exotic diamines sprinkled in throughout. Peripheral group choice has undergone similar variations: from AB 2 to AB 4 to, more recently, AB 3 . The diversity communicated by these groups yields dendrimers ranging from those with a common surface to examples where two groups were presented to those where four orthogonally reactive groups appear. Over time, these groups have grown in complexity from protected amines to tags for biodistribution and drugs like paclitaxel. Herein, strategies adopted and lessons learned are reviewed, intuitions relayed and future directions forecast.