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Proceedings. Mathematical, Physical, and Engineering Sciences

Michael Pan, Peter J Gawthrop, Kenneth Tran, Joseph Cursons, Edmund J Crampin
Mathematical models of cardiac action potentials have become increasingly important in the study of heart disease and pharmacology, but concerns linger over their robustness during long periods of simulation, in particular due to issues such as model drift and non-unique steady states. Previous studies have linked these to violation of conservation laws, but only explored those issues with respect to charge conservation in specific models. Here, we propose a general and systematic method of identifying conservation laws hidden in models of cardiac electrophysiology by using bond graphs, and develop a bond graph model of the cardiac action potential to study long-term behaviour...
June 2018: Proceedings. Mathematical, Physical, and Engineering Sciences
Mrityunjay Kothari, Moon-Hyun Cha, Kyung-Suk Kim
Here, we report the discovery of a new, curvature-localizing, subcritical buckling mode that produces shallow-kink corrugation in multi-layer graphene. Our density functional theory (DFT) analysis reveals the mode configuration-an approximately 2 nm wide boundary layer of highly localized curvature that connects two regions of uniformly but oppositely sheared stacks of flat atomic sheets. The kink angle between the two regions is limited to a few degrees, ensuring elastic deformation. By contrast, a purely mechanical model of sandwich structures shows progressive supercritical curvature localization spread over a 50-100 nm wide boundary layer...
June 2018: Proceedings. Mathematical, Physical, and Engineering Sciences
R Reuvers
A quantum state's entanglement across a bipartite cut can be quantified with entanglement entropy or, more generally, Schmidt norms. Using only Schmidt decompositions, we present a simple iterative algorithm to maximize Schmidt norms. Depending on the choice of norm, the optimizing states maximize or minimize entanglement, possibly across several bipartite cuts at the same time and possibly only among states in a specified subspace. Recognizing that convergence but not success is certain, we use the algorithm to explore topics ranging from fermionic reduced density matrices and varieties of pure quantum states to absolutely maximally entangled states and minimal output entropy of channels...
June 2018: Proceedings. Mathematical, Physical, and Engineering Sciences
E Nolde, A V Pichugin, J Kaplunov
A direct asymptotic integration of the full three-dimensional problem of elasticity is employed to derive a consistent governing equation for a beam with the rectangular cross section. The governing equation is consistent in the sense that it has the same long-wave low-frequency behaviour as the exact solution of the original three-dimensional problem. Performance of the new beam equation is illustrated by comparing its predictions against the results of direct finite-element computations. Limiting behaviours for beams with large (and small) aspect ratios, which can be established using classical plate theories, are recovered from the new governing equation to illustrate its consistency and also to illustrate the importance of using plate theories with the correctly refined boundary conditions...
June 2018: Proceedings. Mathematical, Physical, and Engineering Sciences
L Renson, T L Hill, D A Ehrhardt, D A W Barton, S A Neild
Nonlinear normal modes (NNMs) are widely used as a tool for developing mathematical models of nonlinear structures and understanding their dynamics. NNMs can be identified experimentally through a phase quadrature condition between the system response and the applied excitation. This paper demonstrates that this commonly used quadrature condition can give results that are significantly different from the true NNM, in particular, when the excitation applied to the system is limited to one input force, as is frequently used in practice...
June 2018: Proceedings. Mathematical, Physical, and Engineering Sciences
Mark Stromp, Alexandra Farkas, Balázs Kretzer, Dénes Száz, András Barta, Gábor Horváth
Inspired by the pioneer work of the nineteenth century photographer, William Nicholson Jennings, we studied quantitatively how realistic painted lightnings are. In order to answer this question, we examined 100 paintings and 400 photographs of lightnings. We used our software package to process and evaluate the morphology of lightnings. Three morphological parameters of the main lightning branch were analysed: (i) number of branches N b , (ii) relative length r , and (iii) number of local maxima (peaks) N p of the turning angle distribution...
June 2018: Proceedings. Mathematical, Physical, and Engineering Sciences
Ciarán M Lee, John H Selby
To date, there has been no experimental evidence that invalidates quantum theory. Yet it may only be an effective description of the world, in the same way that classical physics is an effective description of the quantum world. We ask whether there exists an operationally defined theory superseding quantum theory, but which reduces to it via a decoherence-like mechanism. We prove that no such post-quantum theory exists if it is demanded that it satisfy two natural physical principles: causality and purification ...
June 2018: Proceedings. Mathematical, Physical, and Engineering Sciences
A Chatziafratis, G Fikioris, J P Xanthakis
The progress in field emission theory from its initial Fowler-Nordheim form is centred on the transmission coefficient. For the supply (of electrons) function one still uses the constant value due to a supply of plane-waves states. However, for emitting tips of apex radius of 1-5 nm this is highly questionable. To address this issue, we have solved the Schrödinger equation in a sharp paraboloidally shaped quantum box. The Schrödinger equation is separable in the rotationally parabolic coordinate system and we hence obtain the exact eigenstates of the system...
June 2018: Proceedings. Mathematical, Physical, and Engineering Sciences
Ingo Nitschke, Michael Nestler, Simon Praetorius, Hartmut Löwen, Axel Voigt
We consider a thin film limit of a Landau-de Gennes Q-tensor model. In the limiting process, we observe a continuous transition where the normal and tangential parts of the Q-tensor decouple and various intrinsic and extrinsic contributions emerge. The main properties of the thin film model, like uniaxiality and parameter phase space, are preserved in the limiting process. For the derived surface Landau-de Gennes model, we consider an L 2 -gradient flow. The resulting tensor-valued surface partial differential equation is numerically solved to demonstrate realizations of the tight coupling of elastic and bulk free energy with geometric properties...
June 2018: Proceedings. Mathematical, Physical, and Engineering Sciences
Benjamin E Schaffer, Ignacio Rodriguez-Iturbe
In arid or semi-arid ecosystems, water availability is one of the primary controls on vegetation growth. When subsurface water resources are unavailable, the vegetation growth is dictated by the rainfall, and the random nature of the rainfall arrivals and quantities induces a probability distribution of soil moisture and vegetation biomass via the coupled dynamic equations of biomass balance and water balance. We have previously obtained an exact solution for these distributions under certain conditions, and shown that the mapping of rainfall variability to observed biomass variability can be successfully applied to a field site...
June 2018: Proceedings. Mathematical, Physical, and Engineering Sciences
Jaume Giné, Jaume Llibre, Claudia Valls
We study the simultaneous existence of centres for two families of planar <mml:math xmlns:mml=""> <mml:msub> <mml:mrow> <mml:mi>Z</mml:mi> </mml:mrow> <mml:mi>q</mml:mi> </mml:msub> </mml:math> -equivariant systems. First, we give a short review about <mml:math xmlns:mml=""> <mml:msub> <mml:mrow> <mml:mi>Z</mml:mi> </mml:mrow> <mml:mi>q</mml:mi> </mml:msub> </mml:math> -equivariant systems...
May 2018: Proceedings. Mathematical, Physical, and Engineering Sciences
Tom Shearer, Camille Vitali, William J Parnell, I David Abrahams
[This corrects the article DOI: 10.1098/rspa.2015.0450.].
May 2018: Proceedings. Mathematical, Physical, and Engineering Sciences
Valeria Settimi, Giuseppe Rega, Eduardo Saetta
The nonlinear response of a reduced model of an orthotropic single-layered plate with thermomechanical coupling is investigated in the presence of thermal excitations, in addition to mechanical ones. Different issues are addressed via accurate and extended local and global analyses. (i) Assessing the possible occurrence, disappearance or modification of mechanical buckling as a result of thermal aspects; (ii) exploiting global dynamics to unveil the effects of coupling; (iii) highlighting the crucial role played by the slow thermal transient evolution in modifying the fast steady mechanical response; (iv) framing the influence of coupling and underlining the need to use a thermomechanical model to grasp the actual plate dynamics; and (v) getting hints of technical interest as to the outcome robustness with respect to variations in the external/internal thermal parameters...
May 2018: Proceedings. Mathematical, Physical, and Engineering Sciences
O P Bardsley
A novel theory for the origin of the westward drift of the Earth's magnetic field is proposed, based upon the propagation of hydrodynamic Rossby waves in the liquid outer core. These waves have the obscure property that their crests always progress eastwards-but, for a certain subset, energy can nevertheless be transmitted westwards. In fact, this subset corresponds to sheet-like flow structures, extended in both the axial and radial directions, which are likely to be preferentially excited by convective upwellings in the Earth's rapidly rotating outer core...
May 2018: Proceedings. Mathematical, Physical, and Engineering Sciences
Darryl D Holm, Tomasz M Tyranowski
We derive a new variational principle, leading to a new momentum map and a new multisymplectic formulation for a family of Euler-Poincaré equations defined on the Virasoro-Bott group, by using the inverse map (also called 'back-to-labels' map). This family contains as special cases the well-known Korteweg-de Vries, Camassa-Holm and Hunter-Saxton soliton equations. In the conclusion section, we sketch opportunities for future work that would apply the new Clebsch momentum map with 2-cocycles derived here to investigate a new type of interplay among nonlinearity, dispersion and noise...
May 2018: Proceedings. Mathematical, Physical, and Engineering Sciences
Matteo Taffetani, Xin Jiang, Douglas P Holmes, Dominic Vella
Depending on its geometry, a spherical shell may exist in one of two stable states without the application of any external force: there are two 'self-equilibrated' states, one natural and the other inside out (or 'everted'). Though this is familiar from everyday life-an umbrella is remarkably stable, yet a contact lens can be easily turned inside out-the precise shell geometries for which bistability is possible are not known. Here, we use experiments and finite-element simulations to determine the threshold between bistability and monostability for shells of different solid angle...
May 2018: Proceedings. Mathematical, Physical, and Engineering Sciences
Pantelis R Vlachas, Wonmin Byeon, Zhong Y Wan, Themistoklis P Sapsis, Petros Koumoutsakos
We introduce a data-driven forecasting method for high-dimensional chaotic systems using long short-term memory (LSTM) recurrent neural networks. The proposed LSTM neural networks perform inference of high-dimensional dynamical systems in their reduced order space and are shown to be an effective set of nonlinear approximators of their attractor. We demonstrate the forecasting performance of the LSTM and compare it with Gaussian processes (GPs) in time series obtained from the Lorenz 96 system, the Kuramoto-Sivashinsky equation and a prototype climate model...
May 2018: Proceedings. Mathematical, Physical, and Engineering Sciences
Yuxiang Yang, Giulio Chiribella, Masahito Hayashi
Quantum mechanics imposes a fundamental trade-off between the accuracy of time measurements and the size of the systems used as clocks. When the measurements of different time intervals are combined, the errors due to the finite clock size accumulate, resulting in an overall inaccuracy that grows with the complexity of the set-up. Here, we introduce a method that, in principle, eludes the accumulation of errors by coherently transferring information from a quantum clock to a quantum memory of the smallest possible size...
May 2018: Proceedings. Mathematical, Physical, and Engineering Sciences
L Ellam, M Girolami, G A Pavliotis, A Wilson
The building of mathematical and computer models of cities has a long history. The core elements are models of flows (spatial interaction) and the dynamics of structural evolution. In this article, we develop a stochastic model of urban structure to formally account for uncertainty arising from less predictable events. Standard practice has been to calibrate the spatial interaction models independently and to explore the dynamics through simulation. We present two significant results that will be transformative for both elements...
May 2018: Proceedings. Mathematical, Physical, and Engineering Sciences
J Gratus, T Banaszek
Despite being studied for over a century, the use of quadrupoles have been limited to Cartesian coordinates in flat space-time due to the incorrect transformation rules used to define them. Here the correct transformation rules are derived, which are particularly unusual as they involve second derivatives of the coordinate transformation and an integral. Transformations involving integrals have not been seen before. This is significantly different from the familiar transformation rules for a dipole, where the components transform as tensors...
May 2018: Proceedings. Mathematical, Physical, and Engineering Sciences
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