Using a custom-FPGA architecture to extend the scale of atomistic magnetic spin simulations

New computational solutions are required to understand how atomic-scale properties affect magnetic behaviour at micrometer dimensions. We describe a field-programmable-gate-array (FPGA) based simulation of a dilute antiferromagnet with a large number of Ising spins using Glauber dynamics. The simple atomic model qualitatively reproduces experimental findings when scaled up to sufficiently large spatial dimensions, and provides insight into the finite size thresholds separating nanoscale from microscale behaviour. A real-time visualisation module was used to study the dynamics of the fractal domain structure and non-exponential relaxation mechanism. A performance comparison with contemporary GPU and CPU implementations suggests that a FPGA route is a competitive alternative.     

Nonlinear vibration of buckled nanowires on a compliant substrate

Buckling of thin nanowires on a pre-strained compliant substrate has been widely used to make nanowire-based stretchable electronics. On nanometer scale, surface effect plays an important role on a buckled nanowire structure. In addition, as the amplitude of the deflection of the buckled nanowire is larger than its thickness, geometrical nonlinearity should be taken into account. Taking the kinetic energy caused by the out-of-plane motion into account, and on the basis of Euler beam theory, a theoretical model for a nanowire-substrate structure is established, combined with the influences of the nano-scale surface effect and geometrical nonlinearity. By means of Lagrange's equation, the equation of motion is derived and then solved by the Symplectic (Partitioned) Runge–Kutta method (PRK). Several numerical examples are analysed to study the nonlinear vibration of the structure. The analytical expressions of stable and unstable equilibrium points, and the relationship between the vibration amplitude and the natural frequency are obtained. The influences of surface effect and pre-strain on the dynamic behaviour are analysed. Through these numerical results, one can find that when the surface elastic modulus and surface residual stress are considered, the number of unstable equilibrium points would increase to three. The frequency obtained with positive surface elastic modulus is greater than that obtained with negative surface elastic modulus, implying that the positive surface elastic modulus can make the nanowire-substrate structure stiffer. Furthermore, when the pre-strain increases, the locations of stable and unstable equilibrium points move further away from the initial displacement, and the homoclinic orbits become expanded. The results presented in this paper should be useful to guide the design of nanowire-based stretchable electronics.     

Qualitative analysis of a chemostat model with inhibitory exponential substrate uptake

In this paper, we consider a simple chemostat model involving a single species feeding on redundant substrate with a constant yield term. Many experiments indicate that very high substrate concentrations actually inhibit growth. Instead of assuming the prevalent Monod kinetics for growth rate of cells, we use a non-monotonic functional response function to describe the inhibitory effect. A detailed qualitative analysis about the local and global stability of its equilibria (including all critical cases) is carried out. Numerical simulations are performed to show that the dynamical properties depend intimately upon the parameters.     

Using tropical optimization to solve minimax location problems with a rectilinear metric on the line

Methods of tropical (idempotent) mathematics are applied to the solution of minimax location problems under constraints on the feasible location region. A tropical optimization problem is first considered, formulated in terms of a general semifield with idempotent addition. To solve the optimization problem, a parameter is introduced to represent the minimum value of the objective function, and then the problem is reduced to a parametrized system of inequalities. The parameter is evaluated using existence conditions for solutions of the system, whereas the solutions of the system for the obtained value of the parameter are taken as the solutions of the initial optimization problem. Then, a minimax location problem is formulated to locate a single facility on a line segment in the plane with a rectilinear metric. When no constraints are imposed, this problem, which is also known as the Rawls problem or the messenger boy problem, has known geometric and algebraic solutions. For the location problems, where the location region is restricted to a line segment, a new solution is obtained, based on the representation of the problems in the form of the tropical optimization problem considered above. Explicit solutions of the problems for various positions of the line are given both in terms of tropical mathematics and in the standard form.     

Influence of oscillation localization on film detachment from a substrate

In modern constructions, thin-layer coats are often used as protecting or strengthening elements. Deformations of such constructions may cause significant stresses on the interface between the base and the coat because of the difference in their physical-mechanical properties, which leads to the destruction or detachment of the cover. Of special interest is strength analysis under dynamical or vibrational impacts because of the possibility of localizing oscillations in a neighborhood of the initial inhomogeneities (such as inclusions, defects, construction elements, etc.). In this paper, on the example of the detachment of a string from an elastic substrate, the possibility of localizing oscillations on a detachment defect is demonstrated and the effect of this localization on the growth of the detachment zone is analyzed. A simplified setting of the problem is considered. The possibility of localizing oscillations on a detachment defect is demonstrated and an approximate analytical solution is constructed, which takes into account only the first symmetric form of oscillations describing the development of the initial detachment. A numerical modeling of the problem is performed, and the results of modeling are compared with the approximate analytical solution.     

Electro-osmotic flow of a viscoelastic fluid in a channel : Applications to physiological fluid mechanics

The paper presents a comprehensive theoretical study on the electro-osmotic flow of a viscoelastic fluid past a channel having stretching walls. An attempt has been made to investigate the effect of rheological and electro-osmotic parameters on the kinematics of the fluid. Results presented here pertain to the case where the channel height is much greater than the thickness of electrical double layer comprising the Stern and diffuse layers. The study reveals that an increase in electro-osmotic parameter leads to an increase in the axial velocity throughout the channel for a fluid having viscoelastic coefficient equal to that of blood. This aspect provides a source of novel insight into the process of designing bio-sensing and micro-fluidic devices.     

Stochastic algorithm for solving transient diffusion equations with a precise accounting of reflection boundary conditions on a substrate surface

A new random walk based stochastic algorithm for solving transient diffusion equations in domains where a reflection boundary condition is imposed on a plane part of the boundary is suggested. The motivation comes from the field of exciton transport and recombination in semiconductors where the reflecting boundary is the substrate plane surface while on the defects and dislocations an absorption boundary condition is prescribed. The idea of the method is based on the exact representations of the first passage time and position distributions on a parallelepiped (or a cube) with a reflection condition on its bed face lying on the substrate. The algorithm is meshfree both in space and time, the particle trajectories are moving inside the domain in accordance with the Random Walk on Spheres (RWS) process but when approaching the reflecting surface they switch to move on parallelepipeds (or cubes). The efficiency of the method is drastically increased compared with the standard RWS method. For illustration, we present an example of exciton flux calculations in the cathodoluminescence imaging method in semiconductors with a set of threading dislocations.     

From fractions to proportional reasoning: a cognitive schemes of operation approach

Four seventh grade students participated in a constructivist teaching experiment in which manipulatives within a computer microworld were used to solve fractional reasoning tasks followed by tasks that involve concepts of rate, ratio and proportion. Through a retrospective analysis of video tapes, their thinking processes were analyzed from the perspective of the types of cognitive schemes of operation used as they engaged in the given problem situations. One result of the study indicates that the modifications of the students’ available schemes of operation when solving the fractional reasoning tasks formed a basis for the cognitive schemes of operation used in their solutions of tasks involving proportionality.     

A simple method for spherical indentation of an elastic thin layer with surface tension bonded to a rigid substrate

An alternative method is proposed to solve the spherical indentation problem of an elastic thin layer with surface tension bonded to a rigid substrate. Based on the Kerr model, we establish a simple modified governing equation incorporating the surface tension effects for describing the relationship between the pressure and downward deflection of the impressed surface of the layer. This modified governing equation holds both inside and outside the contact zone, making it possible to analyze the whole layer by a unified differential equation. Numerical results are presented for the contact pressure inside the contact zone, the surface deflection of the elastic layer and the load-contact zone width relation to illustrate the present method. The validity and accuracy of the present method are demonstrated by comparing our results with those available in the existing literature.     

Analysis of the radiation properties of a planar antenna on a photonic crystal substrate

This paper is concerned with the rigorous investigation of the radiation properties of a planar patch antenna on a photonic crystal substrate. Under the assumptions that the driving frequency of the antenna lies within the band gap of the photonic crystal substrate and that the crystal satisfies a symmetry condition, we prove that the power radiated into the substrate decays exponentially. To do this, we reduce the radiation problem to the study of the well‐posedness of a weakly singular integral equation on the patch antenna, and to the study of the asymptotic behaviour of the corresponding Green's function. We also provide a mathematical justification of the use of a photonic crystal substrate as a perfect mirror at any incidence angle. Copyright © 2001 John Wiley & Sons, Ltd.     

Dynamical response of a prestrained system comprising a substrate and bond and covering layers to a moving load

The dynamical response of a system consisting of a prestressed substrate and covering and bond layers to a moving load is investigated within the scope of a piecewise-homogeneous body model by using a three-dimensional linearized theory of wave propagation in initially stressed bodies. It is assumed that the materials of the constituents are isotropic and homogeneous, and the subsonic speed of the moving load acting on the covering layer is constant. The investigations are carried out for a plane strain state under complete and in complete contact conditions. For various values of problem parameters, numerical results for the critical speed are presented and discussed.     

Implementation of a primal–dual method for SDP on a shared memory parallel architecture

Primal–dual interior point methods and the HKM method in particular have been implemented in a number of software packages for semidefinite programming. These methods have performed well in practice on small to medium sized SDPs. However, primal–dual codes have had some trouble in solving larger problems because of the storage requirements and required computational effort. In this paper we describe a parallel implementation of the primal–dual method on a shared memory system. Computational results are presented, including the solution of some large scale problems with over 50,000 constraints.     

Using a continuous-time Markov model with Poisson arrivals to describe the movements of geriatric patients

The population of geriatrics in a given hospital district is relatively stable and therefore we may model the movement of geriatric patients by considering both their stays in hospital and subsequent releases back into the community. The care of the elderly in departments of geriatric medicine may be generally classified into two forms of clinical care, acute/rehabilitative and long stay. Our paper describes the movement of pateints through departments of geriatric medicine and subsequent stays in the community by a four-stage continuous-time Markov model, where the stages represent acute/rehabilitative patients, long-stay patients, ex-patients in the community and former patients who are now dead, respectively. Admissions are modelled as a Poisson stream and expressions are calculated for the distribution, mean and variance of numbers of patients in each compartment at any time. Using these expressions the model is then fitted to a large data set of hospital spells containing over 10 000 admissions. © 1998 John Wiley & Sons, Ltd.     

Rethinking the role of algorithms in school mathematics: a conceptual model with focus on cognitive development

Starting from the context of mathematics learning in the East and West, this paper discusses the position and role of algorithms within school mathematics and argues that learning of algorithms has suffered from an alleged dichotomy between procedures and understanding, in that algorithms have been associated with low-level cognition. The paper first introduces a broad perspective about algorithms in school mathematics, and then, partially drawing on Bloom’s taxonomy and Säljö’s categorization of learning, proposes a model for the learning of algorithms with focus on students’ cognitive development. The model consists of three cognitive levels: (1) Knowledge and Skills, (2) Understanding and Comprehension, and (3) Evaluation and Construction. The model suggests that the learning of algorithms does not simply imply a low level of cognition, and provides a new perspective and framework to analyse the learning of algorithms. Following the model, we present examples to demonstrate the three levels and discuss related teaching strategies. We propose that the model can be used as an analysis tool to reconceptualize the role of algorithms in school mathematics and pose some questions for further research and scholarly discourse in this direction.