Wavelet Decomposition of Spherical Vector Fields with Respect to Sources

This article is concerned with an approach of modelling the Earth’s magnetic field as measured by satellites in terms of a special system of vector spherical harmonics and in terms of vector kernel functions, called vector scaling functions and wavelets. The main ingredient is the presentation of a system of vector spherical harmonics which separates a given spherical vector field with respect to its sources, i.e., the spherical vector field is separated into a part which is induced by sources inside the sphere, a part which is induced by sources outside the sphere and a part which is induced by sources on the sphere, which are, for example, currents crossing the sphere. Using this special system of vector spherical harmonics vector scaling functions and wavelets are constructed which keep the advantageous property of separating with respect to sources but which also allow a locally reflected modelling of the respective vector field. At the end of the article, the method is tested on real magnetic field data measured by the German geoscientific research satellite CHAMP.     

Heat transfer enhancement for boundary layer flow over a wedge by the use of electric fields

The effect of an electric field is investigated for heat transfer properties in a laminar, incompressible, non-isothermal boundary layer gas flow over a wedge. The governing boundary layer equations are reduced to an ordinary differential equation system using similarity transformations. The reduced equations are solved numerically for different values of electric and flow field parameters characterizing the ratio of electric force to fluid inertia force, Joule heating and ion kinetic work. For specific electric field function forms, leading to similarity solutions, velocity boundary layers are observed to become thinner and heat transfer properties are shown to be enhanced near the wall. The level of enhancement is controlled by the electric body force with additional effects of Joule heating and ion kinetic work on the bulk flow. The effects of low and high Prandtl numbers are also demonstrated. Heat transfer enhancement is observed to increase with increasing Prandtl number.     

Characterizing the human postural control system using detrended fluctuation analysis

Detrended fluctuation analysis is used to study the behaviour of the time series of the position of the center of pressure, output from the activity of a human postural control system. The results suggest that these trajectories present a crossover in their scaling properties from persistent (for high frequencies, short-range time scale) to anti-persistent (for low frequencies, long-range time scale) behaviours. The values of the scaling exponent found for the persistent parts of the trajectories are very similar for all the cases analysed. The similarity of the results obtained for the measurements done with both eyes open and both eyes closed indicate either that the visual system may be disregarded by the postural control system, while maintaining quiet standing, or that the control mechanisms associated with each type of information (visual, vestibular and somatosensory) cannot be disentangled with this technique.     

SCALE AND TIME EFFECTS ON MATHEMATICAL MODELS FOR TRANSPORT IN THE ENVIRONMENT

The purpose of this paper is to analyse mathematical models used in environmental modelling.Following a brief survey of the development in modelling scale-and time-dependent dispersion processes in the environment,this paper compares three similarity solutions,one of which is a solution of the generalized Feller equation(GF)with fractal parameters,and the other two for the newly-developed generalized Fokker-Planck equation(GFP).The three solutions are derived with parameters having physical significance.Data from field experiments are used to verify the solutions.The analyses indicate that the solutions of both GF and GFP represent the physically meaningful natural processes,and simulate the realistic shapes of tracer breakthrough curves.     

Fuzzy-hybrid modelling of an Ackerman steered electric vehicle

Physical system modelling with known parameters together with 2-D or high order look-up tables (obtained from experimental data), have been the preferred method for simulating electric vehicles. The non-linear phenomena which are present at the vehicle tyre patch and ground interface have resulted in a quantitative understanding of this phenomena. However, nowadays, there is a requirement for a deeper understanding of the vehicle sub-models which previously used look-up tables. In this paper the hybrid modelling methodology used for electric vehicle systems offers a two-stage advantage: firstly, the vehicle model retains a comprehensive analytical formulation and secondly, the ‘fuzzy’ element offers, in addition to the quantitative results, a qualitative understanding of specific vehicle sub-models. In the literature several hybrid topologies are reported, sequential, auxiliary, and embedded.In this paper, the hybrid model topology selected is auxiliary and within the same hybrid model, the first paradigm used is the vehicle dynamics together with the actuator/gearbox system. The second paradigm is the non-linear fuzzy tyre model for each wheel. In particular, conventional physical system dynamic modelling has been combined with the fuzzy logic type-II or type-III methodology. The resulting hybrid-fuzzy tyre models were estimated for a-priori number of rules from experimental data. The physical system modelling required the available vehicle parameters such as the overall mass, wheel radius and chassis dimensions. The suggested synergetic fusion of the two methods, (hybrid-fuzzy), allowed the vehicle planar trajectories to be obtained prior to the hardware development of the entire vehicle. The strength of this methodology is that it requires localised system experimental data rather than global system data. The disadvantage in obtaining global experimental data is the requirement for comprehensive testing of a vehicle prototype which is both time consuming process and requires extensive resources. In this paper the authors have proposed the use of existing experimental rigs which are available from the leading automotive manufacturers. Hence, for the ‘hybrid’ modelling, localised data sets were used. In particular, wheel-tyre experimental data were obtained from the University tyre rig experimental facilities. Tyre forces acting on the tyre patch are mainly responsible for the overall electric vehicle motion. In addition, tyre measurement rigs are a well known method for obtaining localised data thus allowing the effective simulation of more detailed mathematical models. These include, firstly, physical system modelling (conventional vehicle dynamics), secondly, fuzzy type II or III modelling (for the tyre characteristics), and thirdly, electric drive modelling within the context of electric vehicles. The proposed hybrid model synthesis has resulted in simulation results which are similar to piece-wise ‘look-up’ table solutions. In addition, the strength of the ‘hybrid’ synthesis is that the analyst has a set of rules which clearly show the reasoning behind the complex development of the vehicle tyre forces. This is due to the inherent transparency of the type II and type III methodologies. Finally, the authors discussed the reasons for selecting a type-III framework. The paper concludes with a plethora of simulation results.     

A note on operator self-similar Gaussian vector fields

In this work, a general representation for an operator self-similar Gaussian vector field is obtained, and some properties are studied. It is also shown that such a process is the operator scaling limit of a Gaussian vector field.     

Numerical modelling and optimization aspects of soft dielectric actuators

Dielectric elastomer actuators (DEAs) represent a new type of actuator made of electroactive polymers which can undergo large deformations. Some applications for DEAs are: Artificial muscles, haptic feedback, pumps, robotics, deformable surfaces in aerospace, etc. For this reason the numerical modelling and optimization of DEAs represent an active field of research which supports the designing process of these actuators. In this work the basic equations of the electro-mechanic coupled problem are presented in a first step. Here the Neo-Hooke and the Yeoh model have been chosen to model the behaviour of a nearly incompressible elastomer. To study optimization aspects, material inclusions in a sandwich type DEA structure have been taken into consideration. Ceramics whit high electric permittivity and air are taken into account to explore the effect of inclusions on the deformation. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Arguments constructed within the mathematical modelling cycle

Socio-cultural theories in mathematics education field recently emphasize the importance of the collective argumentation within small-group work. Since mathematical modelling tasks require a process in which students search for a solution for real life problems through small-group work, the arguments in this process become an issue of concern. This study examines the arguments constructed within the mathematical modelling cycle by considering the participants’ modelling processes. In this context, four primary pre-service mathematics teachers worked on a modelling task and their arguments were explained through the components of Toulmin’s argumentation schema. Findings revealed that the data and the claims of most of the arguments corresponded to the starting and ending points of the modelling transition in which the current arguments constructed. The existence of the arguments corresponded through warrant-claim originated from inquiring the assumptions in the modelling cycle. In addition, the participants made assumptions as warrants to support their arguments and as rebuttals to show the degree of certainty of claims in intra-group challenging situations. Both the warrants and the backings depended on modelling context as well as mathematics context.     

Decision development in facilitated modelling workshops

While the facilitated modelling literature recognises the importance of the group process within facilitated modelling workshops, published empirical research rarely examines their dynamic nature. In this paper, we address this gap in the literature in two ways. First, we propose to locate facilitated modelling workshops as the main focus of investigation, and adopt decision development as the analytical lens. Second, we provide guidance regarding how to implement a research strategy that is informed by such a focus. We start by mapping the different conceptualisations of decision development that seem embedded within the facilitated modelling tradition, and contrast them with theoretical models from the group communication field. Our analysis identifies a number of potentially useful areas for the study of facilitated modelling workshops from a decision development perspective, and articulates a number tentative research questions and testable propositions amenable to empirical research. Central to our proposal are research methods for the study of dynamic group processes. We thus discuss the steps required to extract group process data from facilitated modelling workshops that are usable and open to analysis. This includes a review of issues regarding research design, coding scheme development, data coding and choice of analytical techniques. Finally, we offer conclusions and briefly discuss some feasibility issues related to the implementation of our proposal.     

Modelling framework for artificial hybrid dynamical systems

Many current industry branches use hybrid approaches to solve complex application problems. Over the last decades, different tools for the simulation of such hybrid systems (e.g. Hysdel and YAMLIP) as well as the identification of hybrid systems (e.g. HIT, MLP and OAF NN) have been developed. The framework presented in this work facilitates the integration of artificial feed-forward neural networks in the modelling process of hybrid dynamical systems (HDS). Additionally, the framework provides a structured language for characterising these feed-forward networks itself. Therefore, an interdisciplinary exchange in the field of neural networks and its integration into hybrid dynamical systems is enabled. Focusing on hybrid systems with autonomous events, two different approaches, namely the artificial hybrid model and the artificial hybrid dynamics, are introduced. Challenges of the modelling process of HDS are reflected and advantages as well as disadvantages are discussed. The case study includes two common examples of HDS and analyses the simulation results and examines limitations of the modelling framework.     

The Concertina Pattern: A Bifurcation in Ferromagnetic Thin Films

The concertina pattern is a metastable stage in the switching process of elongated thin-film elements. It is an approximately periodic structure of domains, separated by walls perpendicular to the long axis of the element. In this paper, we give arguments in favor of our claim that the period is frozen-in at nucleation, i.e., at the critical external field. In prior work, R. Cantero-Alvarez and F. Otto, Journal of Nonlinear Science (2006), we argued that there are four qualitatively different regimes for nucleation. In one of these asymptotic regimes, the unstable mode displays an oscillatory behavior in the direction of the long axis. In this work, we derive a scaling limit of the micromagnetic energy near the bifurcation point in the above regime. We also prove that the scaling limit is coercive for all values of the reduced external field. Because of this coercivity, there exists a branch of nontrivial local minimizers. Numerical minimization of the scaling limit reveals that this branch is indeed a continuous branch of concertina pattern. The scaling limit is derived by Γ-convergence of the suitably rescaled micromagnetic energy. This robust procedure combines the limit of an asymptotic parameter regime with a zoom-in in configuration space. The coercivity of the scaling limit is derived by suitable nonlinear interpolation estimates.     

Clustering dynamics in a class of normalised generalised gamma dependent priors

Normalised generalised gamma processes are random probability measures that induce nonparametric prior distributions widely used in Bayesian statistics, particularly for mixture modelling. We construct a class of dependent normalised generalised gamma priors induced by a stationary population model of Moran type, which exploits a generalised Pólya urn scheme associated with the prior. We study the asymptotic scaling for the dynamics of the number of clusters in the sample, which in turn provides a dynamic measure of diversity in the underlying population. The limit is formalised to be a positive non-stationary diffusion process which falls outside well-known families, with unbounded drift and an entrance boundary at the origin. We also introduce a new class of stationary positive diffusions, whose invariant measures are explicit and have power law tails, which approximate weakly the scaling limit.     

Different choices of scaling in homogenization of diffusion and interfacial exchange in a porous medium

Several choices of scaling are investigated for a coupled system of parabolic partial differential equations in a two‐phase medium at the microscopic scale. This system may be regarded as modelling a reaction–diffusion problem, the Stokes problem of single‐phase flow of a slightly compressible fluid or as a heat conduction problem (with or without interfacial resistance), for example. It is shown that, starting with the same problem on the microscopic scale, different choices of scaling of the diffusion coefficients (resp. permeability or conductivity) and the interfacial‐exchange coefficient lead to different types of macroscopic systems of equations. The characterization of the limit problems in terms of the scaling parameters constitutes a modelling tool because it allows to determine the right type of limit problem. New macroscopic models, not previously dealt with, arise and, for some scalings, classical macroscopic models are recovered. Using the method of two‐scale convergence, a unified approach yielding rigorous proofs is given covering a very broad class of different scalings. Copyright © 2008 John Wiley & Sons, Ltd.     

Formalizing and solving the problem of clustering in MCDA

The topic of clustering has been widely studied in the field of Data Analysis, where it is defined as an unsupervised process of grouping objects together based on notions of similarity. Clustering in the field of Multi-Criteria Decision Aid (MCDA) has seen a few adaptations of methods from Data Analysis, most of them however using concepts native to that field, such as the notions of similarity and distance measures. As in MCDA we model the preferences of a decision maker over a set of decision alternatives, we can find more diverse ways of comparing them than in Data Analysis. As a result, these alternatives may also be arranged into different potential structures. In this paper we wish to formally define the problem of clustering in MCDA using notions that are native to this field alone, and highlight the different structures which we may try to uncover through this process. Following this we propose a method for finding these structures. As in any clustering problem, finding the optimal result in an exact manner is impractical, and so we propose a stochastic heuristic approach, which we validate through tests on a large set of artificially generated benchmarks.     

Limit theorems for supercritical branching random fields with immigration

A measure-valued process which carries genealogical information is defined for a supercritical branching random field with immigration. This process counts the particles present at a final time whose ancestors had specified locations at given times in the past. A law of large numbers and a fluctuation limit theorem are proved for this process under a space-time scaling. The fluctuation limit is a nonstationary generalized Ornstein-Uhlenbeck process. An example of interest in transport theory and polymer chemistry is given.     

On the Derivation of the Quasi-steady Model in Electrochemical Machining

The use is discussed of the quasi-steady model, based on theirrotationality of the electric field between the electroded,in electrochemical machining. The general equations are derivedwhich describe the process. From Maxwell's equations, a non-dimensionalparameter, dependent on the physical characteristics of theprocess, is derived. When this paramter is sufficiently small,the electric field is shown to be appromimately irrotational.A quasi-steady model is then formulated in terms of Laplace'sequation togther with an equation relating the change of themode anode surface with the electric field evaluated at theanode surface. This equation can be derived from Ohm's and Faraday'slaws. Different methods for solving the associated boundaryvalue problem are discussed.     

Numerical modelling of EHD effects on heat transfer and bubble shapes of nucleate boiling

In this article, mathematical and numerical models are developed to study pure electrohydrodynamic (EHD) effects on heat transfer and bubble shapes when an initial bubble attached to a superheated horizontal wall in nucleate boiling. In the modelling of EHD effects on heat transfer, an undeformed bubble is considered; the electric body force and Joule heat are added to the momentum and energy equations; governing equations for heat, fluid flow and electric fields are coupled numerically and solved using a non-orthogonal body-fitted mesh system with necessary interfacial treatments at the gas–liquid boundary. While, to study the pure effect of EHD on the deformation of the bubble, the evaluation of a deformable bubble without heat transfer is simulated by volume of fluid (VOF) method based on an axial symmetric Cartesian coordinate system. The simulations indicate that EHD can effectively enhance heat transfer rate of nucleate boiling by influencing the motion of the ring vortex around the bubble and that bubble can be elongated due to the pull in axial direction and push in the negative radial direction by the electric field force.     

Modelling gas injection of a Peirce–Smith-converter

A commercial CFD-code PHOENICS was used to solve isothermal flow field of gas and liquid in a Peirce–Smith-converter. An Euler–Euler based algorithm was chosen for modelling fluid dynamics and evaluating controlling forces of a submerged gas injection. Predictions were made with a k–ε turbulence model in the body fitted coordinate system. The model has been verified with a 1/4 scale water model, and a parametric study with the mathematical model of submerged gas injection was made for the PS-process and the ladle injection processes. Limits of the modelling technique used were recognised, but calculated results indicate that the present model predicts the general flow field with reasonable accuracy. Predicted bubble distribution, pattern of the flow field and magnitude of flow velocities were used to evaluate scaling factors of physical models and general flow conditions of an industrial PS-converter.     

Some results of high-temperature expansions for the ising model in an arbitrary magnetic field

The high-temperature expansions of the function ?(χ,α), viz., the Legendre transform of the specific logarithm of the statsum of the Ising model in an arbitrary external field, are investigated. The computed surface of the thermodynamic potential ? is well described by the scaling form of the similarity hypothesis. For lattices with second-neighbor interaction the calculations clearly indicate the weak nonuniversality of the critical behavior (the index γ for the fcc-2 and sc-2 lattices with coordination number q=18 is closer to the value 1.21 than to the standard value 1.25). It is noted that the bcc-2 lattice has a universal behavior, i.e., nonuniversality is observed for lattices with coordination number q?18. High-temperature series in temperature-magnetization variables are presented in the Appendix for three-space lattices with both first-neighbor as well as second-neighbor interactions, to within the ninth and the eighth orders, respectively.