How Emergent Models May Foster the Constitution of Formal Mathematics

This article deals with the role that so-called emergent models can play in the process of constituting formal mathematics. The underlying philosophy is that formal mathematics is something that is, or should be, constituted by the students themselves. In the instructional design theory for realistic mathematics education, models always have been employed to foster a process in which formal mathematics is reinvented by the students themselves. This article describes how the use of models became more and more explicated over time and developed into the notion of emergent models. The design of an instructional sequence, which deals with flexible mental computation strategies for addition and subtraction up to 100, is taken as an instance for elaborating what is meant by emergent models and what role they play in fostering the constitution of formal mathematics. The analysis shows that there are 3 interrelated processes. First. at a more holistic level, there is a global transition in which "the model" initially emerges as a model of informal mathematical activity and then gradually develops into a model for more formal mathematical reasoning. Second, the transition from "model of" to "model for" involves the constitution of anew mathematical reality that can be denoted formal in relation to the original starting points of the students. Third, in the series of instructional activities, there is not 1 model, but the model actually is shaped as a series of signs, in which each new sign comes to signify activity with a previous sign in a chain of signification.     

Mathematical models of suspension bridges

In this work we try to explain various mathematical models describing the dynamical behaviour of suspension bridges such as the Tacoma Narrows bridge. Our attention is concentrated on the derivation of these models, an interpretation of particular parameters and on a discussion of their advantages and disadvantages. Our work should be a starting point for a qualitative study of dynamical structures of this type and that is why we have a closer look at the models, which have not been studied in literature yet. We are also trying to find particular conditions for unique solutions of some models.     

Mathematical modelling of a hydraulic accumulator for hydraulic hybrid drives

ABSTRACT

Hydraulic accumulators are used as energy storages in a wide area of applications. In particular, in automotive hybrid drive-trains, this type of energy storage is an interesting alternative to today’s common strategies like chemical batteries or flywheels. This article deals with the mathematical modelling of a hydraulic accumulator for passenger vehicles, which comprises a carbon fibre reinforced plastic (CFRP) body and aluminium piston. The thermodynamical behaviour of the oil and gas as well as the interaction with the CFRP body is investigated in detail. Starting from a complex model, two models of reduced complexity are derived. The validation of these models with measurement data from a test drive with a prototype series hydraulic hybrid drive-train proves their high accuracy.     

Reduced-order modelling of self-excited,time-periodic systems using the method of Proper Orthogonal Decomposition and the Floquet theory

The mathematical models of dynamical systems become more and more complex, and hence, numerical investigations are a time-consuming process. This is particularly disadvantageous if a repeated evaluation is needed, as is the case in the field of model-based design, for example, where system parameters are subject of variation. Therefore, there exists a necessity for providing compact models which allow for a fast numerical evaluation. Nonetheless, reduced models should reflect at least the principle of system dynamics of the original model.

In this contribution, the reduction of dynamical systems with time-periodic coefficients, termed as parametrically excited systems, subjected to self-excitation is addressed. For certain frequencies of the time-periodic coefficients, referred to as parametric antiresonance frequencies, vibration suppression is achieved, as it is known from the literature. It is shown in this article that by using the method of Proper Orthogonal Decomposition (POD) excitation at a parametric antiresonance frequency results in a concentration of the main system dynamics in a subspace of the original solution space. The POD method allows to identify this subspace accurately and to set up reduced models which approximate the stability behaviour of the original model in the vicinity of the antiresonance frequency in a satisfying manner. For the sake of comparison, modally reduced models are established as well.     

Applying a one-shot and infinite repeated inspection game to materials management

An inspection game models a conflict situation between an inspector and an inspectee. The mathematical analysis aims to determine optimal behaviour of the inspectee. In this paper, the controller’s (inspector) job is to audit a planner’s (inspectee) decision with regard to its optimality and to submit a report to the company’s top managers. Based on Fandel and Trockel (Int J Prod Econ 133:256–261, 2011) this inspection game is discussed here as an infinite two-person game under the aspect of costs minimization. The results show that a higher costs deviation because of a competitive behaviour of the strategic players will decrease their payoffs, so that the optimal cooperative behaviour will finally lead to a better solution for the company with respect to cost minimization as normally expected.     

Using half-baked microworlds to challenge teacher educators’ knowing

This article illustrates how four teacher educators in training were challenged with respect to their epistemology and perceptions of teaching and learning mathematics through their interactions with expressive digital media during a professional development course. The research focused on their experience of communally constructing artifacts and their reflections on the nature of mathematics and mathematics teaching and learning with digital media. I discussed three different ways in which this media was used by the teachers; first, as a means to engage in technical-applied mathematics to engineer mathematical models; second, as a means to construct models for students to engage in experimental-constructivist activity; thirdly, as a means to engage in a discussion of a challenging mathematical problem.

Machine utilizations achieved using balanced FMS production ratios in a simulated setting

Stecke [21] has developed mathematical programming approaches for determining, from a set of part type requirements, the production ratios (part types to be produced next, and their proportions) which maximize overall machine utilizations by balancing machine workloads in a flexible manufacturing system (FMS). These mathematical programming (MP) approaches are aggregate in the sense that they do not take into account such things as contention for transportation resources, travel time for work-in-process, contention for machines, finite buffer space, and dispatching rules. In the current study, the sensitivity of machine utilizations to these aggregations is investigated through simulation modeling. For the situation examined, it is found that achieved machine utilizations are a strong function of some of the factors ignored in the MP methodology, ranging from 9.1% to 22.9% less than those theoretically attainable under the mathematical programming assumptions. The 9.1% degradation results from modeling with nonzero work-in-process travel times (i.e. 2 minutes per transfer) and using only central work-in-process buffers. Resource levels (e.g. the number of automated guided vehicles; the amount of work-in-process; the number of slack buffers) needed to limit the degradation to 9.1% correspond to FMS operating conditions which are feasible in practice.     

Stability Analysis of the Cambridge Ring

A ring of I cells rotates past I queues, carrying customers from their origins to their destinations. The system is modelled as a Markov chain, and the exact ergodicity conditions are given. They are shown to depend on the precise travel lengths distributions, that is, not only on their means. Ergodicity is proven through the stability analysis of the associated fluid limits. The arrivals distributions, which in the ergodicity conditions appear only through their means, are more subtly involved in the fluid limits behaviour, in that they determine the probabilities of random bifurcations that occur infinitely often in a simple system of I=2 queues.     

How Emergent Models May Foster the Constitution of Formal Mathematics

This article deals with the role that so-called emergent models can play in the process of constituting formal mathematics. The underlying philosophy is that formal mathematics is something that is, or should be, constituted by the students themselves. In the instructional design theory for realistic mathematics education, models always have been employed to foster a process in which formal mathematics is reinvented by the students themselves. This article describes how the use of models became more and more explicated over time and developed into the notion of emergent models. The design of an instructional sequence, which deals with flexible mental computation strategies for addition and subtraction up to 100, is taken as an instance for elaborating what is meant by emergent models and what role they play in fostering the constitution of formal mathematics. The analysis shows that there are 3 interrelated processes. First. at a more holistic level, there is a global transition in which “the model” initially emerges as a model of informal mathematical activity and then gradually develops into a model for more formal mathematical reasoning. Second, the transition from “model of” to “model for” involves the constitution of anew mathematical reality that can be denoted formal in relation to the original starting points of the students. Third, in the series of instructional activities, there is not 1 model, but the model actually is shaped as a series of signs, in which each new sign comes to signify activity with a previous sign in a chain of signification.     

Advanced Mathematical Thinking

In this article we propose the following definition for advanced mathematical thinking: Thinking that requires deductive and rigorous reasoning about mathematical notions that are not entirely accessible to us through our five senses. We argue that this definition is not necessarily tied to a particular kind of educational experience; nor is it tied to a particular level of mathematics. We also give examples to illustrate the distinction we make between advanced mathematical thinking and elementary mathematical thinking. In particular, we discuss which kind of thinking may be required depending on the size of a mathematical problem, including problems involving infinity, and the types of models that are available.     

Endogenous optimization fisheries models

This paper deals with a special class of fisheries models referred to as endogenous optimization models. The distinctive feature of these models is that behaviour of the agents in the model is not predetermined by exogenous behavioural rules. In endogenous optimization models, the model agents are merely furnished with objectives such as profit or utility maximization. Given these objectives and the various constraints determined by the state of the model at each point of time, the agents solve their maximization problem. The corresponding values of their control variables then constitute their behaviour.Having generated individual agents' behaviour by endogenous optimization, summing over agents yields aggregate behaviour. Aggregate behaviour must conform with the overall constraints of the model, be they physical or otherwise. Within the market system, individual behaviour or rather plans are made compatible via changes in relative prices. Therefore, outside equilibrium, behavioural plans must be repeatedly modified to become mutually compatible. This implies iteratively solving the maximization problem of a number of different agents. Endogenous optimization models therefore tend to be computationally very demanding.Clearly, the basic principles of endogenous optimization are just as applicable to any model involving maximizing agents.     

Modelling Intra-City Time-Dependent Travel Speeds for Vehicle Scheduling Problems

Many research papers have presented mathematical models for vehicle scheduling. Several of these models have been embedded in commercial decision support systems for intra-city vehicle scheduling for launderies, grocery stores, banks, express mail customers, etc. Virtually all of these models ignore the important issue of time-dependent travel speeds for intra-city travel. Travel speeds (and times) in nearly all metropolitan areas change drastically during the day because of congestion in certain parts of the city road network. The assumption of constant (time-independent) travel speeds seriously affects the usefulness of these models. This is particularly true when time windows (earliest and latest stop time constraints) and other scheduling issues are important. This research proposes a parsimonious model for time-dependent travel speeds and several approaches for estimating the parameters for this model. An example is presented to illustrate the proposed modelling approach. The issue of developing algorithms to find near-optimal vehicle schedules with time-dependent travel speeds is also discussed. The modelling approach presented in this paper has been implemented in a commercial courier vehicle scheduling system and was judged to be ‘very useful’ by users in a number of different metropolitan areas in the United States.     

Fast Statistical Surrogates for Dynamical 3D Computer Models of Brain Tumors

Understanding how malignant brain tumors are formed and evolve has direct consequences on the development of efficient methods for their early detection and treatment. Adequate mathematical models for brain tumor growth and invasion can be helpful in clarifying some aspects of the mechanism responsible for the tumor. These mathematical models are typically implemented in computer models, which can be used for computer experimentation to study how changes in inputs, such as growth and diffusion parameters, affect the evolution of the virtual brain tumor. The computer model considered in this article is defined on a three-dimensional (3D) anatomically accurate digital representation of the human brain, which includes white and gray matter, and on a time interval of hundreds of days to realistically simulate the tumor development. Consequently, this computer model is very computationally intensive and only small-size computer experiments can be conducted, corresponding to a small sample of inputs. This article presents a computationally efficient multidimensional kriging method to predict the evolution of the virtual brain tumor at new inputs, conditioned on the virtual brain tumor data available from the small-size computer experiment. The analysis shows that this prediction can be more accurate than a computationally competing model.     

Stochastic Models for Oil Prices and the Pricing of Futures on Oil

Abstract

In this article, we investigate and compare the performance of various one-factor diffusion models in their ability to capture the behaviour of Brent crude oil prices. New proposed models, which have a three-quarters power in the diffusion term, are found to outperform all other popular models tested. Analytic solutions for futures prices under the new models are found and used to calibrate market prices. Results from the calibration show that one of the new three-quarters models with a mean-reverting property outperforms other popular models in fitting and forecasting futures prices.     

Chaotic vibrations in nonlinear problems of bar structures

In the response of nonlinear mathematical models which describe vibrations of structural elements one could observe an irregular behaviour which is called chaos. Loss of the information on initial states in deterministic dynamical systems after a short time of theirs evolution, increasing amplitudes of displacements, velocities and accelerations, sensitive dependency on initial conditions makes chaos dangerous phenomenon in mechanics of construction. In this article quantitative (bifurcation diagrams, Poincare sections and Fourier power spectrum analysis) identication methods of the chaotic dynamics in geometrically nonlinear model of one DOF Mises truss are shown. Main goal of this article is to show and verify dangerous influence of chaos (in the engineering sense) on the analyzed structure. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Criteria sensitivity analysis: a new approach to parameter sensitivity

The problem of parameter sensitivity is not well defined. This paper aims to make it well defined. This is done by requiring the modeller to provide a performance indicator l(X,t,β) which is a measure of how the behaviour of the system is to be judged. This forces the modeller to be precise about what he/she regards as acceptable behaviour. Mathematically we define behaviour to be acceptable at β if l >0 for that β, and we define the system as sensitive if the set of acceptable β's is small. The analysis which is required is termed criteria sensitivity analysis. This method is applicable to many different kinds of models. The basic ideas of criteria sensitivity analysis are presented here. A static model is used to demonstrate major points. The power of this new approach is illustrated by an application to a dynamic model.     

Discrete vs. continuous time implementation of a ring car following model in which overtaking is allowed

Car following models seek to describe the interactions between individual vehicles as they move along a stretch of road where the behaviour of each vehicle is dependent on the motion of the vehicle directly in front and overtaking is typically not permitted. In this work we study a modified version of the traditional car following model in which the vehicles are travelling on a closed loop and the ‘no overtaking’ restriction has been removed. The resulting model is described firstly in terms of a set of coupled continuous time delay differential equations and then in terms of their discrete time equivalents and both forms of the model are then solved numerically to analyse their post transient behaviour under a periodic perturbation. For certain parameter choices both the continuous and discrete forms of the model can exhibit chaotic behaviour but a comparison of the behaviour of the two models over a wide range of parameter values shows that the discretization can dramatically affect the type of post transient behaviour exhibited. This becomes increasingly evident as the time step used in the discrete time model is increased.     

Environmental considerations on the optimal product mix

Several types of regulations limit the amount of different emissions that a firm may create from its production processes. Depending on the emission, these regulations could include threshold values, penalties and taxes, and/or emission allowances that can be traded. However, many firms try to comply with these regulations without a systematic plan, often leading not only to emission violations and high penalties, but also to high costs. In this paper, we present two mathematical models that can be used by firms to determine their optimal product mix and production quantities in the presence of several different types of environmental constraints, in addition to typical production constraints. Both models are comprehensive and incorporate several diverse production and environmental issues. The first model, which assumes that each product has just one operating procedure, is a linear program while the second model, which assumes that the firm has the option of producing each product using more than one operating procedure, is a mixed integer linear program. The solutions of both models identify the products that the firm should produce along with their production quantities. These models can be used by firms to quickly analyze several “what if” scenarios such as the impact of changes in emission threshold values, emission taxes, trading allowances, and trading transaction costs.     

Transmission dynamics of a switched multi-city model with transport-related infections

Multi-city epidemic models with unrestricted travel, transport-related infection, general nonlinear incidence rate, and seasonality are analyzed. First, a multi-city SIR model is investigated. Seasonality is considered by assuming that the model’s parameters are time-varying and switching. Under this construction, the parameters can be smoothly-varying (for example, due to seasonal changes) or abruptly-varying (for example, due to school holiday breaks). The functional form of the incidence rate is assumed to take a general form that can change in time (for example, due to changes in population behaviour). The effects of transport-related infection and time-varying parameters are studied and some threshold conditions are established which guarantee that the disease-free solution is globally attractive. A screening process and pulse control strategies are applied to the multi-city SIR model in order to investigate and compare the benefits of each strategy. In the pulse control scheme, vaccine failure is considered and the inter-pulse period is not required to equal the seasonal period of the model parameters. Finally, some simulations are given as well as conclusions and future directions.     

A generalization of Norton's theorem for queueing networks

A general framework for aggregation and decomposition of product form queueing networks with state dependent routing and servicing is presented. By analogy with electrical circuit theory, the stations are grouped into clusters of subnetworks such that the process decomposes into a global process and a local process. Moreover, the local process factorizes into the subnetworks. The global process and the local processes can be analyzed separately as if they were independent. The global process describes the behaviour of the queuing network in which each cluster is aggregated into a single station, whereas the local process describes the behaviour of the subnetworks as if they are not part of the queueing network. The decomposition and aggregation method formalized in this paper allows us to first analyze the global behaviour of the queueing network and subsequently analyze the local behaviour of the subnetworks of interest or to aggregate clusters into single stations without affecting the behaviour of the rest of the queueing network. Conditions are provided such that:

- the global equilibrium distribution for aggregated clusters has a product form;

- this form can be obtained by merely monitoring the global behaviour;

- the computation of a detailed distribution, including its normalizing constant, can be decomposed into the computation of a global and a local distribution;

- the marginal distribution for the number of jobs at the stations of a cluster can be obtained by merely solving local behaviour.

As a special application, Norton's theorem for queueing networks is extended to queueing networks with state dependent routing such as due to capacity constraints at stations or at clusters of stations and state dependent servicing such as due to service delays for clusters of stations.