The Use of the Concept of Entropy in System Modelling

The concept of "entropy" is widely used in the physical sciences and, recently, has aroused interest in the social sciences, especially in building models of urban and regional systems to be used for planning purposes. Most papers referring to the concept focus on the particular model being developed; this paper focuses on the concept itself. The first section is an introductory one which explains the use of system models in planning processes. In the second section, four views are presented of entropy: as related to probability and uncertainty; as a statistic of a probability distribution; in relation to Bayesian inference; and as a measurable system property. In the third part of the paper, the types of application of the concept are discussed in turn, and a number of other applications of the concept of "entropy" outside the usual physical science range are noted. The final section is a summary and concluding discussion, including a comparison of the approach of the statistician and entropy maximizer to system modelling.     

PARADIGM SHIFTS IN FISHERIES STOCK ASSESSMENT: FROM INTEGRATED ANALYSIS TO BAYESIAN ANALYSIS AND BACK AGAIN

ABSTRACT. I describe integrated analysis and Bayesian analysis, which have been two of the most influential paradigms in fisheries stock assessment during the last two decades of the twentieth century. These two paradigms have generally been considered complementary, rather than competing. However, recent advances in integrated analysis, including the special case of meta‐analysis, have made Bayesian analysis somewhat redundant. I describe how data used to create priors for use in Bayesian analysis can be integrated directly into the analyses. This provides a much more convenient way of accurately including the information and associated uncertainty into the analyses. I discuss how there is still a need to describe the uncertainty and suggest that research should focus on the most appropriate methods for doing this.     

Being a mathematics teacher in times of reform

Within research on mathematics teachers and/or their professional development, the concept of identity emerges as a critique of views of how teaching practice is related to teachers’ ‘internal states’ of knowledge and beliefs. Identity relates teachers’ professional lives to teaching practices and to the contexts in which the teaching and/or professional development occurs. However, what might count as the context still needs in-depth discussion. In order to contribute to the development of a theoretical framework for understanding mathematics teachers’ professional lives, we will draw on one remarkable teacher’s identity as a primary mathematics teacher in relation to one political, sociocultural, and pedagogical context. We use this teacher’s experience to discuss how education policies that create what Ball (2003) called ‘terrors of performativity’ tend to impede the formation of a balanced teacher identity.     

A critique of statistical modelling in management science from a critical realist perspective: its role within multimethodology

Management science was historically dominated by an empiricist philosophy that saw quantitative modelling and statistical analysis as the only legitimate research method. More recently interpretive or constructivist philosophies have also developed employing a range of non-quantitative methods. This has sometimes led to divisive debates. ‘Critical realism’ has been proposed as a philosophy of science that can potentially provide a synthesis in recognizing both the value and limitations of these approaches. This paper explores the critical realist critique of quantitative modelling, as exemplified by multivariate statistics, and argues that its grounds must be re-conceptualized within a multimethodological framework.     

A multi-objective evolutionary algorithm for facility dispersion under conditions of spatial uncertainty

Avoiding concentration or saturation of activities is fundamental in many environmental and urban planning contexts. Examples include dispersing retail and restaurant outlets, sensitivity to impacts in forest utilization, spatial equity of waste disposal, ensuring public safety associated with noxious facilities, and strategic placement of military resources, among others. Dispersion models have been widely applied to ensure spatial separation between activities or facilities. However, existing approaches rely on deterministic approaches that ignore issues of spatial data uncertainty, which could lead to poor decision making. To address data uncertainty issues in dispersion modelling, a multi-objective approach that explicitly accounts for spatial uncertainty is proposed, enabling the impacts of uncertainty to be evaluated with statistical confidence. Owing to the integration of spatial uncertainty, this dispersion model is more complex and computationally challenging to solve. In this paper we develop a multi-objective evolutionary algorithm to address the computational challenges posed. The proposed heuristic incorporates problem-specific spatial knowledge to significantly enhance the capability of the evolutionary algorithm for solving this problem. Empirical results demonstrate the performance superiority of the developed approach in supporting facility and service planning.     

Functions: a modelling tool in mathematics and science

It is difficult for the students to transfer concepts, ideas and procedures learned in mathematics to a new and unanticipated situation in science. An alternative to this traditional transfer method stresses the importance of modelling activities in an interdisciplinary context between mathematics and science. In the paper we introduce a modelling approach to the concept of function in upper secondary school is introduced. We discuss pedagogical and didactical issues concerning the interplay between mathematics and science. The discussion is crystallized into a didactical model for interdisciplinary instruction in mathematics and science. The model is considered as an iterative movement with two phases: (1) the horizontal linking of the subjects: Situations from science are embedded in contexts which are mathematized by the students, (2) the vertical structuring in the subjects: The conceptual anchoring of the students' constructs from the horizontal linking in the systematic and framework of mathematics and science respectively.     

Independence for full conditional probabilities: Structure,factorization, non-uniqueness,and Bayesian networks

This paper examines concepts of independence for full conditional probabilities; that is, for set-functions that encode conditional probabilities as primary objects, and that allow conditioning on events of probability zero. Full conditional probabilities have been used in economics, in philosophy, in statistics, in artificial intelligence. This paper characterizes the structure of full conditional probabilities under various concepts of independence; limitations of existing concepts are examined with respect to the theory of Bayesian networks. The concept of layer independence (factorization across layers) is introduced; this seems to be the first concept of independence for full conditional probabilities that satisfies the graphoid properties of Symmetry, Redundancy, Decomposition, Weak Union, and Contraction. A theory of Bayesian networks is proposed where full conditional probabilities are encoded using infinitesimals, with a brief discussion of hyperreal full conditional probabilities.     

Time traps in supply chains: Is optimal still good enough?

Operations Researchers support Supply Chain Management and Supply Chain Planning by developing adequate mathematical optimization models and providing suitable solution procedures. In this paper we discuss what adequate could mean. Therefore, we may ask several questions concerning “optimality” in Supply Chain Planning under causal and temporal uncertainty: What is an optimal solution? When is it optimal? For how long is it optimal? How should the design of a supply chain be changed when conditions and requirements ask for new structures? In particular, we discuss new approaches to Supply Chain Planning in order to give an optimal transformation from an initial solution over multiple periods to a desired one rather than just specifying an optimal snapshot solution. Time and uncertainty are the factors triggering the whole discussion. In particular, several flaws often found when dealing with these factors result in so-called “time traps”. We look at the impact of recent technological developments like the Internet of Things or Industry 4.0 on operational supply chain planning and control, and we show how online optimization can help to cope with real-time challenges. Moreover, we re-coin the concept of risk in the realm of Supply Chain Planning. Here the question is how to measure supply chain specific risks and how to incorporate them “adequately” into mathematical models.     

An examination for mathematical modelling

First year mathematics degree students at Leicester Polytechnic attend a course in mathematical modelling. The aim is to introduce the students to mathematical modelling concepts and to model development. Work is set and marked during the course and this forms a vital part of the students' assessment. In addition to this, however, the students are assessed by means of a three hour examination at the end of the year. This examination is significantly different from the normal ‘five out of eight’ type. The philosophy and organization of the examination are discussed in this paper. An example of a particular examination, that for June 1986, is included as an appendix to illustrate the points made in the discussion.     

Language and communication in mathematics education: an overview of research in the field

Within the field of mathematics education, the central role language plays in the learning, teaching, and doing of mathematics is increasingly recognised, but there is not agreement about what this role (or these roles) might be or even about what the term ‘language’ itself encompasses. In this issue of ZDM, we have compiled a collection of scholarship on language in mathematics education research, representing a range of approaches to the topic. In this survey paper, we outline a categorisation of ways of conceiving of language and its relevance to mathematics education, the theoretical resources drawn upon to systematise these conceptions, and the methodological approaches employed by researchers. We identify four broad areas of concern in mathematics education that are addressed by language-oriented research: analysis of the development of students’ mathematical knowledge; understanding the shaping of mathematical activity; understanding processes of teaching and learning in relation to other social interactions; and multilingual contexts. A further area of concern that has not yet received substantial attention within mathematics education research is the development of the linguistic competencies and knowledge required for participation in mathematical practices. We also discuss methodological issues raised by the dominance of English within the international research community and suggest some implications for researchers, editors and publishers.     

Evaluating the Effects of Privatizing Electricity

In this text of the 1993 Blackett Memorial Lecture, a critical assessment of the effects of electricity privatization in England and Wales is Undertaken, with a particular emphasis upon understanding the new economic and strategic forces in the market. This leads to some discussion of the type of modelling approaches that can provide insight into these processes. In particular, a key role for business simulation, alongside optimization methods and economic analysis, is identified as a basis for an effective regulatory framework to manage the transition to more efficient market competition. Emphasis is placed upon recognizing the impacts of risk and competition on the decision-making of the new companies in the market, and upon analysing the ‘short-termism’, the ‘dash-for-gas’, the demise of the coal industry, consumer prices and the nuclear issues in these terms.     

Frege,Dedekind, and the Modern Epistemology of Arithmetic

In early analytic philosophy, one of the most central questions concerned the status of arithmetical objects. Frege argued against the popular conception that we arrive at natural numbers with a psychological process of abstraction. Instead, he wanted to show that arithmetical truths can be derived from the truths of logic, thus eliminating all psychological components. Meanwhile, Dedekind and Peano developed axiomatic systems of arithmetic. The differences between the logicist and axiomatic approaches turned out to be philosophical as well as mathematical. In this paper, I will argue that Dedekind’s approach can be seen as a precursor to modern structuralism and as such, it enjoys many advantages over Frege’s logicism. I also show that from a modern perspective, Frege’s criticism of abstraction and psychologism is one-sided and fails against the psychological processes that modern research suggests to be at the heart of numerical cognition. The approach here is twofold. First, through historical analysis, I will try to build a clear image of what Frege’s and Dedekind’s views on arithmetic were. Then, I will consider those views from the perspective of modern philosophy of mathematics, and in particular, the empirical study of arithmetical cognition. I aim to show that there is nothing to suggest that the axiomatic Dedekind approach could not provide a perfectly adequate basis for philosophy of arithmetic.     

New advances in economic modelling and evaluation of environmental issues

This paper gives an overview of recent progress made in modelling economic environmental systems and in environmental policy analysis. In the modelling part attention will be given to new integrating frameworks offered inter alia by materials balance approaches, especially in the context of linkages between physical environmental phenomena and economic production and valuation. These can be relevant for studying materials-product chains, multisectoral materials flows, or even multiple use of complex ecosystems. Modern approaches will be dealt with, such as analysis for sustainable development, and ways of incorporating scenario experiments in environmental modelling approaches. In the context of sustainable development, modelling of multiple use of ecosystems and of spatial dimensions is also discussed. In the last part of the paper new advances in the area of environmental policy analysis will be dealt with. The main focus will be on methods for addressing uncertainty in evaluating environmental policy strategies, in particular fuzzy information and the use of meta-analysis.     

Pointwise Efficiency in Multiobjective Stochastic Linear Programming

In most of the approaches to the Multiobjective Stochastic Linear Programming problem that have been proposed in the literature, the notion of quality of a solution is not adequately defined. We reconsider this problem from a decision point of view, in contexts where either the decision maker's preference structure cannot be described by a utility function, or where this structure is expressed by an unknown non-decreasing utility function and the probability distribution of the random parameters is unknown. We define a fundamental set of ‘pointwise admissible’ solutions, as well as several subsets of particular interest. We discuss the relevance of these various pointwise efficient sets, their interrelations, and their practical identification.     

Mathematical modelling in classroom: a socio-critical and discursive perspective

In this paper, I outline a socio-critical perspective of modelling in mathematics education and discuss implications for analysis of students' activities at the micro level. In particular, a discursive perspective is presented with contributions from discursive psychology. Recent studies and classroom examples are taken into consideration.     

Spatially varying SAR models and Bayesian inference for high-resolution lattice data

We discuss a new class of spatially varying, simultaneous autoregressive (SVSAR) models motivated by interests in flexible, non-stationary spatial modelling scalable to higher dimensions. SVSAR models are hierarchical Markov random fields extending traditional SAR models. We develop Bayesian analysis using Markov chain Monte Carlo methods of SVSAR models, with extensions to spatio-temporal contexts to address problems of data assimilation in computer models. A motivating application in atmospheric science concerns global CO emissions where prediction from computer models is assessed and refined based on high-resolution global satellite imagery data. Application to synthetic and real CO data sets demonstrates the potential of SVSAR models in flexibly representing inhomogeneous spatial processes on lattices, and their ability to improve estimation and prediction of spatial fields. The SVSAR approach is computationally attractive in even very large problems; computational efficiencies are enabled by exploiting sparsity of high-dimensional precision matrices.     

The Basic Theorem of triadic concept analysis

Experiences with applications of concept lattices and the pragmatic philosophy founded by Ch. S. Peirce have suggested a triadic approach to formal concept analysis. It starts with the notion of atriadic context combiningobjects, attributes, andconditions under which objects may have certain attributes.The Basic Theorem of triadic concept analysis clarifies the class of structures which are formed by thetriadic concepts of triadic contexts: These structures are exactly thecomplete trilattices up to isomorphism.     

Analysis of psychometric properties as part of an iterative adaptation process of MKT items for use in other countries

Researchers at the University of Michigan have developed sets of items that can be used to analyze teachers’ mathematical knowledge for teaching (MKT). In this paper, we consider what is required in the adaptation of a set of these items for use in a Norwegian context. We discuss how analysis of item difficulty and point–biserial correlation can be applied in combination with qualitative approaches to ensure a high-quality process of piloting adapted MKT items. Findings indicate that researchers who attempt to adapt MKT items for use in cultural contexts other than those for which they were designed need to use different methods to analyze all aspects of the adaptation process. The results from the different analyses conducted might then be used to inform other parts of the process, and this will mean that the process of adapting and piloting items becomes cyclic and iterative.     

Modelling and analysis of multistage stochastic programming problems: A software environment

In many decision problems, time and uncertainty play important roles and stochastic programming models are well suited for capturing both these aspects. In this paper we focus our attention on the difficulties which arise in developing multi-period stochastic models and we outline those challenging aspects where the use of classical modelling techniques for deterministic problems prove to be inadequate. In particular, we discuss the development of a modelling and analysis environment which combines multidimensional databases (MDDB), declarative modelling languages and procedural languages. Our aim is to develop a versatile tool which generates multi-period stochastic models and supports the modeller in browsing of data and solutions across different time stages and among different scenarios.