Fuzzy concepts and formal methods

It has been recognised that formal methods are useful as a modelling tool in requirements engineering. Specification languages such as Z permit the precise and unambiguous modelling of system properties and behaviour. However some system problems, particularly those drawn from the information systems problem domain, may be difficult to model in crisp or precise terms. It may also be desirable that formal modelling should commence as early as possible, even when our understanding of parts of the problem domain is only approximate. This thesis suggests fuzzy set theory as a possible representation scheme for this imprecision or approximation. A fuzzy logic toolkit that defines the operators, measures and modifiers necessary for the manipulation of fuzzy sets and relations is developed. The toolkit contains a detailed set of laws that demonstrate the properties of the definitions when applied to partial set membership. It also provides a set of laws that establishes an isomorphism between the toolkit notation and that of conventional Z when applied to boolean sets and relations. The thesis also illustrates how the fuzzy logic toolkit can be applied in the problem domains of interest. Several examples are presented and discussed including the representation of imprecise concepts as fuzzy sets and relations, system requirements as a series of linguistically quantified propositions, the modelling of conflict and agreement in terms of fuzzy sets and the partial specification of a fuzzy expert system. The thesis concludes with a consideration of potential areas for future research arising from the work presented here.

Mathematics as a second language : looking for a bridge between mathematical language and the language of thought

Our thoughts are in one language, and mathematical results are expressed in a language foreign to the way we think. Mathematics is a unique foreign language with all the components of a language; it has its own grammar, vocabulary, conventions, synonyms, sentence structure, and paragraph structure. Students need to learn these components to partake in a thorough discussion of how to read, write, speak and think mathematics. Beginning with the students natural language and expanding that language to include symbolism and logic is the key. Providing lessons in concrete, pictorial, written and verbal terms allows the instructor to create a translation bridge between the grammar of the mother language and the grammar of mathematics. This papers presents methods to create the translation bridge for students so that they become articulate members of the mathematics community. The students "mother" language, expanded to include the symbols of mathematics and logic, is the the key to both the learning of mathematics and its effective application to problem situations. The use of appropriate language is the key to making mathematics understandable.

Interpersonal trust and formal controls in hierarchical relationships

This paper examines the influence of the level of interpersonal trust between superior and subordinate managers on the control behaviour of the former. On the basis of a questionnaire survey and interviews of senior managers from business organisations in Sri Lanka, and a survey of managers in Beijing-China the study explores the control behaviour of superior managers when their trust in a subordinate is high or low. Sri Lanka and China, societies in which the dependence on interpersonal trust is believed to be high, were chosen for the study to maximise the effect of interpersonal trust.

The findings of this study indicate that a superior’s high (low) trust in a subordinate is associated with a low (high) level of monitoring and a high (low) level of social interactions. The hypothesis that a superior’s high (low) level of trust is associated with a low (high) level of reliance on formal control was supported only in the Sri Lankan sample. These findings are at least indicative of control behaviour of superior managers in Sri Lanka and China and possibly of other countries in Asia. An understanding of the trust-sensitive control behaviour of managers in this region is particularly important for designing and implementing effective control systems for international organizations operating in the region.

Associations between student pursuit of novel mathematical ideas and resilience

As part of a broader study of characteristics of situations that promote or inhibit spontaneous student exploration of novel mathematical ideas, student inclination to display these spontaneous behaviours was studied in conjunction with resilience. Resilience was operationalised using the dimensions of optimism (Seligman, 1995). Indicators of optimism were identified in post-lesson video-stimulated reconstructive interviews with year eight students. Students who demonstrated the pursuit of novel mathematical ideas were found to also display indicators of resilience.

Interpersonal trust and formal controls in low systemic trust societies

This study is concerned with the role of interpersonal trust in management control. On the basis of a questionnaire survey and interviews of senior managers from business organisations in sri lanka, the study explores the control behaviour of superior managers when they trust or distrust their subordinates. Sri lanka, a society in which the dependence on interpersonal trust is high, was chosen for the study to maximise the effect of interpersonal trust.

The findings of this study indicate that a superior's high trust in a subordinate is associated with a low level of monitoring, a high level of social interactions, and a low reliance on formal controls. In contrast, a superior's low level of trust in a subordinate is associated with a high level of monitoring, a low level of social interactions, and a high reliance on formal controls. Because the data emanate from experienced senior managers, these findings are at least indicative of control behaviour of superior managers in sri lanka and possibly of similar countries in asia. An understanding of the control behaviour of managers in this region is particularly important for designing and implementing effective controls systems for firms, subsidiaries, branches or joint-ventures operating in the region.

Development of DAL and DAPL languages for building distributed applications

A common characteristic among parallel/distributed programming languages is that the one language is used to specify not only the overall organisation of the distributed application, but also the functionality of the application. That is, the connectivity and functionality of processes are specified within a single program. Connectivity and functionality are independent aspects of a distributed application. This thesis shows that these two aspects can be specified separately, therefore allowing application designers to freely concentrate on either aspect in a modular fashion. Two new programming languages have been developed for specifying each aspect. These languages are for loosely coupled distributed applications based on message passing, and have been designed to simplify distributed programming by completely removing all low level interprocess communication. A suite of languages and tools has been designed and developed. It includes the two new languages, parsers, a compilation system to generate intermediate C code that is compiled to binary object modules, a run-time system to create, manage and terminate several distributed applications, and a shell to communicate with the run-tune system. DAL (Distributed Application Language) and DAPL (Distributed Application Process Language) are the new programming languages for the specification and development of process oriented, asynchronous message passing, distributed applications. These two languages have been designed and developed as part of this doctorate in order to specify such distributed applications that execute on a cluster of computers. Both languages are used to specify orthogonal components of an application, on the one hand the organisation of processes that constitute an application, and on the other the interface and functionality of each process. Consequently, these components can be created in a modular fashion, individually and concurrently. The DAL language is used to specify not only the connectivity of all processes within an application, but also a cluster of computers for which the application executes. Furthermore, sub-clusters can be specified for individual processes of an application to constrain a process to a particular group of computers. The second language, DAPL, is used to specify the interface, functionality and data structures of application processes. In addition to these languages, a DAL parser, a DAPL parser, and a compilation system have been designed and developed (in this project). This compilation system takes DAL and DAPL programs to generate object modules based on machine code, one module for each application process. These object modules are used by the Distributed Application System (DAS) to instantiate and manage distributed applications. The DAS system is another new component of this project. The purpose of the DAS system is to create, manage, and terminate many distributed applications of similar and different configurations. The creation procedure incorporates the automatic allocation of processes to remote machines. Application management includes several operations such as deletion, addition, replacement, and movement of processes, and also detection and reaction to faults such as a processor crash. A DAS operator communicates with the DAS system via a textual shell called DASH (Distributed Application SHell). This suite of languages and tools allowed distributed applications of varying connectivity and functionality to be specified quickly and simply at a high level of abstraction. DAL and DAPL programs of several processes may require a few dozen lines to specify as compared to several hundred lines of equivalent C code that is generated by the compilation system. Furthermore, the DAL and DAPL compilation system is successful at generating binary object modules, and the DAS system succeeds in instantiating and managing several distributed applications on a cluster.

Logic and The Open Society : revising the place of Tarski’s theory of truth within Popper’s political philosophy

This chapter retraces the way in which the Austrian philosopher Sir Karl Popper came to accept a Correspondence Theory of Truth from the work of the Polish logician and mathematician Alfred Tarski. It is argued that Popper’s use of Tarski’s semantic theory of truth reveals crucial insights into the fundamental characteristics of Popper’s social philosophy.  Quite deceptively, arguments based upon Tarski’s theory of truth appear implicitly throughout the text of The Open Society and Its Enemies (1945). It is then demonstrated how Popper integrated a correspondence theory of truth into a theory of the functions of communicative language that he received from Karl Bühler.

Deep bed filtration : mathematical models and observations

Numerous mathematical models have been developed to evaluate both initial and transient stage removal efficiency of deep bed filters. Microscopic models either using trajectory analysis or convective diffusion equations were used to compute the initial removal efficiency. These models predicted the removal efficiency under favorable filtration conditions quantitatively, but failed to predict the removal efficiency under unfavorable conditions. They underestimated the removal efficiency under unfavorable conditions. Thus, semi-empirical formulations were developed to compute initial removal efficiencies under unfavorable conditions. Also, correction for the adhesion of particles onto filter grains improved the results obtained for removal efficiency from the trajectory analysis. Macroscopic models were used to predict the transient stage removal efficiency of deep bed filters. The O’Melia and Ali1 model assumed that the particle removal is due to filter grains as well as the particles that are already deposited onto the filter grain. Thus, semi-empirical models were used to predict the ripening of filtration. Several modifications were made to the model developed by O’Melia and Ali to predict the deterioration of particle removal during the transient stages of filtration. Models considering the removal of particles under favorable conditions and the accumulation of charges on the filter grains during the transient stages were also developed. This article evaluates those models and their applicability under different operating conditions of filtration.

Surface coverage of filter medium in deep bed filtration : mathematical modelling and experiments

This paper highlights the importance of surface coverage in modeling the removal of particles in deep bed filtration. A model that considers the saturation of sites on which particle deposition occurs is used. Experimental results obtained with monodispersed suspensions of 0.46 and 0.816 μm latex particles at different influent concentrations and ionic strengths were used to calculate the fraction of filter grain surface (β1) on which actual particle deposition occurs. This will be useful in evaluating the filter performance in terms of the utilization of available surface area of the filter medium. Further, the level of dendrite formation of particles on filter grains during filtration is expressed in terms of β1 and the specific surface coverage, θT (the fraction of a filter grain surface that is covered by particles at time T, assuming that the filter grain is covered by a monolayer of particles). This can be used to compare the contribution of deposited particles in the removal efficiency of deep bed filtration for suspensions with different physical and chemical characteristics.

Mathematical tasks and learning goals: examples from Japanese lesson study

This paper explores how to select, or design, the best mathematical task for a given learning goal. Examples are taken from a recent project in Victorian primary schools that employed Japanese lesson study as the means to provide teachers with professional learning within their own classrooms. The discussions by participating teachers and researchers provide some insights into the difficulties and solutions facing teachers intending to improve the critical thinking skills of their pupils. Examples of tasks and goals are provided.