The Mathematics of Control Theory

This text contains papers presented at the Institute of Mathematics and its Applications Conference on Control Theory, held at the University of Strathclyde in Glasgow. The contributions cover a wide range of topics of current interest to theoreticians and practitioners including algebraic systems theory, nonlinear control systems, adaptive control, robustness issues, infinite dimensional systems, applications studies and connections to mathematical aspects of information theory and data-fusion.

Mathematics as a tool in the Life Sciences

Teaching mathematics to students in the biological sciences is often fraught with difficulty. Students often discover mathematics to be a very 'dry' subject in which it is difficult to see the motivation of learning it given its often abstract application. In this paper I advocate the use of mathematical modelling as a method for engaging students in understanding the use of mathematics in helping to solve problems in the Biological Sciences. The concept of mathematics as a laboratory tool is introduced and the importance of presenting students with relevant, real-world examples of applying mathematics in the Biological Sciences is discussed.

Processing empty categories in a second language: When naturalistic exposure fills the (intermediate) gap

An ongoing debate on second language (L2) processing revolves around whether or not L2 learners process syntactic information similarly to monolinguals (L1), and what factors lead to a native-like processing. According to the Shallow Structure Hypothesis (Clahsen & Felser, 2006a), L2 learners’ processing does not include abstract syntactic features, such as intermediate gaps of wh-movement, but relies more on lexical/semantic information. Other researchers have suggested that naturalistic L2 exposure can lead to native-like processing (Dussias, 2003). This study investigates the effect of naturalistic exposure in processing wh-dependencies. Twenty-six advanced Greek learners of L2 English with an average nine years of naturalistic exposure, 30 with classroom exposure, and 30 native speakers of English completed a self-paced reading task with sentences involving intermediate gaps. L2 learners with naturalistic exposure showed evidence of native-like processing of the intermediate gaps, suggesting that linguistic immersion can lead to native-like abstract syntactic processing in the L2.

Mathematics applied to the climate system: outstanding challenges and recent progress

The societal need for reliable climate predictions and a proper assessment of their uncertainties is pressing. Uncertainties arise not only from initial conditions and forcing scenarios, but also from model formulation. Here, we identify and document three broad classes of problems, each representing what we regard to be an outstanding challenge in the area of mathematics applied to the climate system. First, there is the problem of the development and evaluation of simple physically based models of the global climate. Second, there is the problem of the development and evaluation of the components of complex models such as general circulation models. Third, there is the problem of the development and evaluation of appropriate statistical frameworks. We discuss these problems in turn, emphasizing the recent progress made by the papers presented in this Theme Issue. Many pressing challenges in climate science require closer collaboration between climate scientists, mathematicians and statisticians. We hope the papers contained in this Theme Issue will act as inspiration for such collaborations and for setting future research directions.

Predicting long-term growth in students' mathematics achievement: the unique contributions of motivation and cognitive strategies

This research examined how motivation (perceived control, intrinsic motivation, and extrinsic motivation), cognitive learning strategies (deep and surface strategies), and intelligence jointly predict long-term growth in students' mathematics achievement over 5 years. Using longitudinal data from six annual waves (Grades 5 through 10; Mage = 11.7 years at baseline; N = 3,530), latent growth curve modeling was employed to analyze growth in achievement. Results showed that the initial level of achievement was strongly related to intelligence, with motivation and cognitive strategies explaining additional variance. In contrast, intelligence had no relation with the growth of achievement over years, whereas motivation and learning strategies were predictors of growth. These findings highlight the importance of motivation and learning strategies in facilitating adolescents' development of mathematical competencies.

Interaction between grammatical categories and cognition in bilinguals: The role of proficiency, cultural immersion, and language of instruction

Previous studies have demonstrated that there is a tight link between grammatical concepts and cognitive preferences in monolingual speakers (Lucy 1992, Lucy & Gaskins 2003, Imai & Gentner 1997, Imai & Mazuka 2003). Recent research has also shown that bilinguals with languages that differ in their concepts may shift their cognitive preferences as a function of their proficiency (Athanasopoulos, 2006) or cultural immersion (Cook, Bassetti, Kasai, Sasaki, & Takahashi, 2006). The current short paper assesses the relative impact of each of these variables, and furthermore asks whether bilinguals alternate between two distinct cognitive representations of language-specific concepts depending on the language used in the experiment. Results from an object classification task showed that Japanese–English bilinguals shifted their behaviour towards the second language (L2) pattern primarily as a function of their L2 proficiency, while cultural immersion and language of instruction played a minimal role. These findings suggest that acquisition of novel grammatical categories leads to cognitive restructuring in the bilingual mind and have implications for the relationship between language and cognitive processing.

Diversity in mathematics assessment equals diversity in opinion?

We undertook a study to investigate the views of both students and staff in our department towards assessment in mathematics, as a precursor to considering increasing the diversity of assessment types. In a survey and focus group there was reasonable agreement amongst the students with regards major themes such as mode of assessment. However, this level of agreement was not seen amongst the staff, where discussions regarding diversity in mathematics assessment definitely revealed a difference of opinion. As a consequence, we feel that the greatest barriers to increasing diversity may be with staff, and so more efforts are needed to communicate to staff the advantages and disadvantages, in order to give them greater confidence in trying a range of assessment types.

Relationship between classroom authority and epistemological beliefs as espoused by primary school mathematics teachers from the very high and very low socio-economic regions in Thailand

This article presents findings of a larger single-country comparative study which set out to better understand primary school teachers’ mathematics education-related beliefs in Thailand. By combining the interview and observation data collected in the initial stage of this study with data gathered from the relevant literature, the 8-belief / 22-item ‘Thai Teachers’ Mathematics Education-related Beliefs’ (TTMEB) Scale was developed. The results of the Mann-Whitney U Test showed that Thai teachers in the two examined socio-economic regions espouse statistically different beliefs concerning the source and stability of mathematical knowledge, as well as classroom authority. Further, these three beliefs are found to be significantly and positively correlated.

Inclusive approaches to teaching and learning mathematics

This chapter explores the role of mentors in supporting pre-service teachers to include all children in mathematics teaching, no matter what their individual needs.

Interregional migration 'wage premia': the case of creative and science and technology graduates in the UK

We analyze the migration behavior of graduates from UK universities with a focus on the salary benefits they receive from the migration process. We focus on sequential interregional migration and specifically examine the case of Science, Technology, Engineering and Mathematics (STEM) and Creative subject graduates. Our analysis differs from previous studies in that it accounts explicitly for migrant selectivity through propensity score matching, and it also classifies graduates into different migration behavior categories. Graduates were classified according to their sequential migration behavior first from their pre-university domicile to university and then from university to first job post-graduation. Our results show that ‘repeat migration’, as expected, is associated with the highest wage premium (around 15%). Other migration behaviors are also advantageous although this varies across different types of graduates. Creative graduates, for instance, do not benefit much from migration behaviors other than repeat migration. STEM graduates, on the contrary, benefit from both late migration and staying in the university area to work.

Categories, concepts and units: representing energy demand in and through time

Methods of data collection are unavoidably rooted in some sort of theoretical paradigm, and are inextricably tied to an implicit agenda or broad problem framing. These prior orientations are not always explicit, but they matter for what data is collected and how it is used. They also structure opportunities for asking new questions, for linking or bridging between existing data sets and they matter even more when data is re-purposed for uses not initially anticipated. In this paper we provide an historical and comparative review of the changing categories used in organising and collecting data on mobility/travel and time use as part of ongoing work to understand, conceptualise and describe the changing patterns of domestic and mobility related energy demand within UK society. This exercise reveals systematic differences of method and approach, for instance in units of measurement, in how issues of time/duration and periodicity are handled, and how these strategies relate to the questions such data is routinely used to address. It also points to more fundamental differences in how traditions of research into mobility, domestic energy and time use have developed. We end with a discussion of the practical implications of these diverse histories for understanding and analysing changing patterns of energy/mobility demand at different scales.

Transreal limits expose category errors in IEEE 754 floating-point arithmetic and in mathematics

The IEEE 754 standard for oating-point arithmetic is widely used in computing. It is based on real arithmetic and is made total by adding both a positive and a negative infinity, a negative zero, and many Not-a-Number (NaN) states. The IEEE infinities are said to have the behaviour of limits. Transreal arithmetic is total. It also has a positive and a negative infinity but no negative zero, and it has a single, unordered number, nullity. We elucidate the transreal tangent and extend real limits to transreal limits. Arguing from this firm foundation, we maintain that there are three category errors in the IEEE 754 standard. Firstly the claim that IEEE infinities are limits of real arithmetic confuses limiting processes with arithmetic. Secondly a defence of IEEE negative zero confuses the limit of a function with the value of a function. Thirdly the definition of IEEE NaNs confuses undefined with unordered. Furthermore we prove that the tangent function, with the infinities given by geometrical con- struction, has a period of an entire rotation, not half a rotation as is commonly understood. This illustrates a category error, confusing the limit with the value of a function, in an important area of applied mathe- matics { trigonometry. We brie y consider the wider implications of this category error. Another paper proposes transreal arithmetic as a basis for floating- point arithmetic; here we take the profound step of proposing transreal arithmetic as a replacement for real arithmetic to remove the possibility of certain category errors in mathematics. Thus we propose both theo- retical and practical advantages of transmathematics. In particular we argue that implementing transreal analysis in trans- floating-point arith- metic would extend the coverage, accuracy and reliability of almost all computer programs that exploit real analysis { essentially all programs in science and engineering and many in finance, medicine and other socially beneficial applications.