Researching how professional mathematicians construct new mathematical definitions: a case study

In this work, we study the mathematical practice of defining by mathematics researchers. Since research is an important part of many professional mathematicians, understanding how they do research is a necessary step before thinking about future researchers’ undergraduate and postgraduate education. We focus on the defining process associated with the generalization of existing definitions as a way of constructing new ones. Data of this qualitative study come from a case study whose subject is a mathematics researcher in the area of differential geometry. We have interviewed this researcher and collected her research documents. From our analysis of the data, we have identified four phases in the defining process (Finding an opportunity to generalize an existing concept, Proposing a new definition, Justifying that the new definition is valid and Continuing the chain of definitions), which we will describe in detail in Section 4.     

Solving mathematical problems: a personal perspective

Feel like writing a review for The Mathematical Intelligencer? You are welcome to submit an unsolicited review of a book of your choice; or, if you would welcome being assigned a book to review, please write us, telling us your expertise and your predilections.     

On the customer lifetime value: a mathematical perspective

The customer lifetime value (CLV) is an important concept increasingly considered in the field of general marketing and in the management of firms, of organizations to increase the captured profitability. It represents the total value that a customer produces during his or her lifetime, or better represents the measure of the potential profit generating by a customer. The companies use the customer lifetime value to segment customers, analyze probability of churn, allocate resources or formulate strategies and, therefore, they increasingly derive revenue from the creation and from sustenance of long-term relationships with their customers. For this reason, the customer lifetime value is increasingly considered a touchstone for the management of customer relationships. In this article, the authors deepen the concept and use of customer lifetime value and present some mathematical models for its determination. There is many models for this purpose but most of them are theoretic, complex and not applicable. Though not exhaustive, the major contribution of this paper is that it provides a general mathematical formulation to estimate the CLV and that it has a context less specific compared to papers, present in literature, on the customer lifetime.     

Condition monitoring: a new perspective

We present a novel approach to support preventive replacement decisions based on condition monitoring. It is assumed that the condition of a technical unit is measured continuously, with the measurement indicating in which of k≥2 states the unit is. A new unit is in state S₁, and failure occurs the moment a unit leaves state S _k . A unit will only go from state S _j to state S_j+1, and these immediately observed transitions occur at random times. At such transition moments a decision is required on preventive replacement of the unit, which would require preparation time. Such a decision needs to be based on inferences about the total residual time till failure. We present a method for such inferences that relies on minimal model assumptions added to data on state transitions. We illustrate the approach via simulated examples, and discuss possible use of the method and related aspects.     

A social perspective on technology-enhanced mathematical learning: from collaboration to performance

This paper documents both developments in the technologies used to promote learning mathematics and the influence on research of social theories of learning, through reference to the activities of the International Commission on Mathematical Instruction (ICMI), and argues that these changes provide opportunity for the reconceptualization of our understanding of mathematical learning. Firstly, changes in technology are traced from discipline-specific computer-based software through to Web 2.0-based learning tools. Secondly, the increasing influence of social theories of learning on mathematics education research is reviewed by examining the prevalence of papers and presentations, which acknowledge the role of social interaction in learning, at ICMI conferences over the past 20 years. Finally, it is argued that the confluence of these developments means that it is necessary to re-examine what it means to learn and do mathematics and proposes that it is now possible to view learning mathematics as an activity that is performed rather than passively acquired.     

On the use of integral transform methods: a new perspective

Integral transforms are usually used to ‘solve’ renewal‐type equations. These transforms typically require inversion to be of use to a decision maker. In this paper it is demonstrated that in certain situations, stochastic quantities of interest are expressible in terms of integral transforms, thus avoiding the problems associated with inversion.     

Interaction analysis of mathematical communication in primary teaching: The epistemological perspective

Communication between students and teacher in the mathematics classroom is a form of social interaction which focuses on a specific topic:mathematical knowledge. This knowledge cannot be introduced into classroom interaction “from the outside”, but grows through the communicative process, in the course of interactive exchanges between the participants of discussion. Although mathematical communication must be seen and analysed in the same way as any other form of communication, the particularity of interactive constructions of mathematical knowledge and its specificsocial epistemology within the context of teaching processes has to be taken into consideration. Also, the institutional influences of school institutions and those of teaching (analysed in the frame of general socio-interactive research approaches) must be considered. An epistemology-oriented interaction research approaches the specificity of amathematical classroom and communication culture in its analyses.     

Branching in the {\Sigma^0_2} -enumeration degrees: a new perspective

We give an alternative and more informative proof that every incomplete -enumeration degree is the meet of two incomparable -degrees, which allows us to show the stronger result that for every incomplete -enumeration degree a, there exist enumeration degrees x ₁ and x ₂ such that a, x ₁, x ₂ are incomparable, and for all b  ≤  a, b  =  (b ∨ x ₁ ) ∧ (b ∨ x ₂ ). The first author would like to thank her advisor, Andrea Sorbi, whose guidance made this paper possible. The second author has been supported by a Marie Curie Incoming International Fellowship of the European Community FP6 Program under contract number MIFI-CT-2006-021702.     

The analysis of linear probing sort by the use of a new mathematical transform

A variant of the distribution sort approach which makes use of extra storage to sort a list of n elements in an average of about probes into a table is presented. An accurate analysis of this technique is made by introducing a transform from a Poisson approximation to the exact (finite) distribution. This analysis also leads to the solution of an interesting parking problem.     

Images and preimages of subobjects under the morphisms in a new category of fuzzy sets-I

This paper is the third in a sequence of papers on categories by the same authors. In one of the papers, a new category of fuzzy sets was defined and a few results were established pertaining to that special category of fuzzy sets S. Here, the concept of a fuzzy subset of a fuzzy set is defined under the category S. Besides, the notions of images and preimages of fuzzy sets are also defined under morphisms in the category of fuzzy sets and how smoothly these images and preimages behave under the action of these morphisms is analyzed. Finally, results have been proved on algebra of morphisms of this category S.     

Constructing a mathematical definition: the case of the tangent

We analyze the process of constructing a definition within the theoretical framework of Abstraction in Context. Pairs of students were engaged in a task designed to engender a need for a definition of a tangent to a graph at a given point, and lead to constructing a definition following that need. The results of our analysis point to two characteristics of the process of constructing a definition: a. Constructing the concept does not necessarily include constructing its definition; in particular, students were able to use the concept before constructing its definition. b. Students’ language becomes more precise during the constructing process, not only as a characteristic of the process but also as a means of promoting the process.     

The Society for the Protection of Science and Learning as a patron of refugee mathematicians

The Society for the Protection of Science and Learning (SPSL, which started out as the Academic Assistance Council (AAC) 1933–36) distinguished itself from many other aid organizations set up in response to Nazi policies towards Jews and political dissidents in its focus on academic excellence as a criterion for support. Its archives are deposited in the Bodleian Library. Today, the organization is known as the Council for At-Risk Academics (CARA, http://www.cara1933.org/).

In the archives of the SPSL, there are files on two and a half thousand scientists, victims of persecution in their home countries, who appealed to the Society for assistance in finding refuge and work in Britain. Almost a hundred are mathematicians, of which about one in four found temporary or permanent employment in British academia. Many others found their way to the US, in most cases aided by the Emergency Committee, an American organization with priorities similar to those of the SPSL. The paths of some of the most successful SPSL grantees are described, and contrasted with those of some less fortunate applicants.