New techniques for comparing the volume functions of historical texts

Conclusions The techniques described in this paper are essentially experimental. So far, they have been tried on a limited volume of empirical material, and final verdict of applicability should await more detailed checks and calibration. Yet even preliminary results suggest that it is indeed possible to develop tests for classifying texts into dependent and independent with allowance for their volume.Translated from Problemy Ustoichivosti Stokhasticheskikh Modelei, Trudy Seminara, pp. 33–45, 1986.We are grateful to N. Ya. Rives for his considerable interest in this research and for his willing assistance with computer work. His expert help has enabled us to test a number of hypotheses and to advance new ones.     

An historical example of mathematical modelling: the trajectory of a cannonball

Classroom considerations of the concept and processes of mathematical modelling can do much to strengthen students’ problem solving skills. A systematic exposure to the techniques of mathematical modelling helps students formulate problems, re‐think those problems in mathematical terms, appreciate possible solution constraints and seek solutions that are realistic within the scope and conditions of the problem. While many mathematical modelling situations can be found in today's world, there are special pedagogical values in examining existing mathematical models that have an historical basis. Such an examination should reveal the mechanics of a modelling situation and how a model evolves or is refined to meet ever increasing human demands for accuracy or practicality. The trajectory of a cannonball provides such a modelling example. This topic captures the imagination of students and supplies the basis for a variety of classroom discussions and problem solving encounters.     

Some historical aspects of the development of mathematical analysis in hungary

As is well-known, an extraordinarily rapid development of the mathematical sciences has been taking place in Hungary since the turn of the century, whose center lies in the analytical schools of L. Fejér, F. and M. Riesz and A. Haar. The present paper aims to discuss briefly the questions: What were the deep causes of this progress? How did the work of a few great mathematicians contribute to the actual high level of analytical investigations, to their applications, and, in general, to the intensive mathematical life of today in Hungary?     

Commutativity and collinearity: a historical case study of the interconnection of mathematical ideas. Part II

This paper investigates the discovery of an intriguing and fundamental connection between the famous but apparently unrelated mathematical work of two late third-century mathematicians. This link went unnoticed for well over 1500 years until the publication of two groundbreaking but again ostensibly unrelated works by two German mathematicians at the close of the nineteenth century. In this, the second and final part of the paper, we continue our examination of the chain of mathematical events and the related development of mathematical disciplines, without which the connection might never have been noticed in the first place.     

Commutativity and collinearity: a historical case study of the interconnection of mathematical ideas. Part I

This two-part paper investigates the discovery of an intriguing and fundamental connection between the famous but apparently unrelated mathematical work of two late third-century mathematicians, a link that went unnoticed for well over 1500 years. In this, the first installment of the paper, we examine the initial chain of mathematical events that would ultimately lead to the discovery of this remarkable link between two seemingly distinct areas of mathematics, encompassing contributions by a variety of mathematicians, from the most distinguished to the relatively unknown.     

Contradictions and uncertainty in scientists’ mathematical modeling and interpretation of data

Contradictions have been recognized as important factors in learning (conceptual change), because they require students to engage in deep reflection that leads to accommodation and learning. However, in the face of uncertainty, confirmation bias and the theory-laden nature of observation may not allow the recognition of a situation as harboring a contradiction. In the present study, I analyze a meeting in which a scientific research team presents its results to an informed audience. I show that with hindsight, there are contradictions in the mathematical models that the scientists use and the graph interpretations that they produce. Because the contradictions went unnoticed, they could not become a determinant factor in the process. This has implications for thinking about the role of uncertainty and contradiction as factors in and of mathematical learning.     

Authorship attribution of texts: a review

We survey the authorship attribution of documents given some prior stylistic characteristics of the author's writing extracted from a corpus of known works, e.g., authentication of disputed documents or literary works. Although the pioneering paper based on word length histograms appeared at the very end of the nineteenth century, the resolution power of this and other stylometry approaches is yet to be studied both theoretically and on case studies such that additional information can assist finding the correct attribution.We survey several theoretical approaches including ones approximating the apparently nearly optimal one based on Kolmogorov conditional complexity and some case studies: attributing Shakespeare canon and newly discovered works as well as allegedly M. Twain's newly-discovered works, Federalist papers binary (Madison vs. Hamilton) discrimination using Naive Bayes and other classifiers, and steganography presence testing. The latter topic is complemented by a sketch of an anagrams ambiguity study based on the Shannon cryptography theory.     

The fuzzy decodings of educative texts

An educative text about leprosy was played to two groups of students of the Nursing School of University of São Paulo, one of them having been submitted to a special training program on Hanseniasis. After the hearing session, the students recalled the text, constructed the graph of this recall and pointed out the degree of confidence of each information in this graph. The results showed that the untrained people had a fuzzier decoding than the trained students, and favored the information about stigmas and taboos associated to this disease. On the contrary, the trained students focused their attention on the information concerned with the treatment and self-care to prevent body deformation. The results support the idea that fuzzy graph theory may be used to quantify the decoding of educative texts used on health care programs, and thus to evaluate the adequacy of the information intended to be transmitted.     

An approach to the automatic abstracting of Russian texts

A new method of the automatic abstracting of Russian texts which takes account of their functional style is considered. The texts are broken up into five groups and, for each group, its own scenario of constructing the abstract is developed. A general adjustable algorithm embodying the possibilities of all the scenarios developed is described.     

A Chinese mathematical classic of the third century: The sea island mathematical manual of Liu Hui

The Haidao suanjing [Sea island mathematical manual], written by the Chinese mathematician Liu Hui in 263 a.d., consists of nine surveying problems whose solution schemes involve the use of right triangle theory and result in a variety of techniques and formulas for determining distances to inaccessible points. Liu's results were obtained through the use of a prototrigonometry based on the concept of chong cha. This paper presents a translation of the Haidao's mathematical exercises and solution formulas and considers some of the implications of this early mathematical work.

     

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.