The work of Jesse Douglas on Minimal Surfaces

A study of the life and work of the American Mathematician Jesse Douglas, who won one of the first Fields Medals, in 1936, for his work on minimal surfaces

Poincaré and the idea of a group

In many different fields of mathematics and physics Poincaré found many uses for the idea of a group, but not for group theory. He used the idea in his work on automorphic functions, in number theory, in his epistemology, Lie theory (on the so-called Campbell–Baker–Hausdorff and Poincaré–Birkhoff–Witt theorems), in physics (where he introduced the Lorentz group), in his study of the domains of complex functions of several variables, and in his pioneering study of 3-manifolds. However, as a general rule, he seldom appealed to deep results in group theory, and developed no more structural analysis of any group than was necessary to solve a problem. It was usually enough for him that there is a group, or that there are different groups

Poincaré and complex function theory

Poincaré is still well known for the mathematical work that first made his name : his discovery in 1880--1881 of automorphic functions. New documents and insights were added in [Gray and Walter 1999], which can also be consulted for references to the well-known history of his work in this area. He is also remembered for one further theorem that grew out of that early work : the uniformisation theorem, which he sketched a proof of in 1883 and then proved rigorously in 1907, as did Koebe independently. The rest of his numerous contributions to complex function theory are more scattered and do not seem to have been the focus of much attention. In this paper I survey what he did and argue that they tell an eloquent story not only about the state of the subject in the years around 1900 but about Poincaré's place in the mathematical community of his day. To understand either of these it is necessary to give a quick summary of the prior development of complex function theory, which was growing rapidly into a central topic in all mathematics, and that will occupy the first half of this paper. The second half will consider Poincaré’s contributions. We will see that although he was actively involved in many aspects of the subject, his influence is scarcely to be noticed in the many books that were published, and I will investigate why that was and what it may tell us about relationship between research and teaching in the years around 1900

An investigation into the oral English language proficiency gain of pupils taught by native English-speaking teachers in Hong Kong secondary schools

Purpose This study examines the extent to which Native English-speaking Teachers (NETs) have an impact on the oral English language proficiency gain of pupils taught in secondary schools in Hong Kong i.e. the so-called 'NET effect'. The principal aim was to determine whether the oral proficiency gain of subjects involved in this study was greater in students taught by NET teachers than it was in students taught by local teachers. Through the examination of Time one and Time two oral assessment data, this study also sets out to investigate the nature and strength of other predictor variables for the outcome variable 'Time two oral assessment'. Through a number of different statistical modelling techniques this study also sought to establish the model that would account for or 'explain' as much variance as possible between the Time one and Time two assessment scores. Procedures A randomised, stratified sample of secondary schools that was representative of the whole population secondary students in Hong Kong who are studying English was generated. From this sample, one thousand four hundred and twenty four students from forms one, three and four were selected and an English language oral proficiency test, specifically developed for this study was administered as a pre and post test. The period of data collection was a two year period, from the beginning of the 1998-1999 academic year to the end of the 1999-2000 academic year. The oral proficiency assessment instrument was designed, and piloted by a small team of trained assessors, and a standardised procedure was established for conducting the assessments. Hong Kong NET and local teachers were trained in the procedures and use of materials and techniques required to administer the assessments in specifically dedicated language assessment workshops. The assessments were then administered by the trained group of teachers who taped all of the interviews to allow monitoring to take place and to provide a data source for a second stage interview analysis (not covered in this thesis).The resulting pre-test and post-test data was then analysed through the use of a number of statistical techniques. In the first instance, a descriptive analysis was conducted in order to satisfy the assumptions on which traditional statistical analysis is based. The data analysis then proceeded with a number of scaling processes and was finally analysed to determine whether or not any significant 'NET effect' had been detected. In addition, the analysis also considered whether any of the other variables could be considered good predictors of the final post-test score. Major Findings Analysis of data produced from the Time one and Time two oral assessments revealed a number of important findings. Students did make significant oral English language proficiency gain as measured by the specially developed instrument. This gain was significant regardless of whether the students' results were analysed by whole sample or by separate year/age group. An analysis of means revealed that on average, the mean scores of students attending EMI schools were significantly higher than those attending CMI schools indicating that the medium of instruction is potentially a strong predictor of the Time two assessment score. In the post test analysis of means, students taught by NETs performed better than those taught by 'both' [NETs and local teachers] and in general, students taught by local teachers also performed better than those taught by 'both'. There was little difference between the scores of students taught by NETs and local teachers. The banding of the schools was also found to be an important predictor variable, with the average scores of students in high band schools significantly higher than their peers in medium and low band schools. Multiple regression analysis also revealed some important findings. When the modelling was conducted on the whole sample, the medium of instruction, the school level and NET teacher were all found to be significant predictor variables although in the case of the latter, the effect was small. When modelled by separate form/age group, similar results were found with Form one and with Form three students and again the medium of instruction, the school level and NET teacher were significant predictor variables although in the case of NET teacher, the effect was again small. The findings of this thesis suggest that in terms of measuring value-added between Time one and Time two, there are indeed strong predictor variables such as medium of instruction, school level and student level. However, in trying to evaluate the contribution of NETs to students' oral English language proficiency gain over a two-year period, there is some evidence of a so-called 'NET effect' although this is rather weak, suggesting that more research is required to investigate this question more thoroughly

A formally verified proof of the prime number theorem

The prime number theorem, established by Hadamard and de la Vall'ee Poussin independently in 1896, asserts that the density of primes in the positive integers is asymptotic to 1 / ln x. Whereas their proofs made serious use of the methods of complex analysis, elementary proofs were provided by Selberg and Erd"os in 1948. We describe a formally verified version of Selberg's proof, obtained using the Isabelle proof assistant.Comment: 23 page

Design and Implementation of a Unified Command and Control Architecture for Multiple Cooperative Unmanned Vehicles Utilizing Commercial Off the Shelf Components

Small unmanned systems provide great utility to military applications due to their portable and expendable design. These systems are, however, costly to develop, produce, and maintain, making it desirable to integrate available commercial off the shelf (COTS) components. This research investigates the integration of COTS components through the development of a modular unified command and control (C2) architecture for heterogeneous and homogeneous vehicle teams to accomplish formation flocking and communication relay scenarios. A vehicle agnostic architecture was developed to be applied across different vehicle platforms, different vehicle combination, and different cooperative missions. COTS components consisting primarily of open source hardware and software were integrated and tested based on the positional accuracy, precision, and other qualitative measures. The resulting system successfully demonstrated formation flocking in three of four vehicle combinations, with the forth still demonstrating leader follower behaviors. The system achieved at best a mean positional error of 0.99m, a standard deviation of 0.44m, and a DRMS of 0.59m. The communication relay scenario was also demonstrated with two vehicle combinations for both distance and physical obstructions breaking the C2 link. This system demonstrated the desired capabilities and could easily be adapted to accomplish others through the use of the flexible architecture

Number theory: An approach through history; from Hammurapi to Legendre By A. Weil. Boston (Birkhäuser). 1984. xv + 375 pp

Unknow

Anxiety and abstraction in Nineteenth-Century Mathematics

The paper is in three parts. The first part surveys the current literature in the history of 19th Century mathematics in order to show that the question ‘Did the increasing abstraction of mathematics lead to a sense of anxiety?’ is a new and valid question. I argue that the mathematics of the 19th Century is marked by a growing appreciation of error leading to a note, hesitant at first but persistent by 1900, of anxiety. This mounting disquiet about so many aspects of mathematics after 1850 is seldom discussed. Insofar as this part of the paper is critical of present practice it retains the personal tone of the talk, but also it incorporates some of the helpful suggestions made during the discussion after the talk. The second part explores the issue of anxiety in mathematical life through an interesting account of these issues as they were perceived by one mathematician in 1911, Oscar Perron. The third and final part ventures some conclusions about the value of anxiety as a question for historians of mathematics to pursue