Optimality conditions using sum-convex approximations

The purpose of this paper is to give necessary and sufficient conditions of optimality for a general mathematical programming problem, using not a linear approximation to the constraint function but an approximation possessing certain convexity properties. Such approximations are called sum-convex. Theorems of the alternative involving sum-convex functions are also presented as part of the proof.This work is part of the author's PhD Thesis under the supervision of Professor S. Zlobec at McGill University.     

新型高速视频图像记录判读系统

介绍了新型高速视频图像记录判读系统的特点、组成和功能,并采用了一种改进的自动判读方法.该系统利用集成在高速CMOS摄像机上的存储器记录数字视频序列图像,并利用高速接口下载到计算机移动硬盘或可读写光盘上,以供事后使用判读软件对记录的图像进行分析处理.运用数学形态学图像处理方法滤除二值图像的噪声,较好地实现了清晰提取目标边界的目的.针对自动判读与半自动判读的不同要求,采用不同的细分技术.在现有分辨率的情况下,达到对目标边缘亚像元级定位的目的,显著提高测量准确度.对其他类似系统的研制有一定的借鉴作用.     

Existence and stability of limit cycles in a macroscopic neuronal population model

We present rigorous results concerning the existence and stability of limit cycles in a macroscopic model of neuronal activity. The specific model we consider is developed from the Ki set methodology, popularized by Walter Freeman. In particular we focus on a specific reduction of the KII sets, denoted RKII sets. We analyse the unfolding of supercritical Hopf bifurcations via consideration of the normal forms and centre manifold reductions. Subsequently we analyse the global stability of limit cycles on a region of parameter space and this is achieved by applying a new methodology termed Global Analysis of Piecewise Linear Systems. The analysis presented may also be used to consider coupled systems of this type. A number of macroscopic mean-field approaches to modelling human EEG may be considered as coupled RKII networks. Hence developing a theoretical understanding of the onset of oscillations in models of this type has important implications in clinical neuroscience, as limit cycle oscillations have been demonstrated to be critical in the onset of certain types of epilepsy.     

Investigating written expressions of mathematical reasoning for students with learning disabilities

Assessment results from two open-construction response mathematical tasks involving fractions and decimals were used to investigate written expression of mathematical reasoning for students with learning disabilities. The solutions and written responses of 51 students with learning disabilities in fourth and fifth grade were analyzed on four primary dimensions: (a) accuracy, (b) five elements of mathematical reasoning, (c) five elements of mathematical writing, and (d) vocabulary use. Results indicate most students were not accurate in their problem solution and communicated minimal mathematical reasoning in their written expression. In addition, students tended to use general vocabulary rather than academic precise math vocabulary and students who provided a visual representation were more likely to answer accurately. To further clarify the students struggles with mathematical reasoning, error analysis indicated a variety of error patterns existed and tended to vary widely by problem type. Our findings call for more instruction and intervention focused on supporting students mathematical reasoning through written expression. Implications for research and practice are presented.     

Secondary teachers’ meanings for function notation in the United States and South Korea

This study investigates what teachers in U.S. reveal about their meanings for function notation in their written responses to the Mathematical Meanings for Teaching secondary mathematics (MMTsm) items, with particular attention to how productive those meanings would be if conveyed to students in a classroom setting. We then report South Korean teachers’ responses to see whether the meanings U.S. teachers demonstrated are shared with South Korean teachers. The results show that many U.S. teachers use function notation to name rules instead of to represent relationships. The data from South Korean teachers indicates that the problematic meanings in U.S. teachers’ responses are shared with a minority of South Korean teachers. The results suggest a need for attention to ideas regarding function notation in teacher education for pre-service teachers and professional development programs for in-service teachers.     

Generalized stationary points and an interior-point method for mathematical programs with equilibrium constraints

Generalized stationary points of the mathematical program with equilibrium constraints (MPEC) are studied to better describe the limit points produced by interior point methods for MPEC. A primal-dual interior-point method is then proposed, which solves a sequence of relaxed barrier problems derived from MPEC. Global convergence results are deduced under fairly general conditions other than strict complementarity or the linear independence constraint qualification for MPEC (MPEC-LICQ). It is shown that every limit point of the generated sequence is a strong stationary point of MPEC if the penalty parameter of the merit function is bounded. Otherwise, a point with certain stationarity can be obtained. Preliminary numerical results are reported, which include a case analyzed by Leyffer for which the penalty interior-point algorithm failed to find a stationary point.Mathematics Subject Classification (1991):90C30, 90C33, 90C55, 49M37, 65K10     

A locus for transnational exchanges: European mathematical journals for students and teachers, 1860s–1914

While there were a few mathematical journals aimed at teachers and students as early as the 1840s, it was only in the late 19th century that they became more numerous in Europe. This article is based on the analysis of a corpus of European mathematical journals published between the 1860s and World War I, selected in the first place because they were aimed at high school teachers and high school or/and first two years university students, which are often referred to as “intermediate journals”. All these journals had focused on the teaching of mathematics and, as such, they were shaped by the educational context of the country in which they were published. However, leafing through theses journals, one is struck by the fact that the mathematics they published was in fact highly commensurable, and can see that they were the locus of transnational exchanges on mathematical knowledge. This article shows that several aspects of “internationalisation” were in fact at stake in mathematical journals for students: making knowledge from elsewhere available and of publicizing to the whole world the mathematics produced in one country; making people from different countries collaborate. Finally, it focuses on the effects of transnational exchanges between journals for teachers and students: what was the mathematical knowledge that was circulated through them, and in what respect was it different from that published in other mathematical journals?     

Interplay between mathematical journals on various scales 1850–1950

This paper is meant as an introduction to a special issue on circulating mathematics via journals, from ca 1850 to 1950. It intends to study the practices, actors, places and institutions involved in mathematical circulation. Journals constitute one of the main carriers of mathematical knowledge and hence the main vehicle for its circulation. The focus of the paper is on ‘mathematical’ (understood very broadly) journals and the various forms of interplay between them. In order to understand and analyse the processes of circulation that are an important feature of mathematical production, the interactions that take place between journals are considered across different scales, from local practices and representations to more global appraisals. We shall use the notion of circulation space in order to break down the representations of the mathematical world in terms of the centres and peripheries suggested by the geographical area, with its massive publishing centres at different times – Paris, London, Berlin, Leipzig, New York, etc. Besides the geographical dimensions, the focus is on connections between journals, polarisations, competition, rivalries, etc.     

非均衡宏观调控最优控制模型及应用

本文基于最优控制理论原理 ,从理论上给出宏观调控的最优数学模型 ,为国民经济宏观调控的定量评价提供了理论依据和实证分析 .