Leonia Lerer’s Mathematical Work and Amsterdam Visits

2013 ◽  
pp. 1-7
Author(s):  
M. A. Kaashoek
Keyword(s):  
Mathematical Work

Hugh James Rose’s Anglican Critique of Cambridge: Science, Antirationalism, and Coleridgean Idealism in Late Georgian England

2001 ◽  
Vol 33 (02) ◽  
pp. 218-242
Author(s):  
David A. Valone
Keyword(s):  
Natural Philosophy ◽  
Church Of England ◽  
Natural Theology ◽  
Religious Faith ◽  
Mathematical Work ◽  
Undergraduate Instruction ◽  
John Herschel ◽  
Anglican Theology ◽  
Georgian England ◽  
The University

On Commencement Sunday in the summer of 1826, Hugh James Rose ascended the pulpit of the University Church at Cambridge to deliver a sermon. As Rose surveyed the assembled crowd, he would have been well aware that before him sat the future of English political, religious, and intellectual life—present and future members of Parliament, the leaders and local prelates of the Church of England, and the next generation of Cambridge scholars. While commencement addresses today are rather formulaic in their celebratory character, the sermon Rose had prepared for that day was far from uplifting. Rose had chosen to preach on Ecclesiastes chapter eleven, verse five: “No man can find out the work, which God maketh, from the beginning to the end.” Using this passage as a decree upon the limits of human knowledge, Rose launched into a blistering attack on the University and the educational philosophy that he believed it espoused. Far from praising the University and its graduates, Rose called into question much of what Cambridge had been doing to educate its students. The essence of Rose’s critique was that the University had lost its way as a religious institution and had become dominated by the search for “knowledge of the material Universe.” Pursuing this end, Rose warned, was a tremendous danger, because in so doing Cambridge was failing to provide a proper moral and religious foundation for those who would guide the nation. Naturally, Rose’s sermon came as a shock to many of those gathered before him, especially since it not only took the University to task but also implicitly seemed to indict some of Rose’s closest friends. His sermon battered one of the girders of Cambridge intellectual and religious life, and of Anglican theology more generally: the notion that natural philosophy was an appropriate handmaiden to religion. The tradition of reasoning up from nature to the Creator had long flourished at Cambridge in the hands of both men of science and theologians. Most at Cambridge took for granted the compatibility between the study of God’s creation and religious faith. For the previous three decades Cambridge had made the works of alumnus William Paley, replete with the ways nature manifested the wisdom and goodness of God, a cornerstone of undergraduate instruction. Ironically, many of Rose’s acquaintances from his own undergraduate days at Cambridge were themselves involved in scientific and mathematical pursuits and were generally sympathetic to Natural Theology. His dearest friend at the University was William Whewell, an intellectual polymath who excelled in mathematics, physics, and mineralogy, as well as moral philosophy, history, and theology. Rose also was a close associate of John Herschel and Charles Babbage, men who were renowned for their astronomical and mathematical work. Himself a fairly accomplished mathematician a decade earlier, Rose even had considered publishing some work to support Herschel and Babbage’s efforts to revitalize Cambridge mathematics during his undergraduate days.

Egg-Forms and Measure-Bodies: Different Mathematical Practices in the Early History of the Modern Theory of Convexity

2009 ◽  
Vol 22 (1) ◽  
pp. 85-113 ◽  
Author(s):  
Tinne Hoff Kjeldsen
Keyword(s):  
Convex Bodies ◽  
Mathematical Work ◽  
Late Nineteenth Century ◽  
Modern Theory ◽  
Methodological Framework ◽  
Mathematical Research ◽  
Mathematical Practices ◽  
History Of ◽  
Research Questions ◽  
Epistemic Objects

ArgumentTwo simultaneous episodes in late nineteenth-century mathematical research, one by Karl Hermann Brunn (1862–1939) and another by Hermann Minkowski (1864–1909), have been described as the origin of the theory of convex bodies. This article aims to understand and explain (1) how and why the concept of such bodies emerged in these two trajectories of mathematical research; and (2) why Minkowski's – and not Brunn's – strand of thought led to the development of a theory of convexity. Concrete pieces of Brunn's and Minkowski's mathematical work in the two episodes will, from the perspective of the above questions, be presented and analyzed with the use of the methodological framework of epistemic objects, techniques, and configurations as adapted from Hans-Jörg Rheinberger's work on empirical sciences to the historiography of mathematics by Moritz Epple. Based on detailed descriptions and a comparison of the objects and techniques that Brunn and Minkowski studied and used in these pieces it will be concluded that Brunn and Minkowski worked in different epistemic configurations, and it will be argued that this had a significant influence on the mathematics they developed for those bodies, which can provide answers to the two research questions listed above.

More Dynamical Properties Revealed from a 3D Lorenz-like System

2014 ◽  
Vol 24 (10) ◽  
pp. 1450133 ◽  
Author(s):  
Haijun Wang ◽  
Xianyi Li
Keyword(s):  
Numerical Simulations ◽  
Local Stability ◽  
Equilibrium Points ◽  
Heteroclinic Orbits ◽  
Mathematical Work ◽  
Chaotic Attractors ◽  
Heteroclinic Cycles ◽  
Homoclinic And Heteroclinic Orbits ◽  
Dynamical Properties ◽  
Theoretical Results

After a 3D Lorenz-like system has been revisited, more rich hidden dynamics that was not found previously is clearly revealed. Some more precise mathematical work, such as for the complete distribution and the local stability and bifurcation of its equilibrium points, the existence of singularly degenerate heteroclinic cycles as well as homoclinic and heteroclinic orbits, and the dynamics at infinity, is carried out in this paper. In particular, another possible new mechanism behind the creation of chaotic attractors is presented. Based on this mechanism, some different structure types of chaotic attractors are numerically found in the case of small b > 0. All theoretical results obtained are further illustrated by numerical simulations. What we formulate in this paper is to not only show those dynamical properties hiding in this system, but also (more mainly) present a kind of way and means — both "locally" and "globally" and both "finitely" and "infinitely" — to comprehensively explore a given system.

scholarly journals Mathematical knowledge for teaching: Adding to the description through a study of probability in practice

Pythagoras ◽  
2006 ◽  
Vol 0 (63) ◽  
Author(s):  
Mercy Kazima ◽  
Jill Adler
Keyword(s):  
Problem Solving ◽  
Mathematical Knowledge ◽  
Mathematical Problem ◽  
Mathematical Knowledge For Teaching ◽  
Mathematical Work ◽  
Mathematical Problem Solving ◽  
Knowledge For Teaching ◽  
Scaling Down ◽  
Multilingual Classroom ◽  
Everyday Knowledge

In their description of the mathematical work of teaching, Ball,  Bass & Hill (2004) describe the mathematical problem solving that teachers do as they go about their work. In this paper we add to this description through our study of teaching of probability in a grade 8 multilingual classroom in South Africa. We use instances of teaching to highlight the mathematical problem solving that teachers might face as they work with learners’ ideas, both expected and unexpected. We discuss  the restructuring of tasks as an inevitable feature of teachers’ work, and argue that in addition to scaling up or scaling down of the task as Ball et al. (2004) describe, restructuring can also entail shifting the mathematical outcomes from those intended. We also point out how well known issues in mathematics education, for example working with learners’ everyday knowledge, and the languages they bring to class, are highlighted by the context of probability, enabling additional insights into the mathematical work of teaching.

scholarly journals Correcting Bias in Log-Linear Instrument Calibrations in the Context of Chemical Ionization Mass Spectrometry

2021 ◽  
Author(s):  
Chenyang Bi ◽  
Jordan E. Krechmer ◽  
Manjula R. Canagaratna ◽  
Gabriel Isaacman-VanWertz
Keyword(s):  
Chemical Ionization ◽  
Correction Method ◽  
Operating Conditions ◽  
Oxidation Products ◽  
Mathematical Work ◽  
Ionization Mass ◽  
Individual Compound ◽  
Linear Calibration ◽  
Order Of Magnitude ◽  
Log Linear

Abstract. Quantitative calibration of analytes using chemical ionization mass spectrometers (CIMS) has been hindered by the lack of commercially available standards of atmospheric oxidation products. To accurately calibrate analytes without standards, techniques have been recently developed to log-linearly correlate analyte sensitivity with instrument operating conditions. However, there is an inherent bias when applying log-linear calibration relationships that is typically ignored. In this study, we examine the bias in a log-linear based calibration curve based on prior mathematical work. We quantify the potential bias within the context of a CIMS-relevant relationship between analyte sensitivity and instrument voltage differentials. Uncertainty in three parameters has the potential to contribute to the bias, specifically the inherent extent to which the nominal relationship can capture true sensitivity, the slope of the relationship, and the voltage differential below which maximum sensitivity is achieved. Using a prior published case study, we estimate an average bias of 30%, with one order of magnitude for less sensitive compounds in some circumstances. A parameter-explicit solution is proposed in this work for completely removing the inherent bias generated in the log-linear calibration relationships. A simplified correction method is also suggested for cases where a comprehensive bias correction is not possible due to unknown uncertainties of calibration parameters, which is shown to eliminate the bias on average but not for each individual compound.

scholarly journals Correcting bias in log-linear instrument calibrations in the context of chemical ionization mass spectrometry

2021 ◽  
Vol 14 (10) ◽  
pp. 6551-6560
Author(s):  
Chenyang Bi ◽  
Jordan E. Krechmer ◽  
Manjula R. Canagaratna ◽  
Gabriel Isaacman-VanWertz
Keyword(s):  
Chemical Ionization ◽  
Correction Method ◽  
Operating Conditions ◽  
Oxidation Products ◽  
Mathematical Work ◽  
Ionization Mass ◽  
Individual Compound ◽  
Linear Calibration ◽  
Order Of Magnitude ◽  
Log Linear

Abstract. Quantitative calibration of analytes using chemical ionization mass spectrometers (CIMSs) has been hindered by the lack of commercially available standards of atmospheric oxidation products. To accurately calibrate analytes without standards, techniques have been recently developed to log-linearly correlate analyte sensitivity with instrument operating conditions. However, there is an inherent bias when applying log-linear calibration relationships that is typically ignored. In this study, we examine the bias in a log-linear-based calibration curve based on prior mathematical work. We quantify the potential bias within the context of a CIMS-relevant relationship between analyte sensitivity and instrument voltage differentials. Uncertainty in three parameters has the potential to contribute to the bias, specifically the inherent extent to which the nominal relationship can capture true sensitivity, the slope of the relationship, and the voltage differential below which maximum sensitivity is achieved. Using a prior published case study, we estimate an average bias of 30 %, with 1 order of magnitude for less sensitive compounds in some circumstances. A parameter-explicit solution is proposed in this work for completely removing the inherent bias generated in the log-linear calibration relationships. A simplified correction method is also suggested for cases where a comprehensive bias correction is not possible due to unknown uncertainties of calibration parameters, which is shown to eliminate the bias on average but not for each individual compound.

scholarly journals Piloting a problem solving module for undergraduate mathematics students

2019 ◽  
Vol 17 (2) ◽  
pp. 46
Author(s):  
David McConnell
Keyword(s):  
Problem Solving ◽  
Mathematical Thinking ◽  
Thinking Skills ◽  
Mathematical Work ◽  
Assessment Practices ◽  
Traditional Teaching ◽  
Mathematics Students ◽  
Undergraduate Mathematics ◽  
Students First ◽  
Academic Year

We report on a new problem solving module for second-year undergraduate mathematics students first piloted during the 2016-17 academic year at Cardiff University.  This module was introduced in response to the concern that for many students, traditional teaching and assessment practices do not offer sufficient opportunities for developing problem-solving and mathematical thinking skills, and more generally, to address the recognised need to incorporate transferrable skills into our undergraduate programmes.  We discuss the pedagogic and practical considerations involved in the design and delivery of this module, and in particular, the question of how to construct open-ended problems and assessment activities that promote mathematical thinking, and reward genuinely original and independent mathematical work.  

M. A. LAVRENT'EV'S MATHEMATICAL WORK

1970 ◽  
Vol 25 (6) ◽  
pp. 7-27
Author(s):  
P S Aleksandrov ◽  
L V Ovsyannikov ◽  
B V Shabat
Keyword(s):  
Mathematical Work

scholarly journals The Mathematical Proficiency Promoted by Mathematical Modelling

2021 ◽  
Vol 4 (2) ◽  
pp. 107-131
Author(s):  
Priscila Dias Corrêa ◽  
Keyword(s):  
High School Students ◽  
Mathematical Modelling ◽  
Mathematics Teaching ◽  
Research Study ◽  
Mathematical Work ◽  
School Students ◽  
Design Based Research ◽  
Video Recordings ◽  
Mathematical Proficiency ◽  
Insight Into

This study aims to investigate the mathematical proficiency promoted by mathematical modelling tasks that require students to get involved in the processes of developing mathematical models, instead of just using known or given models. The research methodology is grounded on design-based research, and the classroom design framework is supported by complexity science underpinnings. The research intervention consists of high-school students, from a grade 11 mathematics course, aiming to solve four different modelling tasks in four distinct moments. Data was collected during the intervention from students’ written mathematical work and audio and video recordings, and from recall interviews after the intervention. Data analysis was conducted based on a model of mathematical proficiency and assisted by interpretive diagrams created for this research purpose. This research study offers insight into mathematics teaching by portraying how mathematical modelling tasks can be integrated into mathematics classes to promote students’ mathematical proficiency. The study discusses observed expressions and behaviours in students’ development of mathematical proficiency and suggests a relationship between mathematical modelling processes and the promotion of mathematical proficiency. The study also reveals that students develop mathematical proficiency, even when they do not come to full resolutions of modelling tasks, which emphasizes the relevance of learning processes, and not only of the products of these processes.

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