Opinion survey: the best mathematical books of the twentieth century

The Mathematical Intelligencer encourages comments about the material in this issue. Letters to the editor should be sent to either of the editors-in-chief, Chandler Davis or Marjorie Senechal.     

The chords theorem recalled to life at the turn of the eighteenth century

This paper is a historical account of the chords theorem, for conic sections from Apollonius to Boscovich. We comment the most significant proofs and applications, focusing on Newton's solution of the Pappus four lines problem. Newton's geometrical achievements drew L'Hospital's attention to the chords theorem as a fundamental one, and led him to search for a simple and direct proof, that he finally obtained by the method of projection. Stirling gave a very elegant algebraic proof; then Boscovich succeeded in finding an almost immediate geometrical proof, and showed how to develop the elements of conic sections starting from this theorem.     

Some topics in the history of harmonic analysis in the twentieth century

In December 2015 I gave a series of six lectures at the Indian Institute of Science in which I sketched the thematic development of some of the main techniques and results of 20th-century harmonic analysis. The subjects of the lectures were, briefly, as follows:

1. 1.
   Fourier series, 1900-1950.
    
2. 2.
   Singular integrals (part I).
    
3. 3.
   H ^p , BMO, and singular integrals (part II).
    
4. 4.
   Littlewood-Paley theory: the history of a technique.
    
5. 5.
   Harmonic analysis on groups.
    
6. 6.
   Wavelets.
    

I emphasized interconnections, both the way in which the material in the first lecture provided the roots out of which most of the developments in the other lectures grew, and the ways in which those developments interacted with each other. I included sketches of as many proofs as the time would permit: some very brief, but some fairly complete, especially those whose methodology is an important part of the subject. Much was omitted, of course, and there was a natural bias toward the areas where I have spent periods of my own mathematical life. Many developments, particularly those of the final quarter-century, received at most a brief mention.This paper is a written account of these lectures with a few more details fleshed out, a few topics reorganized, and a few items added. I hope that others may find it an interesting narrative and a useful reference, and that it may lead some of them to share my enjoyment of exploring the original sources. I have tried to provide the references to those sources wherever possible, and for the more recent developments I also provide references to various expository works as the occasion arises. For the pre-1950 results discussed here and their proofs, however, there is one canonical reference, which I give here once and for all: Antoni Zygmund’s treatise [96]. (The more fundamental ones can also be found in Folland [29].)     

The development and substantiation of the kinetic theory of gases in the twentieth century

A new kinetic theory that is close to dynamic processes is constructed. A system of M integro-differential equations and a system of M partial differential equations are obtained. The theory is demonstrated using the examples of the calculation of the structure of intense shock-waves and by calculating turbulent flows in a plane channel. It is shown that the theory of the structure of high-intensity shock-waves agrees with remarkable accuracy with numerous experimental data. Calculations of turbulent flow approximate quite well to experimental data, but it is remarkable that a single theory can describe both the turbulent core at the centre of the channel and the laminar sublayer on the wall so well.     

Half a century of operational research in the RAF

The original Operational Researchers conducted scientific analyses of operational data to help solve operational problems. Following the success to this essentially empirical process, a general belief has grown that all forms of decision problems can be assisted by similar scientific analysis using ‘a priori’ models, even when the issues are beyond experience and the models can never be validated. In this paper the author describes the evolution of Operational Research in the RAF at Command level over half a century, reflects on the value of techniques and concludes that the adaptive OR group itself has proved to be the most enduring and useful “tool” of all.     

Computational science in the eighteenth century. Test cases for the methods of Newton,Raphson, and Halley: 1685 to 1745

Numerical Algorithms - This is an overview of examples and problems posed in the late 1600s up to the mid 1700s for the purpose of testing or explaining the two different implementations of the...     

Mathematics of currency and exchange: arithmetic at the end of the thirteenth century

In western Europe, a sophisticated banking system for the purposes of international trade had evolved by the end of the thirteenth century. It was based upon the ‘bill of exchange’, which enabled an exporter of goods to receive payment in his own currency, by means of a balancing payment made in the currency of the importer. This paper discusses the arithmetical tools that were available for use in accounting for transactions made in different currencies. It is argued that algorithmic methods based on the Hindu-Arabic numerals were used at the higher levels of banking, in order to prepare tables of foreign exchange such as those collected by the Florentine banker, Francesco Pegolotti. On the other hand, the clerks who were responsible for routine book-keeping would have used a simple abacus and counters, and recorded their transactions in Roman numerals. The paper is based on a talk given to the BSHM at Gresham College on 25 April 2008.     

Redesigning the calculus sequence at a research university: issues,implementation, and objectives

The paper discusses the progress and challenges of a new reformed calculus sequence for science, engineering, and mathematics students developed by the Institute of Technology Centre for Educational Programs and School of Mathematics, University of Minnesota. The main objective of the Initiative is to enable undergraduates to better learn calculus and the critical thinking skills necessary to apply it in a variety of science and engineering problems. Changes in content and pedagogy are emphasized, including instructional teamwork and student-centred learning, involving students working cooperatively in small groups and exploring mathematical ideas using appropriate technologies. Achievement and retention of Initiative students are compared with a control group from the standard calculus sequence. Student attitudes about the usefulness of the Initiative's curriculum, pedagogy, and its influence on learning are discussed. Future implications including new uses of distributed learning are also addressed.     

Remarks on the relations between the Italian and American schools of algebraic geometry in the first decades of the 20th century

In this paper we give an overview of the interactions between Italian and American algebraic geometers during the first decades of the 20th century. We focus on three mathematicians—Julian L. Coolidge, Solomon Lefschetz, and Oscar Zariski—whose relations with the Italian school were quite intense. More generally, we discuss the importance of this influence in the development of algebraic geometry in the first half of the 20th century.     

Teaching objectives: the value of using cases in teaching operational research

This paper describes our experience with a case study that was intended to provide business students with a straightforward exercise in operational research/management science (OR/MS) but which consistently produces a learning experience for students that is quite different from that envisaged when the case was originally written. An effort is made to explain the surprising results in the classroom. This example demonstrates the fact that building OR/MS models often reveals important dimensions of a problem that otherwise might remain unexplored. The richness of this case also demonstrates the value of exposing OR/MS students to real problems through the use of cases.