摘要:

In this note the existing proofs of the fact that a real, symmetric, nxn matrix A has a real eigenvalue and that there exists an orthonormal basis for R” consisting of eigenvectors of A are reviewed. A proof of this fact is then given which differs from the existing proofs in that it uses no results from the theory of self‐adjoint operators on complex inner product spaces or from analysis. This makes it possible to prove the spectral theorem for R” in introductory linear algebra courses where, heretofore, such a proof has not usually been given.

ISSN:0020-739X    

DOI:10.1080/0020739790100411

出版商:Taylor & Francis Group    

年代:1979

数据来源: Taylor

摘要:

While most university teachers think they give satisfactory lectures, few claim to give good tutorial classes. This paper describes a teaching method developed for a service course in mathematics for students of science and engineering. The proceedings of one group were videotaped and the tapes subsequently viewed by staff and students. A number of important points of teaching technique emerged from the ensuing discussion.

ISSN:0020-739X    

DOI:10.1080/0020739790100412

出版商:Taylor & Francis Group    

年代:1979

数据来源: Taylor

摘要:

The fascinating interplay between mathematics and mechanics is becoming more popular.

ISSN:0020-739X    

DOI:10.1080/0020739790100413

出版商:Taylor & Francis Group    

年代:1979

数据来源: Taylor

摘要:

This article elaborates on the following aspects of the role played by the history of mathematics in mathematical education: (1) content of curriculum, (2) standard of rigour, (3) interrelation among topics, (4) understanding of subject matter, (5) cultural viewpoint, (6) confidence in doing mathematics.

ISSN:0020-739X    

DOI:10.1080/0020739790100414

出版商:Taylor & Francis Group    

年代:1979

数据来源: Taylor

摘要:

A dimensional extension of complex numbers is used to derive the general solution to the cubic and quartic equations.

ISSN:0020-739X    

DOI:10.1080/0020739790100415

出版商:Taylor & Francis Group    

年代:1979

数据来源: Taylor

摘要:

A theory of strategy games that emphasizes algorithmic aspects will be developed. A partition algorithm that classifies each possible game position to be either a winning position or a loss position or a draw position is then applied to analyse a special game.

ISSN:0020-739X    

DOI:10.1080/0020739790100416

出版商:Taylor & Francis Group    

年代:1979

数据来源: Taylor

摘要:

This paper attempts to explain why we have felt the need to alter the mode of teaching our undergraduate mathematics courses at Victoria University in New Zealand (change is also occurring at Massey University). It attempts to analyse the needs of the students and practical difficulties facing the lecturers. It involves a brief description of the system in New Zealand and the many and various ways in which we have attempted to alter, change and improve our mathematics teaching and some comments on the relative success of these methods.

ISSN:0020-739X    

DOI:10.1080/0020739790100417

出版商:Taylor & Francis Group    

年代:1979

数据来源: Taylor

摘要:

Bayes’ theorem is often applied in problems that essentially involve finding the accuracy of atesting device, given the sensitivity of the device. It is shown how linear programming can be used to determine (in a certain sense) the minimum sensitivity needed for a given prescribed minimum accuracy.

ISSN:0020-739X    

DOI:10.1080/0020739790100418

出版商:Taylor & Francis Group    

年代:1979

数据来源: Taylor

摘要:

Since the days of the colonial colleges, topics in the calculus have been primarily a part of the college and university curriculum in the United States. During the past two decades, however, the traditional first course in calculus has gained a considerable foothold in American secondary schools. This downward shift in the placement of the first calculus course has been accompanied by considerable controversy. The calculus issue is presented here in an historical context, and related empirical studies are reviewed.

ISSN:0020-739X    

DOI:10.1080/0020739790100419

出版商:Taylor & Francis Group    

年代:1979

数据来源: Taylor

摘要:

This paper has two aspects, one mathematical, the other in mathematics education. Mathematically, it investigates the bifurcation set K of the linear differential equation x = Ax on the plane, when the real 2x2 matrix A is regarded as a point in four‐dimensional space R4. The relation with mathematics education is to provide an exposition showing in detail how the material combines ideas from analysis, topology, geometry and algebra in an elementary and meaningful setting that is often not found in undergraduate courses, when these subjects are treated separately. In the hope of cultivating spatial awareness, we replace R4without essential loss of information by the three‐sphere S3, which we then project stereographically into three‐space R3, to draw two‐dimensional sketches of the projection of the set K?S3. This projection divides R3into regions, in each of which the differential equation displays a structurally stable phase‐portrait, and we emphasise the role of ‘degenerate’ cases in allowing each such portrait to change to another qualitatively different one. Use is made of the Jordan canonical form, which is derived for 2x2 matrices through a method avoiding the conceptual difficulties of the usual n X n treatment. The associated changes of variable give a good example of a ‘concrete’ group action on R4, and the isotropy subgroups are calculated; the orbits are the levels of a Morse‐theoretic function and their topology as three‐manifolds is described.

ISSN:0020-739X    

DOI:10.1080/0020739790100420

出版商:Taylor & Francis Group    

年代:1979

数据来源: Taylor