摘要:

This paper describes a series of empirical studies, over a period of three years, designed to investigate the nature of learners’ difficulties with the proof technique of structural induction. The learners concerned were second‐year university students who had all completed a course in discrete mathematics and were currently attending a first course in Formal Methods of Computer Science. The main findings are applicable to both mathematical and structural induction, and can be ascribed to conceptual gaps. The precise nature of the errors made varied with the task and the teaching context, but a constant factor across all the studies was the failure of the majority of the subjects to appreciate the role of implication in induction. We suggest that this may be due, in part, to the conceptual gap between the main goal of the proof and the subgoals; the main goal might give false cues as to the structure of the subgoals. Another difficulty seen concerned the conceptual gap between knowing how to apply the principle of induction and understanding why it is valid. Our data indicate that performance may be improved by shifting teaching emphasis and by drawing learners’ attention to the structure of a recursive definition.

ISSN:0020-739X    

DOI:10.1080/0020739980290201

出版商:Taylor & Francis Group    

年代:1998

数据来源: Taylor

摘要:

This article is based on the work of Jameson who, in 1968, produced a solution of Sylvester's matrix equation. Jameson's method applies only to the case of auniquesolution. In 1993 the present author produced an improved version of Jameson's formula. Here it is proposed to update Jameson's method so as to provideallsolutions when the solution isnot unique.

ISSN:0020-739X    

DOI:10.1080/0020739980290202

出版商:Taylor & Francis Group    

年代:1998

数据来源: Taylor

摘要:

The paper looks at the mathematics of rotational and translational motion and how one can influence the other in the context of cams and cranks. With the aid of a spreadsheet and its graphical capabilities and a simple macro these motions can be animated so that the mathematics can be brought alive. The paper also describes how the individual components can be brought together to simulate a four‐stroke engine and the engine animated again using the same simple macro.

ISSN:0020-739X    

DOI:10.1080/0020739980290203

出版商:Taylor & Francis Group    

年代:1998

数据来源: Taylor

摘要:

The Jacobi method for solving a system of linear equations is applied to a class of problems. A detailed analysis of the relationship between theoretical convergence results and observed monotonic and non‐monotonic convergence is made, with particular consideration of the ?‐ and 1‐norms. A number of specific examples are included, together with additional examples for further consideration.

ISSN:0020-739X    

DOI:10.1080/0020739980290204

出版商:Taylor & Francis Group    

年代:1998

数据来源: Taylor

ISSN:0020-739X    

DOI:10.1080/0020739980290205

出版商:Taylor & Francis Group    

年代:1998

数据来源: Taylor

摘要:

A simple method of finding a biased estimate of the population proportion is introduced. This biased estimate is derived by finding a confidence interval for the population proportion without using the concept of convergence in probability. While all existing biased estimates of the population proportion are based on specific prior distributions, the biased estimate discussed here does not require any prior distribution. The proposed biased estimate is asymptotically unbiased, has smaller variance and smaller maximum mean squared error than the sample proportion. Under certain conditions it also has a smaller margin of error and smaller mean squared error than the sample proportion.

ISSN:0020-739X    

DOI:10.1080/0020739980290206

出版商:Taylor & Francis Group    

年代:1998

数据来源: Taylor

摘要:

The paper shows that basic concepts of discrete mathematics can be introduced to students through gradual elaboration of experiences with iconic spreadsheet‐based simulations of concrete materials. It suggests that the study of homogeneous and heterogeneous patterns of manipulative spreadsheet templates allows for the appreciation of the development of knowledge about discrete structures and offers a vision of mathematics as a human activity.

ISSN:0020-739X    

DOI:10.1080/0020739980290207

出版商:Taylor & Francis Group    

年代:1998

数据来源: Taylor

摘要:

Stochastic differential equations enrich man's modelling capacity to an astonishing degree. To illustrate this in a familiar environment two applications of the stochastic growth equation are discussed. The first has a familiar deterministic analogue. The second uses a stochastic differential with a random number of contributing sub‐differentials. There is no deterministic analogue in this case. Both applications admit of closed form solutions in a tractable form.

ISSN:0020-739X    

DOI:10.1080/0020739980290208

出版商:Taylor & Francis Group    

年代:1998

数据来源: Taylor

摘要:

The article shows how to construct simple numerical examples in two‐way ANOVA, specifically randomized blocks, balanced two‐way layouts, and Latin squares, such that the estimated effects and the residual standard deviation(ssay) are preset whole numbers. Working through simple numerical problems is helpful to students meeting a technique for the first time, to be followed by (computer‐based) analysis of larger, real datasets when the basic mechanics of the method are understood. This work extends that of Read (1985) on one‐way ANOVA and effectively completes the coverage of layouts met in elementary experimental design. First, the user defines an estimated mean and set of effects appropriate to his chosen layout and to the context in which the data are set, along with the value ofs.The estimated effects are essentially arbitrary, and so the significance or insignificance of a set of effects, of contrasts between them or of interactions, may be demonstrated on datasets designed for the purpose. The second step, to find a residual vector satisfying the side‐conditions implied by the estimating equations, is managed by simple table look‐up procedures. Adding the residual vector to the structural effects then gives the data for analysis. An example is given of each layout discussed.

ISSN:0020-739X    

DOI:10.1080/0020739980290209

出版商:Taylor & Francis Group    

年代:1998

数据来源: Taylor

摘要:

An algorithm to find optimal routeing for distribution‐related problems is presented. Usually these problems are formulated as transportation problems (TP) and solved using the stepping‐stone (SS) technique. In practice, however, the SS algorithm encounters major obstacles including difficulties in identifying an initial basic feasible solution, resolving SS degeneracy, and enumerating SS paths. The proposed algorithm solves the linear programming (LP) formulation of TP in a novel way and, unlike simplex or dual simplex methods, does not require artificial or slack/surplus variables. A general‐purpose algorithm is proposed which not only solves the classical TP, but also can effectively handle an unbalanced condition and additional side‐constraints on the routes. The proposed algorithm uses only the Gauss Jordan row operation, as used in the simplex method. As such, it is much easier to understand and apply, and can serve as an effective tool for teaching TP. Numerical examples are presented to illustrate applications of the proposed method. It is assumed that the reader is familiar with simplex terminology.

ISSN:0020-739X    

DOI:10.1080/0020739980290210

出版商:Taylor & Francis Group    

年代:1998

数据来源: Taylor