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Computational Differential Equations
, 1996
"... Introduction This first part has two main purposes. The first is to review some mathematical prerequisites needed for the numerical solution of differential equations, including material from calculus, linear algebra, numerical linear algebra, and approximation of functions by (piecewise) polynomial ..."
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Cited by 56 (4 self)
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Introduction This first part has two main purposes. The first is to review some mathematical prerequisites needed for the numerical solution of differential equations, including material from calculus, linear algebra, numerical linear algebra, and approximation of functions by (piecewise) polynomials. The second purpose is to introduce the basic issues in the numerical solution of differential equations by discussing some concrete examples. We start by proving the Fundamental Theorem of Calculus by proving the convergence of a numerical method for computing an integral. We then introduce Galerkin's method for the numerical solution of differential equations in the context of two basic model problems from population dynamics and stationary heat conduction.
Exploring the regular tree types
 In Types for Proofs and Programs
, 2004
"... Abstract. In this paper we use the Epigram language to define the universe of regular tree types—closed under empty, unit, sum, product and least fixpoint. We then present a generic decision procedure for Epigram’s inbuilt equality at each type, taking a complementary approach to that of Benke, Dyb ..."
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Cited by 21 (4 self)
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Abstract. In this paper we use the Epigram language to define the universe of regular tree types—closed under empty, unit, sum, product and least fixpoint. We then present a generic decision procedure for Epigram’s inbuilt equality at each type, taking a complementary approach to that of Benke, Dybjer and Jansson [7]. We also give a generic definition of map, taking our inspiration from Jansson and Jeuring [21]. Finally, we equip the regular universe with the partial derivative which can be interpreted functionally as Huet’s notion of ‘zipper’, as suggested by McBride in [27] and implemented (without the fixpoint case) in Generic Haskell by Hinze, Jeuring and Löh [18]. We aim to show through these examples that generic programming can be ordinary programming in a dependently typed language. 1
Looking at graphs through infinitesimal microscopes, windows and telescopes
 Math. Gaz
, 1980
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A Graphic Approach to the Calculus
 Sunburst Inc, USA (for I.B.M. compatible computers
, 1990
"... In recent years, reform calculus has used the computer to show dynamic visual graphics and to offer previously unimaginable power of numeric and symbolic computation. Yet the available technology has far greater potential to allow students (and mathematicians) to make sense of the ideas. A sensible ..."
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Cited by 10 (8 self)
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In recent years, reform calculus has used the computer to show dynamic visual graphics and to offer previously unimaginable power of numeric and symbolic computation. Yet the available technology has far greater potential to allow students (and mathematicians) to make sense of the ideas. A sensible approach to the calculus builds on the evidence of our human senses and uses these insights as a meaningful basis for various later developments, from practical calculus for applications to theoretical developments in mathematical analysis and even to a logical approach in using infinitesimals. Its major advantage is that it need not be based initially on concepts known to cause student difficulty, but allows fundamental ideas of the calculus to develop naturally from sensible origins, in such a way as to make sense in its own right for general purposes, support the intuitions necessary for applications, provide a meaning for the limit concept to be used later in standard analysis and further, to provide a sensible basis for infinitesimal concepts in nonstandard analysis.
Derivatives of containers
 of Lecture notes in Computer Science
, 2003
"... Abstract. We are investigating McBride’s idea that the type of onehole contexts are the formal derivative of a functor from a categorical perspective. Exploiting our recent work on containers we are able to characterise derivatives by a universal property and show that the laws of calculus includin ..."
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Cited by 8 (4 self)
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Abstract. We are investigating McBride’s idea that the type of onehole contexts are the formal derivative of a functor from a categorical perspective. Exploiting our recent work on containers we are able to characterise derivatives by a universal property and show that the laws of calculus including a rule for initial algebras as presented by McBride hold — hence the differentiable containers include those generated by polynomials and least fixpoints. Finally, we discuss abstract containers (i.e. quotients of containers) — this includes a container which plays the role of e x in calculus by being its own derivative. 1
Leibniz’s syncategorematic infinitesimals, smooth infinitesimal analysis and Newton’s proposition 6’, forthcoming
, 2006
"... In contrast with some recent theories of infinitesimals as nonArchimedean entities, Leibniz’s mature interpretation was fully in accord with the Archimedean Axiom: infinitesimals are fictions, whose treatment as entities incomparably smaller than finite quantities is justifiable wholly in terms of ..."
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Cited by 5 (0 self)
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In contrast with some recent theories of infinitesimals as nonArchimedean entities, Leibniz’s mature interpretation was fully in accord with the Archimedean Axiom: infinitesimals are fictions, whose treatment as entities incomparably smaller than finite quantities is justifiable wholly in terms of variable finite quantities that can be taken as small as desired, i.e. syncategorematically. In this paper I explain this syncategorematic interpretation, and how Leibniz used it to justify the calculus. I then compare it with the approach of Smooth Infinitesimal Analysis (SIA), as propounded by John Bell. Despite many parallels between SIA and Leibniz’s approach —the nonpunctiform nature of infinitesimals, their acting as parts of the continuum, the dependence on variables (as opposed to the static quantities of both Standard and Nonstandard Analysis), the resolution of curves into infinitesided polygons, and the finessing of a commitment to the existence of infinitesimals — I find some salient differences, especially with regard to higherorder infinitesimals. These differences are illustrated by a consideration of how each approach might be applied to Newton’s Proposition 6 of the Principia, and the derivation from it of the v2/r law for the centripetal force on a body orbiting around a centre of force. It is found that while Leibniz’s syncategorematic approach is adequate to ground a
Visualizing Differentials in Two and Three Dimensions
"... The current calculus curriculum may be very good at teaching the algorithms of differentiation and integration, but it is less successful at giving coherent meanings to the fundamental ideas. For instance, what do the dx and dy mean in the expression dy ..."
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Cited by 2 (2 self)
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The current calculus curriculum may be very good at teaching the algorithms of differentiation and integration, but it is less successful at giving coherent meanings to the fundamental ideas. For instance, what do the dx and dy mean in the expression dy
A COGNITIVE ANALYSIS OF CAUCHY’S CONCEPTIONS OF FUNCTION, CONTINUITY, LIMIT, AND INFINITESIMAL, WITH IMPLICATIONS FOR TEACHING THE CALCULUS
"... In this paper we use theoretical frameworks from mathematics education and cognitive psychology to analyse Cauchy’s ideas of function, continuity, limit and infinitesimal expressed in his Cours D’Analyse. Our analysis focuses on the development of mathematical thinking from human perception and acti ..."
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Cited by 1 (1 self)
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In this paper we use theoretical frameworks from mathematics education and cognitive psychology to analyse Cauchy’s ideas of function, continuity, limit and infinitesimal expressed in his Cours D’Analyse. Our analysis focuses on the development of mathematical thinking from human perception and action into more sophisticated forms of reasoning and proof, offering different insights from those afforded by historical or mathematical analyses. It reveals the conceptual power of Cauchy’s vision and the fundamental change involved in passing from the dynamic variability of the calculus to the modern settheoretic formulation of mathematical analysis. This offers a reevaluation of the relationship between the natural geometry and algebra of elementary calculus that survives in applied mathematics, and the formal set theory of mathematical analysis that develops in pure mathematics and evolves into the logical development of nonstandard analysis using infinitesimal concepts. It counsels us that educational theories developed to evaluate student learning are themselves based on the conceptions of the experts who formulate them. It encourages us to reflect on the principles that we use to analyse the developing mathematical thinking of students, and to make an effort to understand the rationale of differing theoretical viewpoints. 1.