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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 2 (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
354 Book reviews Earth’s Magnetism in the Age of Sail
"... For many centuries, the source, behavior, and even the essential nature of geomagnetism were enigmatic. Despite this, the effect of geomagnetism was familiar, by imparting a directional preference on the magnetized needle of the compass and providing a useful, if somewhat annoyingly complicated, re ..."
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For many centuries, the source, behavior, and even the essential nature of geomagnetism were enigmatic. Despite this, the effect of geomagnetism was familiar, by imparting a directional preference on the magnetized needle of the compass and providing a useful, if somewhat annoyingly complicated, reference for navigators. Although the compass seems to have first been invented in China, it was the Europeans who made the most systematic early studies of magnetism, who made the first elaborate and practical usage of the compass, and who developed most of the early theories as to the cause of the compass needle’s northseeking tendency. From the centuries of the Middle Ages, through the late 16th century of the Renaissance, to the 17th century of philosophical enlightenment and the 18th century of discovery, the subject of magnetism and, more specifically, geomagnetism, evolved from a hodgepodge of mystical beliefs into something that we can today recognize as the object of modern scientific pursuit. Those same centuries witnessed the great transoceanic sailing voyages undertaken by European nations for reasons of exploration, territorial claim, religious mission, and mercantile trade. Naturally, the navigator’s compass, and therefore geomagnetism, played an important role in these developments. This romantic intersection of science and history is the subject of Earth’s Magnetism in the Age of Sail, a pleasantly written and scholarly book by A.R.T.