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When more can be less: book review
No full textAn Introduction to Ordinary Differential Equations first edition By James C. Robinson Cambridge University Press 399pp, Pounds 65.00 and Pounds 24.99 ISBN 0 521 82650 0 and 53391 0Differential Equations with Mathematica third edition By Martha L. Abell and James P. Braselton Academic Press, 876pp, Pounds 40.95 ISBN 0 12 041562 3My seven-year-old returned from school last week waving a mathematics syllabus. Over the next three years, she will learn to multiply two-digit numbers in three different ways. If anything will confuse all but the swiftest of learners, it must surely be having to learn a new algorithm each year. This approach wastes two years during which more advanced material could be covered, and has consequences throughout the mathematical education of a UK student. I therefore sympathise with James Robinson, author of An Introduction to Ordinary Differential Equations. His stated motivation for writing the book is that he could not find a suitable text that covered less-advanced first-year undergraduate mathematics topics at length. Indeed, the first 40 per cent of the book covers material previously firmly embedded in A-level study. Differential Equations with Mathematica by Martha Abell and James Braselton also deals with contemporary first-year undergraduate mathematics courses in differential equations. Both books cover equations of first and second order, numerical solutions, coupled linear and nonlinear systems of equations. There, however, the similarity ends.Robinson's approach is straightforward but includes novel examples. When discussing resonance, he chooses not to discuss the cliched Tacoma Narrows Bridge collapse, but uses the marching-related failure of the Broughton suspension bridge in 1831. Continuing the design theme, London's Millennium Bridge features as one of the many exercises and provides the jacket illustration.The middle sections cover series solutions, numerics and difference equations. To this point, critics might comment that much of the material could be found in the plethora of texts containing "mathematics", "engineering" and "advanced" in the title. However, the style of the final two sections, covering coupled linear and nonlinear equations, with hints of what can be done later, directs the book at mathematicians.The very title of Differential Equations with Mathematica rules out its use on many courses. But the authors have ecumenically written an almost identical book involving Maple. Titles using Matlab also exist.Linking the learning of fundamental topics so firmly with computer packages runs the obvious risk of germinating mathematical couch potatoes. Courses now often involve students "working through" pre-prepared computer notebooks. Theoretically, this "innovative" solution satisfies the nebulous concept of "student-centred learning" while minimising the associated increase in staff workloads.In practice, however, some students sit, chins in hands, rhythmically tapping return keys, staring blankly at the screen. Finding that this is not the Game Boy entertainment they expect from university, they fire up their MP3 players and retreat into the Darkness.Some readers will find the repeated interruption of basic definitions and examples with code in Differential Equations with Mathematica extremely distracting. Conversely, the presence of code tempts the motivated student into active interaction. Rather than passively accepting the complexity of a dynamical system, they can easily plot the trajectories to understand the structure better. Faced with the Fourier series expansion of a discontinuous function, for example, they observe the Gibbs phenomenon experimentally. It is that type of student who will benefit most from this text.Interestingly, this book contains many examples rooted in biological systems. It deals with Fourier series, Laplace transforms and partial differential equations, and so it can be used to teach engineers. One concern is that the nomenclature of notebooks on the accompanying CD does not appear to correspond with the printed book.The contrasting approaches of these two texts indicate that undergraduate mathematics is again on the cusp. A little-noticed statement in the recent Tomlinson report suggests that the A-level syllabus will shrink by a third in the new diploma.If the recommendations of this report are implemented in full, I fear that by the time my overmultiplied daughter leaves school, Robinson will have had to write another book.Christopher Howls is senior lecturer in applied mathematics, Southampton University
Book review. Asymptotics and mellin–barnes integrals (Encyclopedia of Mathematics and its Applications 85) by R.B. Paris and D. Kaminski: 422 pp., £65.00 (US$95.00), ISBN 0 521 79001 8 (Cambridge University Press, 2001)
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Hyperasymptotics
No full textWe develop a technique for systematically reducing the exponentially small (‘superasymptotic’) remainder of an asymptotic expansion truncated near its least term, for solutions of ordinary differential equations of Schrödinger type where one transition point dominates. This is achieved by repeatedly applying Borel summation to a resurgence formula discovered by Dingle, relating the late to the early terms of the original expansion. The improvements form a nested sequence of asymptotic series truncated at their least terms. Each such ‘hyperseries’ involves the terms of the original asymptotic series for the particular function being approximated, together with terminating integrals that are universal in form, and is half the length of its predecessor. The hyperasymptotic sequence is therefore finite, and leads to an ultimate approximation whose error is less than the square of the original superasymptotic remainder. The Stokes phenomenon is automatically and exactly incorporated into the scheme. Numerical computations confirm the efficacy of the technique
Overlapping Stokes smoothings: survival of the error function and canonical catastrophe integrals
No full textWe derive doubly uniform approximations for the remainder in the optimally truncated saddle-point expansion for an integral containing a large parameter. Double uniformity means that the formulae remain valid while distant saddles responsible for the divergence of the expansion coalesce and separate (as described by catastrophe theory) and while the subdominant exponentials they contribute switch on and off (as described by the error-function smoothing of the Stokes phenomenon). Two sorts of asymptotic singularity are thereby united in a common framework. The formula for the remainder incorporates both the Stokes error function and the canonical catastrophe integrals. A numerical illustration is given, in which the distant cluster contains two saddles; the asymptotic theory gives an accurate description of the details of the fractional remainder, even when this is of order exp ( –36)
Fake Airy functions and the asymptotics of reflectionlessness
No full textTwo classes of analytic refractive-index profile P2(z,ε ), whose reflection coefficients r are zero for all values of a parameter in , are studied as in to 0. The aim is to understand why r=0 rather than r varies as exp(-1/ε ) as for generic profiles. The authors find that reflectionlessness is a consequence of the fact that transition points of P2 (zeros or poles in the complex z plane) form tight clusters (whose size vanishes with in ) which can be regarded neither as coalesced nor well separated. Expansion near a cluster yields the local wave not as the usual Airy function, whose Stokes phenomenon generates reflection, but as Bessel functions of half-integer order (fake Airy functions) which are exactly trigonometric functions with no Stokes phenomenon and so no reflection
High orders of the Weyl expansion for quantum billiards: resurgence of periodic orbits, and the Stokes phenomenon
No full textA formalism is developed for calculating high coefficients cr of the Weyl (high energy) expansion for the trace of the resolvent of the Laplace operator in a domain B with smooth boundary ∂B. The cr are used to test the following conjectures. (a) The sequence of cr diverges factorially, controlled by the shortest accessible real or complex periodic geodesic. (b) If this is a 2-bounce orbit, it corresponds to the saddle of the chord length function whose contour is first crossed when climbing from the diagonal of the Möbius strip which is the space of chords of B. (c) This orbit gives an exponential contribution to the remainder when the Weyl series, truncated at its least term, is subtracted from the resolvent; the exponential switches on smoothly (according to an error function) where it is smallest, that is across the negative energy axis (Stokes line). These conjectures are motivated by recent results in asymptotics. They survive tests for the circle billiard, and for a family of curves with 2 and 3 bulges, where the dominant orbit is not always the shortest and is sometimes complex. For some systems which are not smooth billiards (e. g. a particle on a ring, or in a billiard where ∂B is a polygon), the Weyl series terminates and so no geodesics are accessible; for a particle on a compact surface of constant negative curvature, only the complex geodesics are accessible from the Weyl series
Infinity interpreted
No full textDespite the denunciations of the mathematician Abel, if the devil did invent divergent series it was because his creator counterpart chose to build our physical universe so that they are among the more useful ways to describe its finite properties
Appearance of the higher-order Stokes phenomenon in a discrete Airy equation
No full textWe study a discrete variant of the Airy equation and show that discretization produces amore intricate Stokes structure than in the continuous case, inducing the higher-orderStokes phenomenon and infinite accumulations of Stokes and anti-Stokes curves. Thesefeatures are absent in the continuous Airy equation and are typically seen only in solutionsof at least third-order linear homogeneous, second-order or higher linear inhomogeneous, ornonlinear differential equations. Remarkably, this behavior is seen here to arise in asecond-order homogeneous linear difference equation. Using exponential asymptoticmethods, we derive the asymptotic solutions and the corresponding Stokes structure, withnumerical simulations confirming our predictions. We conjecture that the higher orderStokes phenomenon is able to be present in other second order linear difference equations
Dispersive hyperasymptotics and the anharmonic oscillator
No full textHyperasymptotic summation of steepest-descent asymptotic expansions of integrals is extended to functions that satisfy a dispersion relation. We apply the method to energy eigenvalues of the anharmonic oscillator, for which there is no known integral representation, but for which there is a dispersion relation. Hyperasymptotic summation exploits the rich analytic structure underlying the asymptotics and is a practical alternative to Borel summation of the Rayleigh–Schrödinger perturbation series
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