We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
To send content items to your account,
please confirm that you agree to abide by our usage policies.
If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account.
Find out more about sending content to .
To send content items to your Kindle, first ensure no-reply@cambridge.org
is added to your Approved Personal Document E-mail List under your Personal Document Settings
on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part
of your Kindle email address below.
Find out more about sending to your Kindle.
Note you can select to send to either the @free.kindle.com or @kindle.com variations.
‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi.
‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.
We investigate the connectedness properties of the set
$I^{\!+\!}(f)$
of points where the iterates of an entire function
$f$
are unbounded. In particular, we show that
$I^{\!+\!}(f)$
is connected whenever iterates of the minimum modulus of
$f$
tend to
$\infty$
. For a general transcendental entire function
$f$
, we show that
$I^{\!+\!}(f)\cup \{\infty \}$
is always connected and that, if
$I^{\!+\!}(f)$
is disconnected, then it has uncountably many components, infinitely many of which are unbounded.
We show that if $f$ is a transcendental meromorphic function with a finite number of poles and $f$ has a cycle of Baker domains of period $p$, then there exist $C > 1$ and $r_0>0$ such that $\bigg\{z:\frac1C r\lt |z|\lt Cr\bigg\}\cap \mbox{sing} (f^{-p})\ne\varnothing,{\for}r\ge r_0.$ We also give examples to show that this result fails for transcendental meromorphic functions with infinitely many poles.
Let $f$ be a transcendental meromorphic function and $U$ a Baker domain of $f$. We obtain new estimates for the behaviour of the iterates of $f$ in $U$ and we use these estimates to improve an earlier result relating the geometric properties of $U$ to the proximity of $f$ to the identity function in $U$. We also apply these estimates to Baker domains of transcendental meromorphic functions $f$ of the form
where $k \in {\mathbb N},\ a > 1$ and $b > 0$, and show that these Baker domains contain an unbounded set of critical points and an unbounded set of critical values.
Let f be a transcendental meromorphic function with finitely many poles such that the finite singularities of f-1 lie in a bounded set. We show that the Julia set of f has Hausdorff dimension strictly greater than one and packing dimension equal to two. The proof for Hausdorff dimension simplifies the earlier argument given for transcendental entire functions.
It is shown that for any meromorphic function $f$ the Julia set $J(f)$ has constant local upper and lower box dimensions, $\overline{d}(J(f))$ and $\underline{d}(J(f))$ respectively, near all points of $J(f)$ with at most two exceptions. Further, the packing dimension of the Julia set is equal to $\overline{d}(J(f))$. Using this result it is shown that, for any transcendental entire function $f$ in the class $B$ (that is, the class of functions such that the singularities of the inverse function are bounded), both the local upper box dimension and packing dimension of $J(f)$ are equal to 2. The approach is to show that the subset of the Julia set containing those points that escape to infinity as quickly as possible has local upper box dimension equal to 2.
Areas of coastal marshland formed an important and distinctive part of the landscape of Roman Britain, and current work is showing that different wetlands were utilised in very different ways. Some areas, for example in Essex and Kent, were simply exploited for their natural resources to produce salt and support seasonal grazing. Parts of Fenland were also used in this way, though the higher coastal siltlands were modified through the creation of drainage systems in order to improve agricultural opportunities within a landscape that was still liable to tidal flooding. A third strategy towards wetland exploitation is reclamation: a major transformation of the natural environment, involving the construction of a sea wall along the coast to keep tidal waters out and a system of drainage ditches cut into the surface of the former saltmarsh to lower the water table and remove surface run-off from the surrounding uplands.
For a transcendental meromorphic function f, various properties of the set
formula here
were obtained in [8] and [9]. Here we establish
analogous properties for the smaller sets
formula here
introduced in [5], and
formula here
We deduce a symmetry result for Julia sets J(f), and also indicate some techniques for showing that
certain invariant curves lie in I′(f), Z(f) and J(f).
Let D be a bounded region in ℝm, m ≥ 2. We say that a function u defined in D has asymptotic value α if there is a boundary path Γ:x(t), 0≤t<1, in D (that is, dist (x(t), ∂D)→0 as t→1), such that u(x(t))→α as t→1. If in addition, x(t)→ξ as t→1, then u has asymptotic value α at ξ.
The real difference equation an+2 − (λ|an+1| + μan+1) + an = 0 may be interpreted as a dynamical system Φ:(an, an+1) ↦ (an+1, an+2) acting in the plane. The set ΛP of points (λ, μ) for which the mapping Φ is periodic has a rich structure. In this paper, we derive some geometric properties of ΛP (for example, we show that it is unbounded and uncountable), and we derive criteria for Φ to be periodic. We also investigate when Φ is conjugate to a rotation of the plane, and we describe how the rotation numbers of the corresponding circle maps Φ/|Φ| are related to the structure of ΛP.
In this chapter and in Chapter 5 we shall present many mathematical applications of computer graphics. In order to draw the line somewhere (pardon the pun) we shall restrict ourselves to the mathematics associated with the plane and in particular with curves in the plane. There is a whole other realm, just as fascinating, connected with objects, such as curves, polyhedra and surfaces, in three-dimensional space. Nevertheless what the computer actually draws is, in these cases too, a curve or system of curves in the plane – that of the screen. The extra complications come from taking a three-dimensional object and associating with it a curve or system of curves – for example its outline when seen from a distance, or a sequence of such outlines or a sequence of plane sections of the object. We touch on this in a discussion of the swallowtail surface in Section 4.14.
The computer can (with our help) draw curves and collections of curves which are just too complicated to attempt by hand. For some purposes a rough sketch of a curve does very well – you will probably have drawn many such sketches by hand, and we are certain that the art of curve-sketching by hand is still an art well worth acquiring. However, for some other purposes, such as the illustration or discovery of facts connected with the differential geometry of curves and families of curves (not to mention surfaces), accurate drawings are essential.
The interaction between computer and mathematics is becoming more and more important at all levels as computers become more sophisticated. This book shows how simple programs can be used to do significant mathematics. The purpose of this book is to give those with some mathematical background a wealth of material with which to appreciate both the power of the microcomputer and its relevance to the study of mathematics. The authors cover topics such as number theory, approximate solutions, differential equations and iterative processes, with each chapter self-contained. Many exercises and projects are included giving ready made material for demonstrating mathematical ideas. Only a fundamental knowledge of mathematics is assumed and programming is restricted to 'basic BASIC' which will be understood by any microcomputer. The book may be used as a textbook for algorithmic mathematics at several levels, with all the topics covered appearing in any undergraduate mathematics course.
This book is intended for anyone who has some mathematical knowledge and a little experience with programming a microcomputer in BASIC (or any other language). The book shows how simple programs can be used to do significant mathematics.
To spell out our mathematical prerequisites in more detail, some of the chapters assume no more mathematical knowledge than whole numbers, but for the most part we assume some calculus, and the rudiments of algebra (polynomials, equations) and trigonometry (sines, cosines and tangents). Thus, British readers with A or A/S level in mathematics and American readers with a Freshman calculus course behind them will, we hope, have little difficulty in following most of the mathematics here. We have, naturally, included some material for the more mathematically sophisticated reader: those sections requiring closer inspection are, appropriately, printed in smaller type. (We hope that readers who do not immediately recognise the small type material will be intrigued rather than frightened. Surely one of the charms of mathematics is the glimpses it affords of mysterious and fascinating territory which is for the moment just out of reach.)
As for programming, the knowledge we assume is very small, and most programs are given full listings in the text. It seems to us that a very effective way to learn programming is to use it to solve interesting mathematical problems. We have regarded the mathematics as the pre-eminent interest, and have not tried too hard to make the sample programs beautiful or elegant, or even particularly efficient.
Sir Isaac Newton is certainly one of the greatest scientists to have ever lived. He is generally reckoned to have been one of the three most outstanding mathematicians of all time, along with Archimedes and Gauss, and his discoveries in physics are unrivalled in their width and influence. What was Newton's secret? How did he achieve as much as he did? Obviously there is no simple answer, but Newton had one secret, which he guarded jealously, and which he believed to be vital. It was ‘Data aequatione quotainque fluentes quantitoe involuente fluxions invenire et vice versa’ or in English ‘solve differential equations’.
Nowadays this ‘secret’ is entirely unremarkable; we are all aware that many processes and phenomena in the world are governed by differential equations. The very fact that Newton's secret is now common knowledge clearly indicates its worth and power. Of course his secret was rather hard won; he did have to invent differential equations before pronouncing his dictum concerning solving them!
In this chapter we shall see what the microcomputer can do for those intending to follow Newton's advice. Our eventual viewpoint will be considerably more modern than Newton's. It turns out that in certain circumstances solving differential equations is not as useful as watching them.
Differential equations and tangent segments
Much of science is devoted to the problems of predicting the future Differential behaviour of some physical system or other. Often the underlying equations and physical law will describe the rate at which the system evolves; what tangent segments we require is a description of how it evolves.
One of the most important and useful ways in which mathematics can help us to solve problems is by the solution of equations. ‘Let x be the length of the piece of string; then x satisfies the equation x2 – 2x - 3 = 0 and solving the equation gives x = 3.’ We are sure that you have solved many problems using equations; unfortunately all but the simplest equations cannot be solved exactly.
There are two reasons for this. In the first place even for a quadratic equation, unless the solutions are rational numbers (as in the above example), there is a square root such as √2 to be evaluated, and this cannot be done exactly. The decimal expansion does not terminate or recur, so we must be satisfied either with the formal ‘√2’ or with an approximation to so-many decimal places.
The second reason is more profound. Exact formulae analogous to the famous quadratic formula do exist for equations of degrees 3 and 4 – of course these formulae involve cube roots and so on, so are open to the same difficulty as we noted above for quadratics. On the other hand no algebraic formula exists at all for equations of degree 5 or more! In a precise sense, the equation x5 - 6x + 3 = 0 cannot be solved algebraically at all. This is a difficult statement and has an even more difficult proof, in which computers won't help in the least.