Abstract - Writing a Thesis

Mathematical Writing - Vivaldi Franco 2014

Abstract
Writing a Thesis

An abstract is a short summary of a document. If well-written, it complements the title and sustains the reader’s attention. The abstract of a research paper is placed immediately after the title, on the front page. In a thesis, where space constraints are not so severe, the abstract usually appears on a separate page.

It is difficult to compose a convincing abstract before the work has been completed, understood, and written out. So the abstract—the first thing one reads—is typically the last thing to be written. The main results must be in evidence, expressed with precision and clarity; at the same time, specialised jargon should be reduced to a minimum, so as not to alienate potential readers. A delicate balance must be achieved between these conflicting requirements.

Abstracts are short, rarely exceeding 200 words. The use of symbols is to be reduced to a minimum, the space is confined and every word counts. The exercise of writing MICRO-ESSAYs (see Sect. 6.6) is good training for writing abstracts, as both tasks require similar rigour and discipline.

We now examine in detail abstracts taken from mathematical literature. They are less than optimal, and we seek to improve them. (The original text has been edited, mainly to protect anonymity.) We don’t intend to produce templates of well-written abstracts—this would be futile, as no two abstracts are the same. Rather, we want to sharpen our analytical skills, learn how to spot problems, and find ways of solving them. For this purpose, understanding the mathematics is not essential. Indeed, dealing with some unknown words and symbols has a certain advantage, as one must pay attention to the writing’s internal structure. We have met a similar situation in Chap. 7, when we learnt how to organise the beginning of a proof.

Our first example, from a research paper, exemplifies a rather common problem: the use of unnecessary symbols.

ABSTRACT. Let $$F$$ be a rational map of degree $$n\geqslant 2$$ of the Riemann sphere $$\overline{\mathbb {C}}$$. We develop a theory of equilibrium states for the class of Hölder continuous functions $$f$$ for which the pressure is larger than sup $$f$$. We show that there exists a unique conformal measure (reference measure) and a unique equilibrium state, which is equivalent to the conformal measure with a positive continuous density.

The fault is plain: the symbols $$F$$, $$n$$, and $$\overline{\mathbb {C}}$$ are introduced, but not used again in the abstract, breaking the first golden rule of symbol usage—see Sect. 6.2. Unused symbols distract rather than help. Even if these symbols are used again in the main text, the abstract is not the place where the notation is to be established, and these definitions should be deferred. On this account, the symbol $$f$$ in the second sentence is used appropriately. Embedded in the second sentence we find the abbreviation ’sup’, for ’supremum’. This abbreviation (much like ’max’, for maximum, or ’lim’, for limit) is meant for formulae, and should not be mixed with words. So we replace it by the full expression.

Re-writing the first sentence without symbols and amending the second sentence do not require any understanding of the mathematics. We leave the third sentence as it is.

ABSTRACT. We consider rational maps of the Riemann sphere, of degree greater than 1. We develop a theory of equilibrium states for the class of Hölder continuous functions $$f$$ for which the pressure is larger than the supremum of $$f$$. We show that ...

The symbol $$f$$ in the second sentence is not strictly necessary, and can easily be disposed of.

We develop a theory of equilibrium states for the class of Hölder continuous functions for which the pressure is larger than the supremum of the function.

The next abstract belongs to a thesis. The author presents an accessible problem using limited jargon.

ABSTRACT. The logistic map is a well-studied map of the unit interval into itself. However, if we treat $$x$$ as a discrete variable, as is done in any computer, then every orbit is eventually periodic. Thus the aperiodic behaviour that the continuous map displays for some value of the parameter $$r$$ cannot be obtained from computer simulation. We investigated the differences and the similarities between the dynamics of a continuous map and its discrete approximation. We found that the limit cycles of a discrete map follow the unstable periodic orbits of the corresponding continuous map.

There are unknown words (logistic map, orbit, limit cycle), but also sufficiently many familiar terms (map, unit interval, eventually periodic, etc.) to help us discern the subject matter. The first sentence says that this work is about the ’logistic map’, presented as a well-known object of investigation. The second and third sentences motivate the study. The author introduces the symbol $$x$$, but we are not told what it is. We guess that it must represent the argument of the logistic map, namely a point in the unit interval. Furthermore, this symbol is not used again in the abstract, so it is unnecessary. The symbol $$r$$ suffers from similar problems. It’s a parameter, but we are not told which quantity depends on it. We guess that this is the logistic map, which is referred to as the ’continuous map’, presumably meaning that its argument assumes a continuum, rather than a discrete set, of values. The symbol $$r$$ is not used again either.

With two important ideas not in sharp focus, the main message of the thesis—if we discretise the domain of the logistic map, then we obtain new phenomena worth studying—gets a bit lost.

The closing sentence, which summarises the main findings, is vague. What does ’we found’ mean? Presumably this is not a proof, otherwise the author would have stated that clearly. What does ’follow’ mean? Presumably it indicates some form of convergence. Even if in the abstract there isn’t enough room for a complete statement of the results, we must find a way to render these vague statements acceptable. Finally, the author employs the past tense (we investigated, we found). In an abstract, the present tense is more common. Based on the above considerations, we rewrite the abstract as follows.

ABSTRACT. We consider the logistic map, a well-studied map of the unit interval which depends on a parameter. If the domain of this map is discretised, as happens in any computer simulation, then, necessarily, all orbits become eventually periodic. Thus the aperiodic orbits observed for certain parameter values no longer exist. We investigate differences and similarities between the original map and its discrete approximations. We provide evidence that the limit cycles of the discrete map converge to the unstable orbits of the original map, in a sense to be made precise.

The last expression states that the thesis contains experimental data or heuristic arguments that support a clearly formulated notion of convergence. (We hope that this is the case in the present thesis!)

Next we consider the opening sentences of an abstract of a research paper. The problems are more subtle here: there are no superfluous symbols, and we can’t rely on an understanding of the mathematics!

ABSTRACT. In this work we investigate properties of minimal solutions of multi-dimensional discrete periodic variational problems. A well-known one-dimensional example of such a problem is the Frenkel-Korontova model. We select a family ...

The opening sentence begins with the expression ’in this work’; this is redundant and should simply be deleted. Expressions of this type serve a purpose only if the present work is being juxtaposed to other works: ’In 1964, Milnor proved an estimate...In this work, we prove...’

The object of the investigation is described by a long string of attributes: ’multi-dimensional discrete periodic variational problem’. Such a flat arrangement of words requires a pause or a highlight. At this stage however, it’s not clear how this should be done, so we read on.

The second sentence begins with an expression of the type ’an example of this is that’. We have already criticised this format in Sect. 1.3. As written, the emphasis is placed on the word ’example’, but the terms ’one-dimensional’ or ’Frenkel-Korontova model’ surely are more significant. The author’s intentions are now clearer: the term ’multi-dimensional’ in the first sentence is contrasted with ’one-dimensional’ in the second sentence, and a known example is given for clarification. To get across the message that this work generalises ’one-dimensional’ to ’multi-dimensional’, we shall isolate the latter in the first sentence and emphasise the former in the second.

ABSTRACT. We investigate properties of minimal solutions of discrete periodic variational problems, in the multi-dimensional case. These generalise one-dimensional problems, such as the well-known Frenkel-Korontova model. We select a family ...

In our final abstract some symbols are necessary.

ABSTRACT. Let $$f:X\rightarrow X, X=[0,1)$$, be an interval exchange transformation (IET) ergodic with respect to the Lebesgue measure on $$X$$. Let $$f_t:X_t\rightarrow X_t$$ be the IET obtained by inducing $$f$$ to $$X_t=[0,t), \,0<t<1$$. We show that

$$ X_{wm}=\{0<t<1\, : \, f_t \text { is weakly mixing}\} $$

is a residual subset of $$X$$ of full Lebesgue measure. The result is proved by establishing a Diophantine sufficient condition on $$t$$ for $$f_t$$ to be weakly mixing.

The overall content of the abstract is unclear, although we recognise several words introduced in Chaps. 2 and 4. These fragments of understanding are sufficient to clarify the logic of the argument.

The set $$X$$ is the unit interval, and we have a function $$f$$ of this set into itself. This function belongs to a certain class of functions, identified by the acronym IET. The author assumes that $$f$$ has certain properties, specified by some jargon. Then a parameter $$t$$ is introduced, and the same construction is repeated for a one-parameter family of functions $$f_t$$ over sub-intervals $$X_t$$. The functions $$f_t$$, constructed from $$f$$ via a process we need not worry about, are also of type IET. (Evidently the author believes that this assertion does not require justification.)

Using a Zermelo definition, the author introduces a subset $$X_{wm}$$ of the open unit interval, determined by a property of $$f_t$$, called ’weak mixing’. The subscript $$wm$$ in the symbol $$X_{wm}$$ must be an abbreviation for this expression. The paper establishes some properties of the set $$X_{wm}$$. An indication of the argument used in the proof is then given.

Our first task is to remove some symbols, including the displayed equation. This isn’t too difficult; for example, the symbol $$X_{wm}$$ is never used. The first two sentences are formal definitions, which need to be made more readable. The acronym IET is useful, since the long expression it represents is used twice. However, the primary object is ’interval exchange transformation’, not IET, so the latter should follow the former, not precede it. Finally, in the expression ’Diophantine sufficient condition’ the two adjectives compete with each other, so we separate them out.

ABSTRACT. Let $$f$$ be an interval exchange transformation (IET) of the unit interval, ergodic with respect to the Lebesgue measure, and let $$f_t$$ be the IET obtained by inducing $$f$$ on the sub-interval $$[0,t)$$, with $$0<t<1$$. We show that the set of values of $$t$$ for which $$f_t$$ is weakly mixing is a residual subset of full Lebesgue measure. The result is proved by establishing a Diophantine condition on $$t$$, which is sufficient for weak mixing.

More examples of abstracts in need of improvement are given as exercises.