Advice on writing mathematics

Transcription

1 Advice on writing mathematics Brooke Benjamin (original author), revised by Jonathan Wattis December Introduction The following advice is aimed at students starting to write mathematical reports or theses, for undergraduate projects, MSc dissertations, PhD theses, or papers for publication. Most of these notes are informal suggestions, aimed at helping students develop a good written style. They are closely based on Advice on the composition of mathematical theses written by Brooke Benjamin, sometime before These rules are imposed by most publishers of mathematical journals and should be adhered to by those writing dissertations. Reading papers and text books provides valuable experience in learning what makes a good paper. When writing, consideration should be given to presentation, wording, the layout of mathematical expressions as well as scientific content. Taking scientific substance for granted, good written work still needs to be (i) accurate, (ii) concise, (iii) consistent with established conventions about presentation, and (iv) readable. These requirements are interdependent, in particular, readability is generally unattainable without (i) (iii). 2 Structure and organisation 2.1 Layout After a title, there should be an abstract. In theses and longer works, these should be on separate pages, and be followed by a table of contents, table of figures, and Acknowledgements. The funding agency of the work should be acknowledged, possibly with grant numbers quoted, if required by them. There may be an introduction of a few pages giving an overview of the work before chapter 1; also a list of notation or glossary of terminology may be presented before Chapter 1. After the body of work, there should be a short Conclusions section drawing together the project, then appendices, and finally a list of references. In a thesis each chapter should start on a fresh page, in papers and shorter reports, sections can follow on without a page break. When writing for a journal, consult the publisher s guide for authors to find their house style, and where appropriate download their L A TEXstylefile. It is usual for dissertations to be double-spaced. In LATEXthis can be accomplished by the command \renewcommand{\baselinestretch}{1.6} followed by \normalsize (with an argument of 2, text appears triple-spaced). Single-line spacing can be used for the table of contents, the list of figures and references, where normal spacing is acceptable (replace 1.6 by 1 in the above command). 2.2 Title & Abstract Think about the title, it should be informative but brief. not too general, and too long; as a rough guide less than a line in normal font size. The abstract should summarise briefly the entire contents of the paper, not just an outline of the problem. It should include a precis of the results and conclusions. 2.3 List of symbols/glossary Consider whether the reader would benefit from a glossary or list of symbols. In many mathematical reports this will not be necessary, but particularly in mathematical biology, a glossary of biological terminology, including a list of acronyms will be essential. 2.4 Titles Each section and subsection should have a title summarising its content. Repetition of titles should be avoided, and titles should ideally not stretch to more than one line. However, overly short titles, such as Case 3 or t = O(ε 3 ) are insufficiently descriptive. Titles do not need a full stop at their end, but should otherwise appear as a sentence, that is, there is no need to capitalise every word. Chapter/section 1 is nearly always simply Introduction and the final chapter/section is usually Conclusions, though Discussion is also acceptable. Chapters and sections should be divided into subsections, each with number and short title. Mathematical discussions covering many pages can be daunting to the reader. Even experts value natural breaks in the account, and marking these breaks with subtitles highlighting the work aids the reader in seeing the bigger picture. For example, Applications of equation (3.14), Conservation equations. Referring to chap- 1

2 ters, sections, figures should be made, with consistent capitalisation and preferably using the full word, for example see Figure 4, or as in section 3.6. Neither figure numbers or section numbers should be in brackets, only equation numbers need be bracketed, and then equations can be referred to either with or without using the word equation ; for example, see equation (6.13), or by substituting (3.7) into (2.1) we find Introduction Papers and theses should start with a clear statement of the aims and achievements of the research to be reported. This may fill the first paragraph, and should not be a repeat of the abstract. Whilst the abstract should be strictly neutral, in the introduction, there is space to be more expansive, and discuss the difficulties of the research and give the reader an indication of what to expect from the body of the work. Ostentatious openings should be avoided, for example, In recent years, important developments in the theory of X have led to.... A clear focus on the work to be presented is preferable to vague grandstanding. The introduction should include a review of background to the research, review of relevant literature, outline of methodology, and summary of the principle results. The content of later chapters should be outlined. The introduction can be shortened to a handful of pages, with a more extensive literature review and detailed discussion appearing in the second chapter. The introduction is often the hardest part of the thesis to write, and great care and effort should be invested as it is one of the parts most likely to be read, and will be the first impression that any examiner has of the work. With good reason, the introduction is often written last. 2.6 Conclusion Together with the abstract and introduction, this section will probably be read more often and in more detail than the body of the thesis. Although the conclusion should include a summary of the main results of the work, it should be used to comment more broadly on the subject area of the research. It is entirely appropriate to conjecture about unsolved problems, suggest topics and approaches for future work. 2.7 Appendices should be used for the presentation of supporting material that would unduly interrupt the flow of the argument in the main text. They should be relevant to the main topic of the thesis, it is not appropriate to present other, unrelated, research that the author has done. Suitable material includes supporting numerical simulations, statistical analyses, derivations of parameter values, unavoidably lengthy proofs. The material should be presented in a professional style, similar to work in the main body of the thesis or paper, though a more concise explanation may be appropriate, particularly for more routine calculations. Equations should be numbered (A1), (A2),.. in sequence through the appendices. If code is presented, it should be thoroughly commented, and preceded by a brief explanation of its workings. 2.8 Footnotes should be as few and as brief as possible. Almost all incidental remarks can either be accommodated in the text, possibly in parentheses, or put into an appendix. 2.9 References should be made in one style throughout the paper or dissertation. The references should include every paper, report, book, and other material mentioned in the thesis. It should not include references not mentioned in the thesis. The title References is preferable to Bibliography. The most common method of referencing papers is numerical; with the references arranged alphabetically by first author. An alternative is to number references in the order in which they are cited, thus [1] is the paper cited first in the thesis, and [2] the second. References are listed by their number (if used), authors, title, journal, volume, pages, year; or authors, year, title, journal, volume, pages. Authors initials may be placed before or after surnames, but in a consistent fashion. For books, author, title, publishing company, and city of publication should be quoted. Book titles should be capitalised throughout, but paper titles should not. It is common for journal names to be in italic font and volume numbers to be in bold. For papers which appear as chapters in a book, both the author of the chapter and the editor of the book should be cited, and both the titles of the book and the chapter given. References to websites, links, use of wikipedia, etc. should be kept to a minimum. It is acceptable to cite papers from preprint servers, such as the arxiv, giving appropriate reference numbers. However, it is preferable to cite a refereed version of the paper if available. Most wikipedia pages have lists of references detailing more authoritative sources where material has been obtained, it is better to cite one of these references rather than the wikipedia page itself. If citing a webpage is essential, it is common to quote the date it was accessed. As well as the web address (URL), you should also give the name of the organisation that is 2

3 responsible for the webpage, for example, Institute of Physics, University of Nottingham. For example, [1] CM Bender and S Orzsag. Advanced Mathematical Methods for Scientists and Engineers. Springer, New York, [2] Zabusky, N.J., Kruskal, M.D. Interaction of solitons in a collisionless plasma and the recurrence of initial states. Phys. Rev. Lett. 15, , (1965). [3] Fermi, E., Pasta, J., Ulam, S.: Studies of nonlinear problems. Los Alamos Scientific Laboratory report LA-1940, Reprinted in: Lect. in Appl. Math. 15, , (1974) [4] Eilbeck, J.C.: Numerical studies of solitons on lattices. In: Remoissenet, M., Peyrard, M. (eds.), Nonlinear Coherent Structures in Physics and biology. In: Lect. Notes in Phys. Vol. 393, Berlin, Heidelberg, New York: Springer 1991, pp [5] M Gleiser, SI Walker. The chirality of life: from phase transitions to astrobiology. arxiv: [astro-ph], (2008) 2.10 In papers, personal communications can be cited in a similar way as a paper. For example, these phenomena have also been observed by Benjamin [11], which is associated with an item in the references of the form [11] Benjamin, TB, personal communication (1987). However, in theses, personal communications should be cited simply in the text as... have also been observed by Benjamin (personal communication, 1987) References to books should include the page number, or section number where the appropriate material can be found (however, there is no need to give both). Even when citing a whole book, it can be helpful to note which sections are particularly relevant. 3 Style 3.1 All mathematics should be presented as part of sentences which obey the usual rules of English grammar, and correct punctuation should be used. More punctuation is required in mathematical work than in ordinary writing. A useful rule is that every displayed equation or other mathematical expression must end with a full stop, or a comma, for example when the following phrase is where.... Do not introduce displayed mathematical expressions with a colon, unless this punctuation is required by the syntax. Whilst The transformation may be represented as: is correct; it is preferable to write The transformation has the form The variety of English sentences that can be used to present mathematics is virtually limitless, and written work allows the author great scope for originality of expression. It is customary to use we in writing, to denote both the author and the reader as a pair. The only context where use of I is appropriate is in personal statements, such as, I consider this result to be the main accomplishment of the thesis. Excessive use of any standard construction should be avoided. For example, long passages in which most sentences start with We... are turgid. Use variety of expression to promote readability and give due points emphasis; intersperse active and passive constructions; occasionally use one as an alternative to we. For example, It is found by induction that..., Suppose that x > y,... Whilst mathematical symbols can have the role of a verb, for example =, >, in any sentence, the main verb should be an English word. Generally this rule should only be broken in exceptionally short sentences, such as Therefore x = ct. 3.3 All displayed equations should be numbered. Even if the author does not refer to an equation, another reader or examiner may need to refer to it. Systems of equations need only be numbered once if desired. If an equation occupies the whole line, the number should appear on the next line rather than extending into the margin. Displayed items can be numbered (1), (2) etc throughout, or numbered from 1 in successive chapters or sections, for example (4.3.1), (4.3.2), (4.3.3), etc. Equations can be referred to simply by their number, as in by solving (6.3) we obtain (6.5). 3.4 When putting mathematical expressions in text care should be taken so that small fonts are not needed. Thus avoid subscripts on subscripts. Preferred Avoid a a/b b ( 13) or Q 1/5 Q 1 5 exp( x 2 ) x u or u/ x e x2 u x However, simpler fractions such as 1 2, 1 3 can be displayed as such in text. Even in displayed formulae, where a smaller font is required, a/b is preferable to a b, for example, in the limits on integrals, and when writing e nπix/l. The 3

4 usual sequence of brackets is { [ ( ) ] } To avoid following every equation with where..., try ending the equation with a full stop, and start a new sentence with Here.... This is particularly appropriate where the explanation of the equation is long. 3.5 When letters of the English alphabet are used as mathematical symbols, they should be written in italic font, vectors should be in bold, the same form should be used in displayed equations and in text. There are many common functions which should appear in normal font (i.e. not italicised), for example, sin, log, exp, lim, det, etc. In L A TEX, remember to use \sin etc. 3.6 The end of a proof needs to be indicated clearly, and distinguished from the discussion that follows. A simple statement such as the proof of Theorem1 is thus complete is adequate, alternatively, use the symbol. 3.7 Mathematical theses typically contain a great deal of jargon, much of it peculiar to narrow subject areas. A good thesis will include explanations of terms and make it easy for a reader from another area to follow the argument. It is often simplest to give explanations of terms in parentheses as terms are encountered. 3.8 All figures should be numbered in order of introduction and should be given captions and discussed in the the text; for example, Figure 3. Plot of the function y(x) against x, as defined by (2.4) for the parameters a = 1, b = 2. Choose one of the styles figure 3, Fig. 3, or Figure 3, and use it consistently throughout. Never place a figure into a sentence, eg this function has the form [gap in text filled with figure]. 3.9 Be careful with the use of since ; use because, for or as instead. The principle meaning of the word since relates to time Do not start a sentence with And. Inside a paragraph, it is acceptable to start a sentence with But, however, do not start a paragraph with But Strictly whose should refer only to persons, though it may be applied to mathematical objects, for example, Diffusion of heat is described by (5), whose general solution is (6), The domain is bounded by the arc CD, whose radius of curvature is given by (4). Note that in the second example, the use of whose avoids the awkward repetition of of Avoid this, these, that and those as unrelated pronouns. These words should always be followed by a qualifier giving further explanation, e.g. these parameters..., this equation Another way of avoiding the overuse of where following an equation is to use in which. For example,... establishes the inequality θ > qφ(1 + φ 2 ), in which q is the coefficient of To aid readability, consider following displayed formulae with the occasional use of: whence (from which it follows that), whereupon (as a result of which), or whereby (by means of which) Take care in splitting infinitives, avoid where possible, but do not fear to split an infinitive if it is the clearest and most natural expression. Eg The generalised theory is needed in order to fully explain the property. ( fully splits the infinitive to explain, but acceptable) The generalised theory is needed in order fully to explain the property. ( to explain is not split, but sounds slightly awkward) The generalised theory is needed in order to explain fully the property. ( to explain is not split, the preferred option). The last estimate seems to just fail the test. (preferred option, though just splits the infinitive) The last estimate seems just to fail the test. ( to fail is not split, but not as natural as above). The last estimate seems to fail just the test. ( to fail is not split, but sense of the sentence is lost) 3.17 Make sure the subject of the verb is clear and appropriate. For example, Assuming λ to be positive, multiply (6) by λ and integrate... (poor) Assuming λ to be positive, we multiply (6) by λ and integrate... (preferred) We assume λ to be positive and we multiply (6) by λ and we integrate... (too repetitive). Substituting (2) for x in this equation and cancelling the factor c then common to both sides, the result (1) is recovered. (poor grammatical style, as the result (1) is not the subject of the verbs substitute or cancel. Only the author and the reader are able to cancel and make substitutions). Substituting (2) for x in this equation, we cancel the factor c then common to both sides, and recover the result (1). (preferred) 3.18 Words to be avoided very and quite are not effective in giving emphasis. Removing them rarely alters the meaning of a sentence. 4

5 To be avoided utilize amongst save re can but deem divers implement of late peruse thrice gotten / got exceeding delve alright viable (doable) transpire tantamount nigh well nigh between... or none... are neither... are sophisticated invariably hopefully preferred use among except with regard to can only suppose several/various fulfil complete recently read three times obtained / found / yields exceedingly/extremely dig correctly practicable happen equivalent near nearly between... and none... is neither... is subtle, refined always it is hoped Table 1: Awkward words and safer alternatives. rather, when used as an alternative to quite, e.g. a rather difficult proof. However, rather can be used to contrasting two options, as in rather than a derive a general solution, we simply quote a particular integral. As far as... is concerned,... - a cliched phrase considerable - does not add impact. beautiful, novel overuse has dulled their power. classical refers to ancient Greek culture, (refer to highly regarded texts as classic) symmetrical - use symmetric instead asymptotical - use asymptotic or asymptotes to iff,,,, etc, write therefore, hence, etc Awkward words and safer alternatives. There are many words that should be avoided on the basis of them being pretentious, archaic, pedantic or awkward. Table 1 lists words to be avoided and preferred alternatives Whilst repetition can help emphasise a point, one should avoid pointless repetition, such repetition may distract or bore the reader. This applies not only to repeated words, as above, but meanings, for example, in saying we plan ahead, the ahead is redundant since one cannot plan back. Similarly, the phrases a new invention, the reason is because, absolutely essential, short summary, would all be improved by removing superfluous words Beware of false comparisons, for example, more impossible, less major, more universal Exercise care in the use of a priori, a posteriori, a fortiori, check that you are using them with the correct meaning The Latin abbreviations i.e. and e.g. should be used sparingly, the English phrases that is, and for example, are preferable. In L A TEX, remember that a double space appears after a full stop, to force a single space after abbreviations, use, that is, e.g.~should Do not start a sentence with a mathematical symbol, even if it is a capital letter. There are many ways to avoid this inelegance, for example, The parameter A represents The abbreviation const. is preferable to constant in displayed formulae and mathematical expressions. However, in text, the full word should be used When using compound adjectives, be sure to hyphenate appropriately; for example, second-order approximation, two-dimensional problems, to avoid confusion with the phrase two dimensional problems, which has a different meaning Avoid frequent use of quotation marks to emphasise words used with special meanings Resist temptations to make ironic or deprecatory remarks about your own work, and the work of others. Where elementary work needs to be recapped, or where a derivation is particularly easy and straightforward, do not waste space justifying the need to present the argument, simply present a concise summary of the work In English, Mathematics is always singular, for example mathematics and its applications, ( its not their ). 5