MATH36032 Problem Solving by Computer

Mathematical Writing

PART I: General Academic Writing

Academic writing: how it is different?

formality

objectivity

Main characteristics (compared to business or personal writing):

• Passive verb forms

• Qualifying words/phrases (suggest,imply...)

• Complex sentence structures

• Specialised vocabulary

• Citation

• Large amount of experimental details, data and figures

Different stages of the writing process

Assuming that you have more or less finished the coding part,

I Exploring, Planning and Organising

I Drafting: just the draft, no grammar or other details

I Revising, Editing

I Proofreading

The focus should be on the last two stages!

Different stages of the writing process

Revising and Editing

I Edit at sentence, paragraph, and text level: the choice ofwords, length of sentences

I Focus on content, organisation, grammatical accuracy,puncuation and spelling

I Transition between paragraphs and sections

I View the draft as a reader - be a critic of your work

Different stages of the writing process

Proofreading

I Read your draft report in printed form?

I Look at it globally (sections, pages, layout) as well as locally(words, symbols, spelling).

I Check word, letter, space and punctuation mark

I Incomplete/incorrect punctuation, incorrect line or paragraphbreaks, unmatched parentheses

I Wrong font, missing words, phrases

Important tips: Get someone to read your writing

Get some honest feedback on

I Overall structure

I Organisation of ideas

I Grammatical accuracy

I Writing style

I Clarity of the presentation

I Coherence of argument

I Ways of demonstrating data

PART II: Mathematical Writing

Dos and Don’ts

I Punctuate all equations.Now consider the quadratic equation

ax2 + bx + c = 0.

(do not forget the period at the end of the equation!)

I Don’t start a sentence with a symbol.The function f is continuous on its domain.

I Separate symbols by punctuation marks or words.... where a, b and c are constants.

Dos and Don’ts

Add words to explain the logical flow of the manipulation(instead of just using math symbols).

Bad: As n→∞, xn → x∗ = 1− 1/r .Good: As n goes to infinity, xn converges to x∗ = 1− 1/r .

Bad: ∀x > 0, ∃y > 0 such that y > x .Good: For all x > 0, there exists y > 0, such that y > x .

Dos and Don’ts: notations

Do not use unnecessary notations (especially the notation isnot used again).

Bad: Every differentiable function f is continuous.

Good: Every differentiable function is continuous.

Dos and Don’ts: notations

Avoid overloading meaning into notation.

Bad: Let x > 0 ∈ Z.

Good: Let x be an integer, with x > 0.

Good: Let x be a positive integer.

Dos and Don’ts: logical symbols

Try to use words instead of logical symbols like ∀, ∃,∧,∨, · · · ,unless it is about logic.

Bad: We have x2 ≥ 0 ∀ x ∈ R.

Good: We have x2 ≥ 0 for all x ∈ R.

Good: We have x2 ≥ 0 for any real number x .

Fonts for math functions

Math functions are in Roman

Bad: : sinx , tanx , sincxGood: : sin x , tan x , sincx

$\sin x, \cos x, \mbox{sinc} x$

Double negative?1

Bad: Red skin is not uncommon in patients treated withhydroxocobalamin.

Good: Red skin is common in patients treated withhydroxocobalamin.

Bad: Nausea and vomiting were not infrequently reported duringphase II trials of that drug.

Good: Nausea and vomiting were frequently reported during phaseII trials of that drug.

Bad: It is not unusual for writers of case reports to speculate tooreadily.

Good: It is usual for writers of case reports to speculate tooreadily.

Eliminating double negatives will enhance the clarity of yourmessage to readers and make your writing more active.

1From: Double Negatives Do Not Make a Positive Editorial Experience by Mark B. Mycyk,J Med Toxicol. 2011 Sep; 7(3): 254.

Check your spelling

I Use a dictionary and a thesaurus to check meanings, spellingsand synonyms

I Do not assume the spellchecker is always right!

I Printed

I Read aloud

Check your spelling

I Use a dictionary and a thesaurus to check meanings, spellingsand synonyms

I Do not assume the spellchecker is always right!

I Printed

I Read aloud

MATLAB commands/functions/output

I Use typewriter font for MATLAB code, names, variablesand outputBad: Before invoking int, we first set up symbolic variable xand f.Good: Before invoking int, we first set up symbolic variablex and f.

I Make MATLAB M-files and output clearly distinct fromsurrounding narrative

I You should not materially change MATLAB output ( but youcan remove blank lines and ”tiny up” output)

MATLAB commands/functions/output can be handled usingverbatim environment, or extra packages like mcode.

Common mistakes and confusing pairs of words

I singular vs plural: phenomenon, phenomena; criterion, criteria

I loose vs lose

I supersede vs supercede

I its vs it’s

I few, less: few iterations, less computation

I practice, practise

I affect, effect; compare with, compare to

I comprise, compose, constitute

Citations and References (if any)

I Should be cited (or referred) in the main text, not just listsome references at the very end.

I Make sure the format or style is consistency. (Commoncitation styles: APA, MLA, Chicago, ...)

Recommended style (for book and journal articles):

N. J. Higham, Handbook of writing for the mathematical sciences.Second edition. Society for Industrial and Applied Mathematics (SIAM),Philadelphia, PA, 1998. xvi+302 pp. ISBN: 0-89871-420-6

R. C. Fetecau, Y. Huang and T. Kolokolnikov, Swarm dynamics and

equilibria for a nonlocal aggregation model. Nonlinearity.

24(10):2681—2716, 2011.

Cite internet sources (if any)

Not as popular as you may think in the academic world.

Recommended style:

Prime number theorem.https://en.wikipedia.org/wiki/Prime_number_theorem

(you may also have to include access date)

Cite internet sources (if any)

Not as popular as you may think in the academic world.

Recommended style:

Prime number theorem.https://en.wikipedia.org/wiki/Prime_number_theorem

(you may also have to include access date)

Bibliography using bibtex (if using LaTeX)

Further details can be found in the book by N.Higham [1].

produced by

Further details can be found in the book by

N. Higham~\cite{higham1998handbook}.

An entry in bibtex file that looks like:

@book{higham1998handbook,

title={Handbook of writing for the mathematical sciences},

author={Higham, Nicholas J},

year={1998},

publisher={Siam}

}

Proofreading Test (1)

I One of the best unknown methods for determining all theeigenvalues and eigenvectors of (2.1) was developed bythe nineteenth century mathematician Jacobi.

I It would be hard to underestimate the importance ofoptimization to scientific computing.

I (from my submitted draft2)... which can not be deriveddirectly from the criteria given in the seminar paper byMcCann [25].

2JA Carrillo, Y Huang, MC Santos, JL Vazquez. Exponential convergencetowards stationary states for the 1D porous medium equation with fractionalpressure. Journal of Differential Equations, 258(3):736–763

Proofreading Test (2)

I (from last year’s description) We can get some primariesunderstanding using the following code block (changing thevalue of r).

I Acknowledgements. I thank the anonymous referees,particularly Dr. J. R. Ockendon, for numerous suggestions andfor the source of references.

I The ability of physics to explain similar phenomena over scalesof centimeters to 1018 centimeters is why it is such a powerfulscience. (From the book Very Large Scale Computation in the21st Century)

More examples from Prof Nick Higham’s blog post PublicationPeculiarities: Acknowledgements.

References

I Strunk, William. The elements of style. Penguin, 2007.

I Chicago Manual of Style. University of Chicago Press, 2010.

I Higham, Nicholas J. Handbook of writing for themathematical sciences (Chapter 3, 4, 5 and 13). SIAM,1998.

I Krantz, Steven. A Primer Of Mathematical Writing. AMS,1997.