Research Matters

February 25, 2009

Nick HighamDirector of Research

School of Mathematics

1 / 6

How to Write Scientific Papers

Nick HighamSchool of Mathematics

The University of Manchester

nick.higham@manchester.ac.ukhttp://www.maths.manchester.ac.uk/~higham/

Chinese Academy of Sciences, Beijing

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Outline

Writing

Workflow

Mathematical Writing

English Usage

Writing a Paper

Presenting Experimental Results

ProductionNick Higham How to Write Scientific Papers 2 / 80

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For Examples of Good Writing & TypesettingThe Princeton Companion to Applied Mathematics (PCAM)

Pages xvii + 995Articles 186

Authors / countries 165 / 23Figures 196

Cross references 733Index pages / entries 33 / 2842

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Why is it Important to Write Well?Promote your research (globally).Persuade the reader to agree with your views.Improve chance of your papers being published.Help your papers be read.Demonstrate your communication skills.Feeling of satisfaction from job done well.

Most worthwhile careers involve writing!

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What is Difficult?Having something to say.

Getting started (see next slide)

Improving a draft.

Convince reader of value of your work.

Less difficult:English grammar.

Maths-specific writing.

LATEX.

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Getting StartedStart writing earlier rather than later: writing is part ofthe process of understanding.

Procrastination and writer’s block:Plan and write first in your head.Lower your standards until there is no threshold togo over in writing. Just start typing!“The poor words may be the necessary path to thegood words” (R. P. Clark).

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Getting StartedStart writing earlier rather than later: writing is part ofthe process of understanding.

Procrastination and writer’s block:Plan and write first in your head.Lower your standards until there is no threshold togo over in writing. Just start typing!“The poor words may be the necessary path to thegood words” (R. P. Clark).

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Begin Sentences with Subjects and VerbsR. P. Clark:

Make meaning early, then let weaker elements branchto the right.Don’t separate the subject too far from its verb.

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Example“Motivated by the works of Feireisl-Rocca-Schimperna [10]and Feireisl-Fremond-Rocca- Schimperna [6], where thenon-isothermal model with the penalty term f (d) isconsidered, and those of Lin-Lin-Wang [17], Hong [18], andHong-Xin [19], where the isothermal model with term|∇d |2d is considered, we consider the non-isothermalmodel, that is, systems (1.1)–(1.4) and (1.8)–(1.11), and weare going to prove the global existence of weak solutions.”

“Motivated by the works of Feireisl-Rocca-Schimperna [10]and Feireisl-Fremond-Rocca- Schimperna [6], where thenon-isothermal model with the penalty term f (d) isconsidered, and those of Lin-Lin-Wang [17], Hong [18], andHong-Xin [19], where the isothermal model with term|∇d |2d is considered, we consider the non-isothermalmodel, that is, systems (1.1)–(1.4) and (1.8)–(1.11), and weare going to prove the global existence of weak solutions.”

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Example“Motivated by the works of Feireisl-Rocca-Schimperna [10]and Feireisl-Fremond-Rocca- Schimperna [6], where thenon-isothermal model with the penalty term f (d) isconsidered, and those of Lin-Lin-Wang [17], Hong [18], andHong-Xin [19], where the isothermal model with term|∇d |2d is considered, we consider the non-isothermalmodel, that is, systems (1.1)–(1.4) and (1.8)–(1.11), and weare going to prove the global existence of weak solutions.”

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AdverbsUse adverbs to change the meaning of the verb, not toexpress a meaning already in it.

Original: The algorithm completely failed to provide anycorrect significant digits in the computed solution.

Better: The algorithm failed to provide any correctsignificant digits in the computed solution.

Original: . . . the realization that higher resolutionsimulations will inevitably require the replacement of thewhole HPC software stack.

Better: . . . the realization that higher resolution simulationswill require the replacement of the whole HPC softwarestack.

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AdverbsUse adverbs to change the meaning of the verb, not toexpress a meaning already in it.

Original: The algorithm completely failed to provide anycorrect significant digits in the computed solution.

Better: The algorithm failed to provide any correctsignificant digits in the computed solution.

Original: . . . the realization that higher resolutionsimulations will inevitably require the replacement of thewhole HPC software stack.

Better: . . . the realization that higher resolution simulationswill require the replacement of the whole HPC softwarestack.

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AdjectivesOriginal: The numerical experiments show that thealgorithm produces very accurate results.

Better: The numerical experiments show that the algorithmproduces results with errors at the level of the unit roundoff.

Minimize use of adjectives and adverbs.See Cluttered writing: Adjectives and adverbs inacademia by A. Okulicz-Kozaryn (2013).

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Avoid (Double) NegativesOriginal: The convergence of (1.1) is not guaranteedwithout additional assumptions on the function g and thestarting vector x0.

Better: To guarantee convergence of (1.1) assumptionsmust be made on the function g and the starting vector x0.

Original: If the input argument closure is set to true, allthe dependencies are wrapped if they are not explicitlymarked as non-exportable.

Better: If the input argument closure is set to true, allthe dependencies are wrapped unless they are explicitlymarked as non-exportable.

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Avoid (Double) NegativesOriginal: The convergence of (1.1) is not guaranteedwithout additional assumptions on the function g and thestarting vector x0.

Better: To guarantee convergence of (1.1) assumptionsmust be made on the function g and the starting vector x0.

Original: If the input argument closure is set to true, allthe dependencies are wrapped if they are not explicitlymarked as non-exportable.

Better: If the input argument closure is set to true, allthe dependencies are wrapped unless they are explicitlymarked as non-exportable.

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Vary Sentence Length

Example: D. S. Broomhead (PCAM, 2015):

The timetable for a rail network has to coordinate themovements of many independent trains in order to providea safe and predictable service. A range of issues have to beaddressed: railway stations have limited numbers ofplatforms; parts of the network may have single-track lines;passengers need to make connections; etc. Max-plusalgebra provides useful tools to do this.

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Vary Sentence LengthExample: D. S. Broomhead (PCAM, 2015):

The timetable for a rail network has to coordinate themovements of many independent trains in order to providea safe and predictable service. A range of issues have to beaddressed: railway stations have limited numbers ofplatforms; parts of the network may have single-track lines;passengers need to make connections; etc. Max-plusalgebra provides useful tools to do this.

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Name When You Cite

Original: Previous work [4], [9] has shown that. . .

Better: Smith [9] and Jones [4] have shown that . . .

Original: . . . as shown in [2].

Better: as shown by Vaughan and Williams [2].

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Name When You Cite

Original: Previous work [4], [9] has shown that. . .

Better: Smith [9] and Jones [4] have shown that . . .

Original: . . . as shown in [2].

Better: as shown by Vaughan and Williams [2].

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Say What You MeanContext: a nonlinear eigenvalue problem known to havereal eigenvalues.

Original: Our new algorithm guarantees real eigenvalues.

Better: Our new algorithm guarantees real computedeigenvalues in floating point arithmetic.

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Say What You MeanContext: a nonlinear eigenvalue problem known to havereal eigenvalues.

Original: Our new algorithm guarantees real eigenvalues.

Better: Our new algorithm guarantees real computedeigenvalues in floating point arithmetic.

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Revising“Begin by cutting out the big stuff. Brevity comes fromselection, not compression” (R. P. Clark).

Editor Maxwell Perkins reduced one four-page passageof Thomas Wolfe’s about his uncle to six words:

‘Henry, the oldest, was now thirty’.

Wait a few hours or days between writing and revising.

Revise by hand on a printout?

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Recommended Reading on WritingRoy Peter Clark, Writing Tools: 55 EssentialStrategies For Every Writer, Little, Brown andCompany, 2016.

Stephen King, On Writing Well: A Memoir of theCraft, xv+367, Hodder and Stoughton, 2000

William Zinsser, On Writing Well: An Informal Guideto Writing Nonfiction, xiii+288, HarperCollins, 1990

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ExerciseOriginal: The editors strongly encourage authors todeposit, in a permanent repository or as supplementarymaterials, any software and data that allow the results ofpublished articles to be reproduced, and every effort shouldbe made to include sufficient information in manuscripts toenable this.

Better: Authors should include sufficient information in themanuscript to enable the results of the manuscript to bereproduced. To this end, authors should deposit in apermanent repository, or as supplementary materials,relevant software and data.

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ExerciseOriginal: The editors strongly encourage authors todeposit, in a permanent repository or as supplementarymaterials, any software and data that allow the results ofpublished articles to be reproduced, and every effort shouldbe made to include sufficient information in manuscripts toenable this.

Better: Authors should include sufficient information in themanuscript to enable the results of the manuscript to bereproduced. To this end, authors should deposit in apermanent repository, or as supplementary materials,relevant software and data.

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ExerciseOriginal: Combining the previous two inequalities together,we obtain . . .

Better: Combining the previous two inequalities, we obtain. . .

Original: The maximal value grows larger for larger m.

Better: The maximal value grows with m.

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ExerciseOriginal: Combining the previous two inequalities together,we obtain . . .

Better: Combining the previous two inequalities, we obtain. . .

Original: The maximal value grows larger for larger m.

Better: The maximal value grows with m.

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ExerciseOriginal: Combining the previous two inequalities together,we obtain . . .

Better: Combining the previous two inequalities, we obtain. . .

Original: The maximal value grows larger for larger m.

Better: The maximal value grows with m.

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ExerciseOriginal: Combining the previous two inequalities together,we obtain . . .

Better: Combining the previous two inequalities, we obtain. . .

Original: The maximal value grows larger for larger m.

Better: The maximal value grows with m.

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Outline

Writing

Workflow

Mathematical Writing

English Usage

Writing a Paper

Presenting Experimental Results

ProductionNick Higham How to Write Scientific Papers 19 / 80

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WorkflowTools, techniques for whole research process, esp.typesetting, numerical experiments.

See chapter Workflow from The Princeton Companion toApplied Mathematics.

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The Old DaysProducing maths on typewriters and wordprocessors

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LATEX Tips

See my LATEX Tips document in my GitHub repositoryhttps://github.com/higham/latex-tips

The document also exists in the form of one of myLATEX blog posts

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Plain Text is KingMaximize use of plain text.

Minimize use of proprietary formats such as .doc,.docx, .xls, .xlsx.

For writing can use markup languages:Markdown (cheatsheet).

Emacs Org mode (cheatsheet).

LATEX.

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Version ControlAvoid paper1.tex, paper2.tex, . . .

Database of different versions of files and changes.

A form of backup.

Integration with many editors.

Recommend Git.

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Outline

Writing

Workflow

Mathematical Writing

English Usage

Writing a Paper

Presenting Experimental Results

ProductionNick Higham How to Write Scientific Papers 25 / 80

Writing Workflow Math Writing Usage Write a Paper Experiments Production

Summarize Results in a TheoremBe precise and do not overstate your claims.

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Summarize Results in a TheoremBe precise and do not overstate your claims.

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Fine-Tuning a TheoremWord as simply and clearly as possible.

TheoremLet A ∈ Cn×n have no eigenvalues on R−. Then

acosA + asinA =π

2I.

Better.

TheoremIf A ∈ Cn×n has no eigenvalues on R− then

acosA + asinA =π

2I.

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Fine-Tuning a TheoremWord as simply and clearly as possible.

TheoremLet A ∈ Cn×n have no eigenvalues on R−. Then

acosA + asinA =π

2I.

Better.

TheoremIf A ∈ Cn×n has no eigenvalues on R− then

acosA + asinA =π

2I.

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Dos and Don’ts (1)Bad:

x2 + bx − c = 0(x + b/2)2 = c + b2/4

x + b/2 =√

c + b2/4

x = −b/2 +√

c + b2/4.

Much better: We have

0 = x2 + bx − c = (x + b/2)2 − c − b2/4.

Rearranging and taking the positive square root gives

x = −b/2 +√

c + b2/4.

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Dos and Don’ts (2)Watch for notational synonyms.

Original: If λi 6= λj for all i 6= j . . .Alternative: If the eigenvalues are distinct . . .

Original:

A = X

λ1

λ2. . .

λn

X−1

Alternative: A = Xdiag(λi)X−1.

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Dos and Don’ts (3)Punctuate all equations (as in the handouts). E.g.

We consider the initial value problem

y ′(t) = 2y(t), y(0) = 1,

over the range 0 ≤ t ≤ 1.

Don’t start a sentence with a symbol.Bad: A is an ill-conditioned matrix.Good: The matrix A is ill-conditioned.Separate symbols by punctuation marks or words.Bad: If x = 1 f (x) = 0.Fair: If x = 1, f (x) = 0.Good: If x = 1 then f (x) = 0.

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Dos and Don’ts (4)Don’t put “big” equations in-line.

Bad: We find that f (x) = 11−x −

11+x .

Good: We find that f (x) = 1/(1− x)− 1/(1 + x).Good: We find that

f (x) =1

1− x− 1

1 + x.

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ExampleFrom a textbook on linear algebra:

Define a square matrix Eij by

[Ei,j ]kl =

0, k 6= i , k 6= j , l 6= k,1, k 6= i , k 6= j , l = k,0, k = i , l 6= j ,1, k = i , l = j ,0, k = j , l 6= i ,1, k = j , l = i .

Better:Let Ei,j be the matrix obtained by swapping the ithand jth rows of the identity matrix.

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ExampleFrom a textbook on linear algebra:

Define a square matrix Eij by

[Ei,j ]kl =

0, k 6= i , k 6= j , l 6= k,1, k 6= i , k 6= j , l = k,0, k = i , l 6= j ,1, k = i , l = j ,0, k = j , l 6= i ,1, k = j , l = i .

Better:Let Ei,j be the matrix obtained by swapping the ithand jth rows of the identity matrix.

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Study Other WritersLearn by studying the work of writers you admire.

What makes their papers stand out?

What vocabulary do they use?

How do they weave together text and mathematics?

How do they tell the story?

What stops their writing from being boring?

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Outline

Writing

Workflow

Mathematical Writing

English Usage

Writing a Paper

Presenting Experimental Results

ProductionNick Higham How to Write Scientific Papers 34 / 80

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Commonly Confused Words (1)alternative, alternateaffect (verb), effect (noun)compare with (similarities, differences), compare to(use only in poetry and love letters)comprise (to consist of), compose (to make up),constitute: The exam comprises 7 questions, thecourse is composed of 3 topics, these 3 topicsconstitute the course.

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Commonly Confused Words (2)fewer (use w/plural nouns), less (use w/singularnouns): fewer iterations, less computationpractice (noun), practise (verb)which (informs and does not restrict), that (definesand restricts):

Consider the Hilbert matrix that is positive definite.Consider the Hilbert matrix, which is positivedefinite.

Use a dictionary! Other examples: annex–annexe,especially–specially, licence–license.

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Commonly Confused Words (2)fewer (use w/plural nouns), less (use w/singularnouns): fewer iterations, less computationpractice (noun), practise (verb)which (informs and does not restrict), that (definesand restricts):

Consider the Hilbert matrix that is positive definite.Consider the Hilbert matrix, which is positivedefinite.

Use a dictionary! Other examples: annex–annexe,especially–specially, licence–license.

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Distinctionsphenomenon (singular), phenomena (plural)

criterion (singular), criteria (plural)

discrete (distinct, not continuous), discreet(unobtrusive).

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Watch Out for These Mistakessupersede (incorrect: supercede)

loose instead of lose

MATLAB (incorrect: Matlab).Do not write “MATLAB’s expm function”.

its is a possessive (“its solution”),it’s is a contraction of it is (“it’s clear that”).

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Watch Out for These Mistakessupersede (incorrect: supercede)

loose instead of lose

MATLAB (incorrect: Matlab).Do not write “MATLAB’s expm function”.

its is a possessive (“its solution”),it’s is a contraction of it is (“it’s clear that”).

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For Non-Native WritersIf in doubt, include the article a or the.

Read good expository technical English writing,e.g. SIAM News, Scientific American.

Identify writers whose papers you find enjoyable toread and make notes of their phrases, vocabulary, etc.

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Take a look at this book, available here.

Leonardo da Vinci programme European Commission

Writing in English

A Practical Handbook for Scientific and Technical Writers

A Pilot Project

Project Partners

Zuzana Svobodova, Technical University Brno, Czech Republic Heidrun Katzorke and Ursula Jaekel, Technische Universität, Chemnitz, Germany Stefania Dugovicova and Mike Scoggin, Comenius University, Bratislava, Slovakia

Peter Treacher, ELT Centre, University of Essex, England

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Dangling ParticipleI From a textbook:

When writing a program that uses arrays, the DI-MENSION statement should reserve the largestamount of memory that might be needed.

Intended subject of participle “writing” is not present!

Dangling participles usually not ambiguous given context,but distracting.

I A bug was found in the program using random testdata.

I Hastily summoning an ambulance, the corpse wascarried to the mortuary.

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Dangling ParticipleI From a textbook:

When writing a program that uses arrays, the DI-MENSION statement should reserve the largestamount of memory that might be needed.

Intended subject of participle “writing” is not present!

Dangling participles usually not ambiguous given context,but distracting.

I A bug was found in the program using random testdata.

I Hastily summoning an ambulance, the corpse wascarried to the mortuary.

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Dangling ParticipleI From a textbook:

When writing a program that uses arrays, the DI-MENSION statement should reserve the largestamount of memory that might be needed.

Intended subject of participle “writing” is not present!

Dangling participles usually not ambiguous given context,but distracting.

I A bug was found in the program using random testdata.

I Hastily summoning an ambulance, the corpse wascarried to the mortuary.

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Bastard EnumerationThe Basic Linear Algebra Subprograms (BLAS) haveseveral advantages. They

Lead to shorter and clearer codes;Improve modularity.Machine dependent optimizations can be confined tothe BLAS, aiding portability, andTuned BLAS have been provided by manufacturers.

Entries should be grammatically parallel.

Variation:There are three reasons.(a) . . .(b) . . .And thirdly . . .

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Bastard EnumerationThe Basic Linear Algebra Subprograms (BLAS) haveseveral advantages. They

Lead to shorter and clearer codes;Improve modularity.Machine dependent optimizations can be confined tothe BLAS, aiding portability, andTuned BLAS have been provided by manufacturers.

Entries should be grammatically parallel.

Variation:There are three reasons.(a) . . .(b) . . .And thirdly . . .

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Active versus PassiveActive: X did Y.Passive: Y was done by X.

Active voice is more lively and often more concise.Passive voice is more vague and indirect.

Passive: Estimation of the error was achieved byapplication of the Peano kernel formula.

Active: We estimated the error using the Peano kernelformula.

Passive: The specimen was accidentally strained duringmounting.

Active: I dropped the specimen on the floor.

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Active versus PassiveActive: X did Y.Passive: Y was done by X.

Active voice is more lively and often more concise.Passive voice is more vague and indirect.

Passive: Estimation of the error was achieved byapplication of the Peano kernel formula.

Active: We estimated the error using the Peano kernelformula.

Passive: The specimen was accidentally strained duringmounting.

Active: I dropped the specimen on the floor.

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Active versus PassiveActive: X did Y.Passive: Y was done by X.

Active voice is more lively and often more concise.Passive voice is more vague and indirect.

Passive: Estimation of the error was achieved byapplication of the Peano kernel formula.

Active: We estimated the error using the Peano kernelformula.

Passive: The specimen was accidentally strained duringmounting.

Active: I dropped the specimen on the floor.

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Naked ThisConsider the sentence

This raises the question of whether the method al-ways converges.

“This” could refer tomost recent noun in previous sentence,subject of previous sentence,idea of previous sentence,something else.

Qualify “this” with the appropriate noun: “this result”, “thisargument”, . . . to avoid ambiguity.

Even better: rewrite to avoid the overused “this” completely.

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Lively Writing (1)Try not to begin a sentence with

It is . . .It can be . . .There is . . .There are . . .Also . . .

Never use the word “thing”—be more specific.

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Lively Writing (2)Bad: It can be seen in Table 4 that the residual ismonotonically decreasing.Good: Table 4 shows that the residual decreasesmonotonically.

Bad: There are several possibilities that can be used tobreak ties.Good: Ties can be broken in several ways.

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Lively Writing (2)Bad: It can be seen in Table 4 that the residual ismonotonically decreasing.Good: Table 4 shows that the residual decreasesmonotonically.

Bad: There are several possibilities that can be used tobreak ties.Good: Ties can be broken in several ways.

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Online ResourcesOxford English Dictionary

The King’s English, H. W. Fowler and F. G. Fowler,Second edition, 1908.The Elements of Style, William Strunk, Jr., 1918.

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Style GuidesJournals of the London Mathematical Society: housestyle and instructions for copy-editors andtypesettersSIAM Style GuideWikipedia:Manual of Style/MathematicsIEEE Editorial Style ManualThe Chicago Manual of Style (subscription required)The Best Style Guides (web article)

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ExerciseOriginal: We are neither hoping to reconstruct the datain its full entity nor to achieve the high qualitycompression of specialized methods.

Better: We are neither hoping to reconstruct the datain its entirety nor aiming to achieve the high qualitycompression of specialized methods.

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ExerciseOriginal: We are neither hoping to reconstruct the datain its full entity nor to achieve the high qualitycompression of specialized methods.

Better: We are neither hoping to reconstruct the datain its entirety nor aiming to achieve the high qualitycompression of specialized methods.

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Outline

Writing

Workflow

Mathematical Writing

English Usage

Writing a Paper

Presenting Experimental Results

ProductionNick Higham How to Write Scientific Papers 51 / 80

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The TitleThe title should

Give a terse description of the content.Be catchy.

J. L. Kelley:Before: A decomposition of compact continua and

related results on fixed sets under continuousmappings

After: Simple links and fixed sets

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Some Good TitlesHow and how not to check Gaussian quadrature formulae

ALGOL 68 with fewer tears

Nineteen dubious ways to compute the exponential of amatrix

Iterative refinement implies numerical stability for Gaussianelimination

Can you count on your calculator?

Structured polynomial eigenvalue problems: Goodvibrations from good linearizations

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No So Good TitlesPerturbation bounds for the nonsymmetric eigenvalueproblem

A BIE method for a nonlinear BVP

A Report on Harmonic Maps (1978)Another Report on Harmonic Maps (1988)

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Outline

Writing

Workflow

Mathematical Writing

English Usage

Writing a Paper

Presenting Experimental Results

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BasicsGive enough detail to enable interpretation andrepetition of experiment. State, e.g.,

Machine precision.Compiler & options, optimizations; BLAS library.Type of random numbers.

State only as many digits as needed.Be wary of extrapolating from results.Distinguish between objective statements andspeculation.

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Reproducible ResearchWikipediahttp://en.wikipedia.org/wiki/Reproducibility

“Reproducibility . . . refers to the ability of an entireexperiment or study to be reproduced, or by someoneelse working independently.”“The term reproducible research was first proposedby Jon Claerbout at Stanford University and refers tothe idea that the ultimate product of research is thepaper along with the full computational environmentused to produce the results in the paper such as thecode, data, etc. necessary for reproduction of theresults and building upon the research.”

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Table or Graph?Table better for small sets of numbers.Graph better for large data sets & to show trends.

Tables:Keep simple.Minimum number of rules.Like quantities to be compared better in columns thanrows.

Graphs:Ensure readable.Iterate on design.

See Ten Simple Rules for Better Figures (2014).

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MATLABUse typewriter font for MATLAB code, functionnames, variables, and output.Wrong: Before invoking int we first set up symbolicvariables x and f.Right: Before invoking int we first set up symbolicvariables x and f.

Make MATLAB M-files and output clearly distinct fromsurrounding narrative, perhaps by using smaller fontsize and narrower margins.

You should not materially change MATLAB output, butyou can remove blank lines and “tidy up” output.

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Outline

Writing

Workflow

Mathematical Writing

English Usage

Writing a Paper

Presenting Experimental Results

ProductionNick Higham How to Write Scientific Papers 61 / 80

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Nick Higham How to Write Scientific Papers 62 / 80

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Proofreading Test (1)One of the best unknown methods for determining allthe eigenvalues and eigenvectors of (2.1) wasdeveloped by the nineteenth century mathematicianJacobi.

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Proofreading Test (1)One of the best unknown methods for determining allthe eigenvalues and eigenvectors of (2.1) wasdeveloped by the nineteenth century mathematicianJacobi.

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

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Proofreading Test (2)Acknowledgements. I thank the anonymous referees,particularly Dr. J. R. Ockendon, for numeroussuggestions and for the source of references.

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Proofreading Test (2)Acknowledgements. I thank the anonymous referees,particularly Dr. J. R. Ockendon, for numeroussuggestions and for the source of references.

The ability of physics to explain similar phenomenaover scales of centimeters to 1018 centimeters is why itis such a powerful science.

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Proofreading Test (2)Acknowledgements. I thank the anonymous referees,particularly Dr. J. R. Ockendon, for numeroussuggestions and for the source of references.

The ability of physics to explain similar phenomenaover scales of centimeters to 1018 centimeters is why itis such a powerful science.

All these approaches lead to nontrivial loss of accuracywhen larger CNN models were trained for ILSVRCclassification tasks (Zhou et al. (2016)).

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Proofreading Test (3)

324 Chapter 17. Special Algorithms for Special Functions

functions are all real, simple, and well separated. Perturbation theory gives good approx-imations for the larger roots; the Chebyshev-proxy algorithms have not the slightest dif-ficulty with the smaller zeros.

Thus, the casual user need make no distinction between special functions and every-thing else. However, special function libraries make heavy use of special algorithms, notmerely for computing zeros, but also for calculating the special functions themselves. El-liptic functions have rapidly convergent Fourier series with explicit coefficients, but theAGM iteration is faster still. Bessel functions have rapidly convergent power series, butthree-term recurrences simultaneously compute Bessel functions of different orders in asingle for/next loop. And so on for many other special functions, as cataloged by Gil,Segura, and Temme and many earlier authors whom they cite. A library-writer is not acasual user, however, nor is the architect of a big time-marching code that needs to com-pute Bessel zeros a billion times per timestep. So a brief overview of special rootfinders iswarranted.

17.2 What Is a Special Function?The narrow definition is: A function is special if it appears in the NIST Digital Library ofMathematical Functions [284] or the five-volume Bateman Manuscript Project [162]. Weshall refer to these as canonical special functions.

A broader definition is: A function is special if someone has written a paper about itspower series, asymptotic expansions, singularities, and so on. The Blasius function is animportant similarity solution in fluid mechanics, defined as the solution to

2 fx x x + f fx x = 0, f (0) = fx (0) = 0, fx(∞) = 1. (17.1)

It is noncanonical, but definitely special, in that its power series, asymptotic expansions,singularity location and type, dilation group invariances, and rational Chebyshev seriesaccurate to 15 decimal places have been published over the past century [86]. Similarly,the balanced oscillator gain function defined by

k(z)≡ 2

z

∫ 1

0tanh(9z) sin(2πt ) sin(2πt )d t (17.2)

is not a canonical special function either, but Boyd and Visser have derived power se-ries and asymptotic approximations to the inverse, a blend of these that is a never-failingNewton’s initialization (NFNI) [108] plus a pair of Chebyshev series that approximatethe inverse z(k) to eight decimal places over its entire range. The significance of non-canonical special functions is that for these functions, perturbation series and valuableformulas and insights are only a few clicks away.

George Carrier, who was a rare double electee to both the National Academy of Sci-ence and the National Academy of Engineering, advised his students that there were threeways to learn:

Precept 2 (Carrier’s Three Modes of Information Acquisition). 1. Run up and downthe hallway, asking, until you find a human source. [BEST]

2. Scour the library and look it up in a book or journal.3. Work it out yourself. [WORST]

Repeating previously published research yourself is the best, not worst, in terms ofdeeply learning a subject, but it is the worst in terms of time expenditure. Albert Migliori’s

Dow

nloa

ded

11/1

1/14

to 1

30.8

8.49

.66.

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; see

http

://w

ww

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urna

ls/o

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php

Should be

k(z) ≡ 2z

∫ 1

0tanh

(z sin(2πt)

)sin(2πt)dt .

Nick Higham How to Write Scientific Papers 65 / 80

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Proofreading Test (3)

324 Chapter 17. Special Algorithms for Special Functions

functions are all real, simple, and well separated. Perturbation theory gives good approx-imations for the larger roots; the Chebyshev-proxy algorithms have not the slightest dif-ficulty with the smaller zeros.

Thus, the casual user need make no distinction between special functions and every-thing else. However, special function libraries make heavy use of special algorithms, notmerely for computing zeros, but also for calculating the special functions themselves. El-liptic functions have rapidly convergent Fourier series with explicit coefficients, but theAGM iteration is faster still. Bessel functions have rapidly convergent power series, butthree-term recurrences simultaneously compute Bessel functions of different orders in asingle for/next loop. And so on for many other special functions, as cataloged by Gil,Segura, and Temme and many earlier authors whom they cite. A library-writer is not acasual user, however, nor is the architect of a big time-marching code that needs to com-pute Bessel zeros a billion times per timestep. So a brief overview of special rootfinders iswarranted.

17.2 What Is a Special Function?The narrow definition is: A function is special if it appears in the NIST Digital Library ofMathematical Functions [284] or the five-volume Bateman Manuscript Project [162]. Weshall refer to these as canonical special functions.

A broader definition is: A function is special if someone has written a paper about itspower series, asymptotic expansions, singularities, and so on. The Blasius function is animportant similarity solution in fluid mechanics, defined as the solution to

2 fx x x + f fx x = 0, f (0) = fx (0) = 0, fx(∞) = 1. (17.1)

It is noncanonical, but definitely special, in that its power series, asymptotic expansions,singularity location and type, dilation group invariances, and rational Chebyshev seriesaccurate to 15 decimal places have been published over the past century [86]. Similarly,the balanced oscillator gain function defined by

k(z)≡ 2

z

∫ 1

0tanh(9z) sin(2πt ) sin(2πt )d t (17.2)

is not a canonical special function either, but Boyd and Visser have derived power se-ries and asymptotic approximations to the inverse, a blend of these that is a never-failingNewton’s initialization (NFNI) [108] plus a pair of Chebyshev series that approximatethe inverse z(k) to eight decimal places over its entire range. The significance of non-canonical special functions is that for these functions, perturbation series and valuableformulas and insights are only a few clicks away.

George Carrier, who was a rare double electee to both the National Academy of Sci-ence and the National Academy of Engineering, advised his students that there were threeways to learn:

Precept 2 (Carrier’s Three Modes of Information Acquisition). 1. Run up and downthe hallway, asking, until you find a human source. [BEST]

2. Scour the library and look it up in a book or journal.3. Work it out yourself. [WORST]

Repeating previously published research yourself is the best, not worst, in terms ofdeeply learning a subject, but it is the worst in terms of time expenditure. Albert Migliori’s

Dow

nloa

ded

11/1

1/14

to 1

30.8

8.49

.66.

Red

istr

ibut

ion

subj

ect t

o SI

AM

lice

nse

or c

opyr

ight

; see

http

://w

ww

.sia

m.o

rg/jo

urna

ls/o

jsa.

php

Should be

k(z) ≡ 2z

∫ 1

0tanh

(z sin(2πt)

)sin(2πt)dt .

Nick Higham How to Write Scientific Papers 65 / 80

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Proofreading Test (4)

Nick Higham How to Write Scientific Papers 66 / 80

Writing Workflow Math Writing Usage Write a Paper Experiments Production

Spot the Error

Nick Higham How to Write Scientific Papers 67 / 80

Writing Workflow Math Writing Usage Write a Paper Experiments Production

Spot the Error

Nick Higham How to Write Scientific Papers 68 / 80

Writing Workflow Math Writing Usage Write a Paper Experiments Production

Spot the Error

Nick Higham How to Write Scientific Papers 69 / 80

Writing Workflow Math Writing Usage Write a Paper Experiments Production

Spot the Error

Nick Higham How to Write Scientific Papers 70 / 80

Writing Workflow Math Writing Usage Write a Paper Experiments Production

Spot the Error

Nick Higham How to Write Scientific Papers 71 / 80

Writing Workflow Math Writing Usage Write a Paper Experiments Production

Spot the Error

Nick Higham How to Write Scientific Papers 72 / 80

Writing Workflow Math Writing Usage Write a Paper Experiments Production

Spot the Error

Nick Higham How to Write Scientific Papers 73 / 80

Writing Workflow Math Writing Usage Write a Paper Experiments Production

Spot the Error

Nick Higham How to Write Scientific Papers 74 / 80

Writing Workflow Math Writing Usage Write a Paper Experiments Production

Spot the Error

Nick Higham How to Write Scientific Papers 75 / 80

Writing Workflow Math Writing Usage Write a Paper Experiments Production

Spot the Error

Nick Higham How to Write Scientific Papers 76 / 80

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Spot the Error

PBOOE BEADEBS

A large Printing Company in Blantyre, Malawi, requires the services of two fully qualifeid Proof Readers.

Nick Higham How to Write Scientific Papers 77 / 80

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Spot the Error

Nick Higham How to Write Scientific Papers 78 / 80

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Spot the Error

Vice-chance1lors, according to the Association of University Teachers, are paid in the region of £13,000 depending roughly on the size of the university. They also enjoy a range of fringe benefits, notably a horse and use of a university car.

Nick Higham How to Write Scientific Papers 79 / 80

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Spot the Error

Vice-chance1lors, according to the Association of University Teachers, are paid in the region of £13,000 depending roughly on the size of the university. They also enjoy a range of fringe benefits, notably a horse and use of a university car.

Nick Higham How to Write Scientific Papers 79 / 80

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For more information about any of the topics discussedhere seeNicholas J. Higham, Handbook of Writing for theMathematical Sciences, Second edition, SIAM,Philadelphia, 1998and the links athttp://www.maths.manchester.ac.uk/~higham/hwms

Nick Higham How to Write Scientific Papers 80 / 80

References IRoy Peter Clark.Writing Tools: 55 Essential Strategies For Every Writer.Little, Brown and Company, New York, 2016.10th Anniversary Edition.ISBN 978-0-316-01499-1.

George Grätzer.More Math Into LATEX.Fourth edition, Springer-Verlag, New York, 2007.ISBN 978-0-387-32289-6.

Nick Higham How to Write Scientific Papers 1 / 3

References IINicholas J. Higham.Handbook of Writing for the Mathematical Sciences.Second edition, Society for Industrial and AppliedMathematics, Philadelphia, PA, USA, 1998.xvi+302 pp.ISBN 978-0-898714-20-3.

Donald E. Knuth, Tracy Larrabee, and Paul M. Roberts.Mathematical Writing.MAA Notes Number 14. Mathematical Association ofAmerica, Washington, D.C., 1989.115 pp.Also Report STAN-CS-88-1193, Department ofComputer Science, Stanford University, January 1988.

Nick Higham How to Write Scientific Papers 2 / 3

References IIIISBN 0-88385-063-X.

Steven G. Krantz.A Primer of Mathematical Writing: Being a Disquisitionon Having Your Ideas Recorded, Typeset, Published,Read, and Appreciated.American Mathematical Society, Providence, RI, USA,1997.xv+223 pp.ISBN 0-8218-0635-1.

Nick Higham How to Write Scientific Papers 3 / 3