Tutorial on Paper-Writing in Applied Mathematics (Preliminary Draft of Slides)
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This document discusses best practices for writing applied mathematics papers. It covers paper structure, including introductions, sections, and conclusions. It also discusses writing for different audiences and journals. Examples are provided by summarizing and discussing a published applied mathematics paper. The goal is to provide guidance to help write clear, informative papers.
Tutorial on Paper-Writing in Applied Mathematics (Preliminary Draft of Slides)
- 1. Writing Papers in Applied Mathematics: An Opinionated Tutorial and Discussion Mason A. Porter (@masonporter), Department of Mathematics, UCLA
- 2. Notes • Audience questions, discussion, and other participation will be hugely important. • Interrupt early and often • I expect we’ll have some ideas by the end of today’s tutorial about some things that may be good to add to the slides. • My writing style is rather different from the scientific norm, but it’s also true that one has to know the rules in order to break them effectively. • Other people can decide for themselves whether I break them effectively. :)
- 3. A Few Resources • Peter Dodds, “The Narrative Hierarchy—Stories and Storytelling on all Scales” • http://www.uvm.edu/pdodds/writings/2015-06-04narrative-hierarchy/ • Steven G. Krantz, A Primer of Mathematical Writing, Second Edition • Available for free online at arXiv:1612.04888 (or you can buy it at a cheap price) • Simon Peyton Jones, “How to Write a Great Research Paper” • https://www.microsoft.com/en-us/research/academic-program/write-great-research-paper/ • Martin H. Krieger, “Don’t Just Begin with ‘Let A be an algebra…’”, Notices of the AMS, March 2018 • Choice quote: “A paper should have an informative introduction.” • (I expect that I will add more resources when I get feedback. That will make a nice appendix.)
- 4. Applied Mathematics versus Theoretical (“Pure”) Mathematics • The most useful papers in both fields have informative introductions and conclusions, but the fact that applied mathematics touches so deeply on other subjects has a major effect on how you write. • Diversity of audience • Past work in diverse scientific fields and cultures • In applied mathematics, our papers usually draw from a larger background and also branch out in more diverse directions. • Take-home message: Who is your audience?
- 5. What’s in a Paper? (besides blood, sweat, tears, and snark) • Title: Possibly with a Subtitle • Abstract • Keywords and subject-classification codes (e.g., math or physics, depending on journal) • Introduction • Various intermediate sections (further subdivided into subsections or even further), which can include background information, methodology of various types, results of various types (e.g., theory versus computation), applications, case studies, and so on. • In some fields, it is very common to have dry sections like “Methods” and “Results”, and sometimes I do that too. However, I prefer to avoid this, unless we really can’t think of a better organization for a given paper. Also, in mathematics, the new methodology is often the most important part of a paper. • (continued on next page)
- 6. What’s in a Paper? (continued) • Conclusions and Discussion • Sometimes too separate sections but often combined. The latter typically includes some thoughts about future work. • Acknowledgements • Both people and funding. Be generous! • Appendices • Depends on the paper • References • Aggregators make loads of errors, so this needs thorough manual polishing, or else people will know that you’re sloppy. (And when I see sloppy references, I tend to assume that other important things in the paper will also be sloppy.) • Supplementary Material • Not cool enough even for an appendix (and/or we ran out of space) • Depends on the paper • Public data, code, outputs of computation • Make as much of this public as possible (based on what privacy considerations and so on allow)
- 7. Nested Storytelling Hierarchies • Read what Peter Dodds wrote • Paper, sections, subsections, subsubsections (sometimes), paragraphs, sentences, phrases and words (and punctuation) • The whole paper also fits within a larger hierarchy • What came before (from multiple fields) • What hopefully comes hence (from multiple fields) • Related work • Including other choices in related scientific work besides the ones you made • You had to make tough choices in practice. Surely other people make different ones. Scientific communities need to explore a diversity of them to optimally understand the landscape of a problem.
- 8. You are telling a story. • It is a factual story, but it is still a story. • What is the moral (main conclusion) of the story? • Why is the moral important? How does it change the previous state of the art? Where might it lead us into the future? • How is it different and similar from the morals of similar stories (related work)? • Ponder: Romeo and Juliet versus West Side Story • Now ponder: Space Oddity (David Bowie) versus Major Tom (Peter Schilling) versus Ashes to Ashes (Bowie) versus Hallo Spaceboy (Bowie, remixed by Pet Shop Boys) versus … • Now ponder: Discrete opinion spaces versus continuous opinion spaces for opinion models on networks
- 9. No Bullshit. • Ever. • Although you are telling a story, do not ever include bullshit in your paper. • Many people break this rule. It is annoying. • You have to back up everything (both your claims and your source files), and you have to state very clearly when something is opinion/speculation. • Do not mislead.
- 10. Starting Out • As I told you in our prior discussions, start with a shell. The shell includes • An attempt at an abstract, though likely with placeholders for some stuff you’re figuring out. • In each section that you think you will have, include one sentence or so that indicates what will be in its text and one sentence indicating what figures it will have. • If any of these figures already exist (even in preliminary form), put them there. • Indicate explicitly whenever something is a placeholder so we keep this in mind when we’re drafting the paper. • We gradually fill in more, as I give you first primarily big-picture feedback and then ever-more-microscale comments as we continue drafting the paper. • This is also consistent with a nested hierarchy for storytelling. • And, eventually, I also almost always go through the source file myself as well.
- 11. A Snarky, yet Strangely Accurate, Abstract
- 12. Now with a Subtitle (and my correct affiliation)
- 13. Paper from which I created my snarky abstract.
- 14. Who is your audience? • For a typical applied-mathematics paper: smart people, with some basic relevant background, who are not experts in your specific subarea • Different for different journals (e.g., physics journal versus applied-math journal) • Look at what is on the journal’s website • Example, for Physical Review E, I can assume some basic background in statistical physics. Nowadays, I will tersely give definitions for standard networks stuff, but I will give more details about more advanced or specific things (with a focus, of course, on those related directly to the topic of my paper). • This is something that we’ll need to have more detailed discussions about for your specific papers, as you are working on different topics from each other. This balance requires much thought.
- 15. Scrolling Through my Paper (X. F. Meng et al., PRE, 2018) • I. Introduction • II. Background • Short paragraph, subsection A (short), subsection B, subsection C • III. Methods (D’oh! I just said I prefer not to do that.) • Short paragraph, subsection A, subsection B • IV. Numerical Simulations and Results • A few paragraphs, subsection A, subsection B, subsection C, subsection D (I swear this section will end eventually), subsection E • V. Conclusions and Discussion • (continues on next page)
- 16. Scrolling Through my Paper (X. F. Meng et al., PRE, 2018), Part II • Acknowledgements • I am spelling it with an ‘e’, damnit! • Appendix A: Statistical Analysis • Brief paragraph, subsection 1, subsection 2 • Appendix B: Additional Examples • Subsection 1, subsection 2, subsection 3, subsection 4 • Appendix C: Best-Fit Parameters in Regression Models • A short paragraph and a large number of tables • References
- 17. Introduction: What did we do in X. F. Meng et al? • Note: I tend to write longer and more thorough introductions than others. • Referees sometimes complain to me about this, but in my opinion, what I am doing is a feature rather than a bug. • Answers: Why on earth am I writing this paper, and where does it fit into the literature? • Let’s scroll through that part of the paper in more detail • (If we didn’t do it earlier a couple of slides ago.)
- 18. The Introduction of X. F. Meng et al. (scrolling through it; it’s longer than usual, even for me) • Paragraph 1: Introduction to social interactions, opinion dynamics, etc. (and why we care about them) • Paragraph 2: There are various methods for studying this stuff. We connect this to an idea in statistical physics and mention (in a chain) and cite work on several models. • Paragraph 3 (long): We now introduce bounded-confidence models, the specific type that concerns us in this paper, and some types of questions that people consider. • Paragraph 4: Discrete versus continuous opinion spaces, and the fact that bounded-confidence models employ the latter • Paragraph 5: Some prior results (and associated difficulties) with the Deffuant model, the particular bounded-confidence model that we study • Paragraph 6: Prior work Deffuant model on networks • Paragraph 7: Some prior analytical work on Deffuant model • Paragraph 8: Past work on various extensions of Deffuant model • Paragraph 9: Here’s what we do in this paper (big picture) • Paragraph 10 (long): Here’s more specific stuff that we do in this paper (trying to state what we do to accomplish what we wrote in paragraph 9). • Paragraph 11: The rest of this paper is organized as follows… (Some people don’t like these paragraphs; I do like them.)
- 19. What goes in a ‘Conclusions and Discussion’ Section? • In order, one roughly does the following: • A bit of big picture about the problem studied, in the context of what one was trying to look at • Some specific results, in summary form • What one learns, especially in a big-picture context, from those results • Some discussion of future work, especially ideas that arose from the work and results on this paper • Possibly also other future work in order to highlight the important of specific things • Relevant discussion of big-picture limitations • Other, more specific, discussions of limitations (from assumptions, etc.) are usually elsewhere in the paper, though sometimes some of them show up here as well • Let’s look at what we did in X. F. Meng et al. • Again, we were longer than usual in this paper (including in comparison to some of my other papers), but the basic structure is still here,
- 20. Hey, I just went through a paper that appeared in a physics journal! • That’s right: I’m an applied mathematician. I often publish in physics journals (and biology journals and general “glossy” journals and others). • Applied mathematicians need to be able to write multiple types of papers.
- 21. A Paper in a SIAM Journal S. H. Piltz et al., “A Predator–2 Prey Fast–Slow Dynamical System for Rapid Predator Evolution”, SIAM J. App. Dyn. Sys., 2017 • 1. Introduction • Text, unnumbered subsection called “Rapid evolution”, unnumbered subsection called “Our approach”, unnumbered subsection called “Outline of our paper” • 2. The fast–slow 1 predator–2 prey model • A paragraph • 2.1. Ecological dynamics • 2.2. Evolutionary dynamics • 2.3. Coupled ecological and evolutionary dynamics • 3. Analytical setup • A very short paragraph • 3.1. Rescaling of the system (2.5) • 3.2. Linearization around the coexistence equilibrium of (3.2) • 3.3. Analysis of the system (3.2) as a fast–slow system • A short paragraph • 3.3.1. Slow reduced system • 3.3.2. Fast reduced system • 3.3.3. Critical manifold • 3.3.4. Slow flow on the hyperplane M0 (3.9) • 3.3.5. Slow flow on the hyperplane M1 (3.10) • (continues on next page)
- 22. A Paper in a SIAM Journal (Part II) S. H. Piltz et al., “A Predator–2 Prey Fast–Slow Dynamical System for Rapid Predator Evolution”, SIAM J. App. Dyn. Sys., 2017 • 4. Construction of approximate periodic orbits • Short paragraph • 4.1. Construction of a singular periodic orbit • 4.1.1. Geometric analysis of the fast reduced system • 4.1.2. Geometric analysis of the slow reduced system • 4.1.3. Combining the fast and slow reduced dynamics • 4.2. Existence conditions and solution families • 4.3. Approximate periodic orbits • Includes a displayed “Result 4.1” and a “Remark 4.1” that is noted as a paragraph • 5. Ecologically relevant qualitative aspects of the constructed periodic orbits • Discussion that includes a table that summarizes possible behaviors, linked to associated figures that illustrate each one (and then subsequent subsections discuss them in more detail) • 5.1. Synchronization • 5.1.1. Prey–prey synchronization • 5.1.2. Predator–prey synchronization • A paragraph, text highlighted for “Synchronization between all three species”, text highlighted for “Synchronization between predator and one prey” (including a summarizing enumeration at the end) • (continues on next page)
- 23. A Paper in a SIAM Journal (Part III) S. H. Piltz et al., “A Predator–2 Prey Fast–Slow Dynamical System for Rapid Predator Evolution”, SIAM J. App. Dyn. Sys., 2017 • (more Section 5) • 5.2. Clockwise and counterclockwise cycles • 6. Numerical continuation of the singular periodic orbits • 7. Conclusion and discussion • (We separated it analogously to what we did in the Introduction.) • Highlighted text on “Summary and time scales”, highlighted text on “Beyond piecewise-smooth formulations: Various ways to smoothen a jump”, highlighted text on “Obtaining analytical results using time-scale separation”, highlighted text on “Rapid evolution versus phenotypic plasticity”, highlighted text on “Cryptic and out-of-phase cycles”, highlighted text on “Future work and comparison with experiments and field observations” • Appendix A. Geometric singular perturbation theory • Appendix B. Finding solution families • Acknowledgements • References • Supplementary materials: .zip file with Mathematica notebooks for existence conditions (of periodic orbits) and solution visualization, solutionset.csv, solutionset.wdx
- 24. Some Referee Instructions from SIADS (SIAM J. App. Dyn. Sys.)
- 26. Other Notes • Make pointers to figures, tables, algorithms, equations, subsections, etc. explicit whenever possible. • Phrases like “see above” often have much less self-evident locations for paper readers than for paper authors. • (There are probably other notes that we’ll think of that should go here.)
- 27. Exercise • Pretend you are reviewing a paper, and see what you notice. • Do you understand why they undertook their study? • Do you understand what is new that they contributed to the literature? • Is it clear where they will go next? • Did they explain the technical details correctly, effectively, and honestly? • Are their figures clear and easy to understand? • Is anything crucial missing? • Etc. • What did the authors do well? • Not just for science, but for writing • What did they do poorly? And how would you have done certain things differently?
- 28. What else should we discuss? • Questions, questions, questions…
- 29. Many More Resources • Hey, Twitterverse! • When you see these slides (and, eventually, the video of a presentation and discussion) online, please let me know about links to others resources that I can add to these slides. • Please also let me know about any salient points that I missed.
- 30. Acknowledgements • Thanks to Mariano Beguerisse Díaz and Aaron Clauset for helpful comments. • (I expect to add thanks to others once I get their comments.)
- 31. Conclusions • Hopefully this helps! • Now let’s go write some fantastic papers! • Note: Much of what we’ve discussed also applies outside of applied mathematics, but I have geared my discussions and examples towards applied math.