Searching Writing Citing mathematics statistics mecanics. Karoline Moe

Searching Writing Citing mathematics statistics mecanics Karoline Moe

Previously vs. now Textbook and exercises Exams with problems Exact answers Unknown syllabus Unknown literature Unknown results Present it Written thesis Oral presentation

Research in mathematics Solve a problem Build a theory Simulations Combinations Publish Write Problem Get an overview Aquire skills Experiment New results!

You have a topic now what? Get topic and starting points from supervisor You have to find literature about your topic Broad perspective History Motivation What have others done Core material / textbook Review article Specific results Your level Other points of view

How to find literature? Get books or articles from supervisor Ask «older» students Ask on a web page Google it You want to be scientific

Scientific material and peer review Ensure that you build upon scientific material Peer review Results in an article is evaluated by another expert prior to publication Anonymously Gives an insurance no guarantee - that the result is credible Always small errors Fraud and misconduct Retractions and corrections (Retraction Watch) Pre-print publication (www.arxiv.org) Research and publication bias

Retracted 2004 and 2010 https://en.wikipedia.org/wiki/mmr_vaccine_controversy «Perhaps the most damaging medical hoax of the last 100 years»

Alternative facts (2014) https://en.wikipedia.org/wiki/mmr_vaccine_controversy

How to find scientific works in mathematics Use peer reviewed sources of information Books published by renowned publishing houses Articles published in renowned peer reviewed journals Reference hunting from good sources Beall s blacklist Use subject databases Subject databases index renowned journals But Google Scholar works well in mathematics More problems elsewhere Check the Norwegian database NSD Format Mathematicians write in LaTeX Easy to see with experience But pretty LaTeX doesn t always mean correct mathematics Use the library access to 100MNOK/year worth of research Ask your supervisor Stalk the author (Google Scholar, MathSciNet) Professor? Master s student? No university affiliation? Check reviews on MathSciNet Beware of pre-print-archives arxiv.org Stay away from alternative groups vixra.org http://vixra.org/pdf/1312.0082v4.pdf

https://math.stackexchange.com/questions/348198/best-fake-proofs-a-m-se-april-fools-day-collection

About the Science Library Physical The books are mostly sorted by topic browse the shelf Math collection Entrance wall New book collection Undergraduate collection Collection 42 In order of appearance Search online Deposited in storage rooms SL-mag, VB-mag, GSH-mag Online The University Library ub.uio.no Science terms ub.uio.no/matematikk Databases MathSciNet ZBMath Web of Science Google Scholar arxiv (pre-prints)

Examples: Fermat s last theorem Ask at the front desk or book a librarian Use the library catalogue online Search your topic Beware of special characters Request the book or article Search in Emneord Find your MSC and Dewey-number Use Wikipedia as a starting point Search in MathSciNet Wiles, Andrew Search in Google Scholar or Google Fermat s last theorem Go from one article or book Reference hunting (Backwards in time) Citation hunting (Forwards in time) What if you can t find it? Can t find it?-button realfagsbiblioteket@ub.uio.no

How to read an article Read the abstract and the introduction first Be patient It might take years to understand one single article Everything is easier in small bites Put it away and pick it up again Ask for help from your supervisor Start with the most sentral source Find the most sentral result / techique Find other sources that can shed light on the result Make notes in the margin and underline / highlight Work your way through important proofs with pen an paper

Critical reading Is this high quality research? Ethical issues? Evaluate the results and the data Evaluate the numbers Do you have access to the data? What is not commented? Small part of the puzzle or entire picture? Part of a proof left to the reader? Loose claims Lack of references? Quality of the references Groups of authors each show the benefits of their own preferred treatment in an uncontrolled way. https://pure.uva.nl/ws/files/2785453/170706_08.pdf

Structuring your research 101 Find your style Always carry an article that you can read Identify a bunch of small equations or tasks that you can work on anywhere Test ideas on blank sheets of paper Use colours and a sharp pencil Sum up in a notebook or folder What works What doesn t work Find inspiration Write on a LaTeX-document from day 1 Writing is a catalyst for results

Data management 101 Save files a safe place with backup systems M-disk on UiO Safe cloud Save and share on f.ex. Zenodo Keep it tidy Name Folders Use systems with version control F.ex. GitHub / BitBucket F.ex. ShareLaTeX / Overleaf / Authorea

http://www.relativelyinteresting.com/how-to-write-good/

Writing in mathematics Mathematics happens NON-linearly, typically back and forth Mathematics is presented linearly Precise, concise and focused Strong contrast to a novel or popular science Every word has a specific meaning Stay focused when you describe results Avoid describing your results with adjectives A result is never «wonderful» «interesting» «clever» «important»

General advice Write for your audience Yourself, sensor, supervisor, future employer, other students, researchers in your field Invite the reader into your text Make it easy for him/her Don t over- or underestimate the reader Make your intentions clear Build a clear structure with peaks Work on transitions Find your voice

The process Write immediately and all the time Write to create text and structure Write to learn Write to find results Write to clarify how your results are connected Write to correct yourself Write to get the details right Write to be efficient Expect to write everything at least 2-3 times The 30%-rule Always let your supervisor in on your 30% draft Expect feedback accordingly Academic writing centre

Choosing language English Most science is in English Easy to find technical terminology Easily shared with other researchers Norwegian Spend less time on grammar and spelling and more time on math Better understanding Ask your supervisor!

A paragraph Claim Argument Examples Discussion Conclusion

Transitions Make logical transitions between sentences and paragraphs Many different types and subtleties Beware of precision level Keep a list of these in front of you when you write text Heavily used in mathematics In addition to Furthermore Moreover First, second, third Similarly Thus Therefore Consequently Additionaly Hence However On the other hand In particular

Who does what? And it s hardly «cool» Write «we» always Except in the acknowledgements Be conscientious about who did what Important to distinguish between your own and others results Never take credit for something someone else did Take all credit for the things you did on your own (with help from supervisor) «We define» vs. «Is defined as» Who proved the theorem? «We have the following theorem» «We use the following theorem» «We show the following theorem» Avoid adjectives be specific Don t use adjectives to describe results Be specific and describe how they can be used

Sentences Always write in full sentences A mathematical expression is a part of a sentence A citation is a part of a sentence Beware of punctuation In mathematics Commas Build your sentence with logical order The important part first Objects and properties must match

Words and abbreviations Write it out It is, can not, do not Use abbreviations correctly i.e. means complete list of examples or explanation e.g. means incomplete list of examples etc. means and so on Never combine with e.g. cf. means confer (compare) See means find more information

Using symbols Two golden rules Don t introduce unneccessary symbols Define all symbols before they are used Two typical mistakes One thing is referred to by two different symbols One symbol means two things

Alpha and Omega A lot of symbols: Latin alphabet Greek alphabet Small or capital letters Caligraphic letters Upright letters Boldface letters Math symbols In math environment Looks almost like italic Detexify Symbols can be decorated: Indexes, subscripts, superscripts Hats, curls, arrows, lines, stars, dots Be consistent Avoid the christmas tree

Special characters and parenthesis Follow conventions Funksjons (f,g,h) Sets (A,B) Indexes (i,j,k) See the textbook / main reference Avoid special characters in title and abstract Watch the parenthesis Right number {(([((())))])} Right size Right place Readability Line breaks Look out for the indexes Subscript and superscript need ^{}

Figures og tables List of figures Figures and pictures Explanatory caption Does the figure / picture communicate the right idea? White background Less is more Use one program (and cite it) Consistent design Print resolution (300dpi) Think through use of colours (CMYK) Same sizes Same line thickness List of tables Tables Consistent layout Readability Text justification How many lines do you need?

Results and proof Theorem (Strong result) Proposition (Smaller result) Lemma (Intermediate or technical result) Corollary (Follows from a theorem or proposition) Conjecture (Not proven) Remark Proof Introductory and transitional text Prepare notation and concepts Assumptions State the result in an appropriate environment Continuous numbering Type of proof Direct Induction Contradiction Logical structure of proof Outline the steps first Step by step

Sensor guidance on censorship of master theses The sensor assesses the extent to which the candidate has achieved these goals: 1. Professional anchoring 2. Theoretical insight 3. Target description 4. Skill Level 5. The work 6. Analysis and discussion 7. Critical reflection 8. Own contribution / goal achievement 9. Structure 10. Language 11. Form

The problem, broad terms What others have done What you will do and how The bulk of the thesis Preliminaries, notation Methodology Results Summary of your work Limitations, weakness Kent Andre Mardal What others have done, significance

The problem, broad terms What others have done Goal description What you will do and how Professional anchoring Skill level, theoretical insight, effort The bulk of the thesis Preliminaries, notation Methodology Results Kent Andre Mardal Goal achievement Summary of your work Limitations, weakness What others have done, significance Analysis, discussion, crtical reflection

Structure your thesis (or article) Mathematics Title Abstract Introduction Notation and definitions Results and proofs Simulations and pseudocode Applications and examples Conclusive chapter List of references Experimental sciences IMRaD Title Abstract Introduction Method Results Analysis and Discussion List of references

Title Different kinds of titles Relatively short One sentence Descriptive Provocative Motivating Engaging Very rarely a newspaper headline or clickbait Presice Searchable So that other researchers will find it Avoid special characters

Abstract Is read first To read or not to read Put the topic and your results in context quickly Main results only Mention techniques used Avoid special characters Do not cite anyone in the abstract Abstract In this article we explore some wonderfully intricate relationships between three mathematical objects, each of which is associated in some way with the Fibonacci sequence. One of these objects is a particular finite series comprising n terms, and, via the other two, our mathematical journey culminates in the derivation of an expression for the sum of this series in terms of sums and products of certain Fibonacci numbers.

Introduction Explain the problem Put the problem in context Background and motivation Why is this interesting? What have others done? Present what you will do and how (thesis) Present your main results (article) Enough technical background to make it readable for an interested reader Present the outline Very important Everyone reads it Set the «feel» of the thesis Adjust the language and mathematics Tip! Write the introduction several times! The first you write The last you write

Notation and definitions Introduce and explain most technical terms and symbols that are used To make sure that your text can be read by others (including yourself a few years from now) Some terms and notation must be considered to be well known Depends on your audience Be precise, concise og focused Use standard notation Tip! Mathematical Writing by Franco Vivaldi

Simulations and pseudocode You have simulations and code written in a particular program Presented indirectly and referred to in the text Presented in form of pseudocode Complete code included in an appendix? Store and make complete code available online (Zenodo or Github) Main aim: Make it possible to reconstruct your results now and in the future Lost files New software Incompatibility

Applications and examples Applications and examples supporting your results Choose them wisely Argue for your choices! Complete overview Systematic overview Randomly Presented with table and figure

Conclusive chapter Summary of your work Broader perspective Significance and limitations Identify open problems Point to future research possibilities

If I have seen further it is by standing on ye sholders of Giants. Sir Isaac Newton, 1675 https://en.wikipedia.org/wiki/standing_on_the_shoulders_of_giants

Citing in mathematics All results and claims should be supported by Your own mathematical argument A citation of a result by someone else Different kinds of citations The entire source Specific results Quotes are rare Paraphrase Where do we cite After a claim - at the end of sentences or paragraphs In theorems, propositions and so on All sources that are cited is presented in a list of references Different styles Every journal has a preferred style numerical [1] alphabetic [Bor17] Only very well known results are cited by their name: Pythagoras theorem, Triangle inequality, Cauchy Schwarz inequality General citations without page numbers [1] or [Bor17] Borgan in [Bor17] Specific results with numbering or page numbers Maugesten and Moe [MM17, Theorem 1.1] Maugesten and Moe [MM17, p. 12] Maugesten and Moe [MM17, pp. 12--15]

Citation code \cite{ran17} \cite[p.~24]{ran17} \cite[pp.~24--26]{ran17} \cite[theorem 4.2]{Ran17} \cite[see][p.~32]{ran17} I teoremer og liknende {} Tip! Use bibliography style alphabetic and automatically generated citation as citation key

Cite to make your research verifyable Main point with citations is that the reader should be able to check and recreate or verify your results Always use the source in front of you! Be precise, be ware of editions, pre-prints etc. Use the primary source If you don t have the primary source or it can not be used, cite the statement about the result in the seconday source that you have in front of you!

Give credit to the right person Who did this? Clearly state this in the text! Other people s results used directly A combination of other people s results You did it! Well known results Not everything needs a citation Depends on level, audience and goal Pictures, figures, graphs, tables and drawings belong to someone Ask for permission to use them Cite them correctly You may recreate graphs and figures, but cite the data / software you use

Paraphrasing Common to make changes / paraphrase You still have to cite and give credit to the person behind the result From general to special Level on what you write vs level on the source A result that covers a more general case than you are working with Notation must be consequent! Different notations must be merged Declare notation early on and in running text «With this notation, Pythagoras theorem reads»

General or specific?

Choosing notation 1 2 3

Numbering and cross references Numbering Figures, tables, theorems, lemmas and so on should be numbered Equations and formulas that you use different places in the text need a number Central equations and formulas should have a number so that others can cite them What is cross references? A way to connect different parts of the text using numbers Why use cross references? Helps the reader navigate in your thesis Use \usepackage{hyperref} for digital navigation

Cross references code \label{abel} \ref{abel} gives number only \cref{abel} gives type and number \vref{abel} gives type, number and page number Tip! Use prefix ex: or for: or thm: or lem: when you make labels, and use \usepackage{showlabels} when you work on your document \usepackage{showkeys}

Summing up: Using sources in your thesis Find the result you write about Use primary source Find more sources about the topic Book Thesis Article Web page All sources and programs that you use must be cited in the text List of references with complete infomation about your sources at the end of your thesis Clearly convey: What have you done yourself? What has been done by others? Your text should be Scientific Verifiable Results, figures and tables belong to someone Plagiarism never copy text Rules on cheating at UiO Special situations in mathematics

Reference list Detailed information about all sources cited in the text Surjective Only these sources no more, no less Enough information to find exactly the text that has been used Gives information about your text What you work on The level you work on Where the text comes from Where the text is going Helps you get an overview

Reference list / Bibliography Metadata Information about the material Book Author, title, publisher, place and year Article Author, title, journal, volume, year, page numers Web page Author, title, address, visiting date arxiv-artikler has own set of rules Important to be precise Styles numerical [1] alphabetic [Ran17] Any others! (Borgan 2017) Sorting

Citation and reference management in LaTeX Make the file bibliografi.bib Find metadata in BibTeX-format MathSciNet (Preferred) Google, Scholar or Oria (Must be checked) Manuelt (Demanding) Put all metadata in the bibliography file Be tidy Alphabetic Citation key Each item has a nick name Use it to connect the citation with the bibliography Proof read the bibliography! Tip! Use bibliography style alphabetic and automatically generated citation as citation key Beware: All posts in the bibliography must be tidied up and checked for errors and superfluous details

Book Author Title Publisher Place Year

Thesis Author Title Degree University Year May contain Digital Object Identifier (DOI) Uniform Resource Locator (URL)

Article Author Title Journal Often abbreviated Volume Year Page numbers May contain DOI URL

Webpages Author Title Link Visiting date Special cases arxiv-articles wtih numbers (and version)

Two systems for reference mangement in LaTeX BibTeX Old system Few reference classes Widely used in journals Problems with special characters {A} for capital A i title field \usepackage{cite} \bibliographystyle{alpha} \bibliography{bibliografi} BibLaTeX - Biber New system Compatible with BibTeX Many reference classes Smarter with special characters Not that much used in journals \usepackage[style=alphabetic]{biblatex} \addbibresource{bibliografi} \printbibliography

General tips Be consistent Cheatsheet Abbreviate first name Show surname first \DeclareNameAlias{author}{last-first} Use URL for digital navigation DOI Jstor Link to repositories f.ex. DUO arxiv-number Avoid unscientific links: Google books etc.

Proofreading Use a spell checker Not picked up by the spell checker to or too or two flaw or flow plain or plane Mathematics Formulas Indexes Table of contents The reference list Check figures Ensure flow between paragraphs and sections Beware of words repeating themselves Extra words and mistakes after rewrites Punctuation Notation No unneccessary symbols All symbols are defined One-to-one and onto Typography

Done. Now what? Tidy up in your folders and notes «Final» CODESurname.tex A tidy bib.bib All figures in one folder CODESurname.pdf Program code Data «Portfolio» CODESurname.pdf

Exercise Searching for literature What kind of material are the different references? Search for [BK86] in Oria Does the library have it? If so where? Does it exist online? If so download it to your computer Search for [BG97] in: Oria. What do you find? Google Scholar. How many people have cited this article? Find one of them. MathSciNet. Read the review and find the BibTeX-post. How many articles has E. Ballico written? Find [MM17]

Exercise Citing literature You get an excerpt from an article 1. Read the text quickly 2. Mark where you would put: Citations (with or without page numbers) Cross references 3. Discussions