SUPPORTED LATEXFEATURES Authorea features LaTeXML for the web view. The continually updated list of packages supported by LaTeXML includes, at the time of writing, 286 class and style files. This is an up to date list. How to include a LaTeX package in your document To add a LaTeX package to your document, follow these instructions. In a nutshell: click on the Document Settings and select EDIT MACROS. What is supported? When writing LaTeX inside Authorea documents (here’s how), the following class and style files _should_ be supported (rendering LaTeX on the web is not easy so we cannot always guarantee everything will work): -- -------------------- ------------------ --------------------- ----------------- ----------------- ----------------- AMSAfontencoding AMSBfontencoding AmSTeX.pool BibTeX.pool JHEP.cls JHEP2.cls JHEP3.cls LGRfontencoding LaTeX OmniBus.cls SIunits TeX a0poster a0size a4 a4wide aa aasupport aasmacros.sty aassupport.sty aasms aaspp aastex acronym amssupport amsart amsbook amsbsy amscd amsfonts amsgen.sty.ltxml amsmath amsopn amsppt.sty amsproc amsrefs amssymb amstex amstex.tex amstext amsthm amsxtra apjfonts applemac.def array article attachfile.sty authblk.sty avant babel babel.def balance.sty bbm bbold beton bm book bookman booktabs braket breakurl.sty calc.sty cancel caption.sty ccfonts chancery charter circuitikz.sty cite citesort cmbright color colordvi colortbl comment concmath courier cp852.def crop.sty cropmark csquotes dcolumn deluxetable.sty doublespace dsfont eTeX ellipsis elsart.cls elsart.sty elsartsupport elsarticle.cls emulateapj emulateapj5 enumerate.sty epigraph epsf epsfig epstopdf esint eucal eufrak euler eulervm eurosym.sty euscript exscale fancyhdr.sty fix-cm fixltx2e flafter fleqn float floatfig floatflt floatpag fontenc fontspec footmisc fourier framed frenchb.ldf fullpage gen-j-l gen-m-l gen-p-l geometry german graphics graphicx grffile helvet hepunits here hhline html hyperref hyperxmp ifluatex ifpdf ifthen.sty ifvtex ifxetex inputenc insertfirst instsupport iopams.sty iopart.cls iopartsupport keyval latexml latexsym latin10.def lineno listings listingsutf8 llncs.cls lmodern.sty longtable lscape luximono lxRDFa ly1fontencoding makeidx marvosym.sty mathbbol.sty mathpazo mathpple mathptm mathptmx mathrsfs mathtools mn.cls mn2e.cls mn2esupport multicol.sty multido.sty multirow nameref natbib newcent newlfont.sty.ltxml ngerman nicefrac ntheorem numprint ot4fontencoding palatino paralist pdfTeX pdflscape pdfsync pgf.sty pgfkeys.code.tex pgfplots.sty pgfsyslatexmldriver pifont placeins preview psfig psfig.tex pslatex pspicture pst-grad.sty pxfonts relsize report revsymb.sty revtex.cls revtex.sty revtex3support revtex4.cls revtex4.sty revtex4support rotate rotating rsfs scalefnt setspacing showkeys.sty siunitx slashed slides soul srcltx stmaryd subfig.sty subfigure.sty subfloat supertabular svg svjour.cls svmult.cls svsupport t1enc.def t1enc.sty t1fontencoding t2afontencoding t2bfontencoding t2cfontencoding tabularx tabulary textcomp texvc theorem threeparttable tikz-3dplot.sty tikz.sty times tocbibind transparent ts1fontencoding txfonts type1cm.def ulem units upgreek upref url -- -------------------- ------------------ --------------------- ----------------- ----------------- -----------------

PYTHAGORAS’ THEOREM a² + b² = c² _Pythagoras, 530 BC_ -------------------------------- -------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------ --------- LOGARITHMS logxy = logx + logy _John Napier, 1610_ CALCULUS $ f}{ t} = {h}$ _Newton, 1668_ LAW OF GRAVITY $F = G {r^2}$ _Newton, 1687_ THE SQUARE ROOT OF MINUS ONE i² = −1 _Euler, 1750_ EULER’S FORMULA FOR POLYHEDRA V − E + F = 2 _Euler, 1751_ NORMAL DISTRIBUTION $\psi(x) = {} e^{2~\rho^2}$ _C. F. Gauss, 1810_ WAVE EQUATION ${\partial t^2} = c^2 {\partial x^2}$ _J. D‘Ambert, 1746_ FOURIER TRANSFORM f(ω)=∫−∞∞f(x)e−2 π i x ωdx _J. Fourier, 1822_ NAVIER-STOKES EQUATION $\rho \left ( }{\partial t} + \cdot \nabla \right ) = - \nabla p + \nabla \cdot T + f$ _C. Navier, G. Stokes, 1845_ MAXWELL’S EQUATIONS ∇ ⋅ E = 0 _J. C. Maxwell, 1865_ $\nabla \times E = - {e} {\partial t}$ ∇ ⋅ H = 0 $\nabla \times H = {e} {\partial t}$ SECOND LAW OF THERMODYNAMICS dS ≥ 0 _L. Boltzmann, 1874_ PAYWALL RELATIVITY E = mc² _Einstein, 1905_ PAYWALL SCHRÖDINGER’S EQUATION $ \hbar {\partial t} \psi = H \psi$ _E. Schrödinger, 1927_ PAYWALL INFORMATION THEORY H = −∑p(x)logp(x) _C. Shannon, 1949_ PAYWALL CHAOS THEORY xt + 1 = k xt(1 − xt) _Robert May, 1975_ PAYWALL BLACK-SCHOLES EQUATION ${2} \sigma^2 S^2 {\partial S^2} + r S {\partial S} + {\partial t} - r V = 0$ _F. Black, M. Scholes, 1990_ PAYWALL EULER’S TRANSFORMATION $^\infty (-1)^n a_n = ^\infty (-1)^n {2^{n+1}}$ _Euler, 1755_ PAYWALL RUSSELL’S PARADOX Let R = {x ∣ x ∉ x}, then R ∈ R ⇔ R ∉ R _Russell, 1902_ GÖDEL’S INCOMPLETENESS THEOREM G(x):=¬Prov(sub(x, x)) ⇒ PA ⊢ G(⌜G⌝) ↔ ¬Prov(⌜G(⌜G⌝)⌝) _Gödel, 1931_

1SADF ASDF some stuff: 1 sadf sadf asdf asf sadf asdf asdfasdfasdfasd fasdf asdf asdf asdf

ADVICE TO PHD / DPHIL STUDENTSADVICE TO PHD / DPHIL STUDENTS Or What I wish someone had told me PhDs vary wildly from country to city to uni to department to group to individual. All the advice here may be perfect for you, or it may miss the mark by miles. Pick and choose what you need, or create your own. Before you start - Self-proposed PhD projects are much more risky - It is best to pick an advisor because you feel comfortable with and trust the person. This person is going to have to tell you some tough information (that you may not like). - Pick a supervisor and group you can work with and like. You’re going to be spending a lot of time with them and this can really make or break the experience - If the professor you really want to work for does not particularly work in your field of interest, talk to them. Professors have vast interests. They might have actually done their PhD work in your interest and are really excited to advise you! - I wish I’d known how much I would enjoy it - I would have done it sooner. - I wish I’d known my supervisor was going to leave the Uni at very little notice. (I have a new sup, but now I wish I’d started with him. At the beginning - Imposter syndrome is normal - You don’t need to prove how smart you are. We know you’re smart. You wouldn’t be here if you weren’t.   - Supervisors talk about your PhD as if you have done one before and know how its going to turn out   - Self-discipline is good. Try and work sensible office hours, although take advantage of the flexibility when you can - Set a schedule. This is your job. You should be working 8-10 hours a day, at least 5 days a week.   - Be prepared for overtime. No one finishes on time, yet no one prepares you for the stress of overtime   - Making annotated bibliographies is inordinately helpful when it comes time to write - Start compiling refs and writing summaries as early as possible. - Start using EndNote or another reference software early on. Write brief summaries to help you remember what you have read. - You probably haven’t been compiling refs and writing summaries, don’t worry, lots of people don’t and still get their PhDs. ;) - I wish I’d known how much my vision of what I’m doing would change over time. - Writing is hard. Presenting is hard. You’re not supposed to be great at either right from the get go, but it does get easier with time.   - Take opportunities to go to seminars etc   - If you have the chance to take professional development training, particularly for writing and presenting, do it. - If you have the chance to do things outside of research, do it. - The purpose of a PhD is to learn _how to do research_. It’s not just about your project, it’s about learning how to do a project. So there are things you won’t know how to do yet (writing, speaking, time management, whatever). Accept that you will be crap at these things to start with, but do them anyway. You’ll get better with time. - Check and double check related work early on a paper - it is absolutely devastating to find a similar paper close to deadline. Backing up... - Always backup your data, papers, etc. Computers particularly like to crash and break near deadlines. - BACKUP YOUR PAPER DOCUMENTS. Writing - Write progress reports - Write stuff down. All the time - Every time you complete some subproject, write it up properly, you’ll never remember all the methodology when you come to write your thesis otherwise - Come up with a system for indexing things you read. Index cards/folders/binders/If you use bibliographic software, don’t wait until you are on deadline to learn how to use it. General advice - Expect the unexpected - Not having results by the end of your second year is normal - Make friends with your supervisor’s PA. She guards his diary - The academic world is extremely effective at wasting your time. Don’t allow other people’s delays to give you reason to procrastinate. Hassle them daily. - Don’t assume your supervisor is responsible for telling you what you need to do to get a PhD - Don’t assume your supervisor understands your project, or indeed cares about it - The only person interested in the outcome of your PhD is you - unless your supervisor has a very strong direct vested interest in the outcome of your work - Business cards are hugely useful. - Go to conferences. Conferences are interesting and fun. Travel grants are available - Don’t be afraid to reach out to other researchers in the field - People generally respond positively to requests for information / data etc - Perhaps the most important thing is to seek the company of other researchers in your field - Exercise and cook/eat well. Make these things become hobbies so you are getting some enjoyment out of the things that you have to/should be doing. - You might not succeed with what you set out to do - but those mistakes can be GOLDEN, if you use them. - Have hard deadlines and strive to meet them, productivity is way higher when there is a plan and deadlines to meet Libraries - Getting a British Library Reader’s Card is very easy, but makes you feel like a Proper Academic - Never trust the electronic library catalogue - If the catalogue seems to be lying, consult a librarian. Many of them actually _enjoy_ exercising their library-fu! - Get your know the reference librarian in your area. Bring him cookies. They can save your ass when you are on deadline because they can search way more quickly than you. The dreaded administration - Remember the paperwork! - No-one expects your Gantt charts to be realistic - Befriend the cleaning staff, the admin staff and the catering folk. - "DON’T PISS OFF THE ADMINISTRATORS!“ (from an administrator!) When the going gets tough - Always remember that you’re free to leave. There are lots of cool careers out there for folk smart enough to get into a doctoral program - you’re _not_ trapped here! - You can quit. - You can quit. That needs repeating. It is possible to be happy, successful and respected after quitting a PhD. If it’s not working and it’s making you miserable there is no merit in staying. - It’s ok to ask for help. - Remember that you’re here to produce a thesis, not to change the world or make a major breakthrough. No-one ever makes major breakthroughs in their PhD thesis. - Sometimes the most productive thing you can do is take a break and do something non-academic. - The PhD process is almost guaranteed to break you down before it builds you up, so expect to feel down in the dumps some days (or weeks, or even months). To counter this it is important to have a support network (other students, friends, family,…) that you can rely on - There will be times when you and your advisor will argue and disagree. This is part of the process. - Every student follows a different path and makes decisions for different reasons. Follow what is best for you and do not worry about everyone else. - Make sure to make time for the things you love doing (knitting for me). You’re going to be working on the PhD for a long time. - Avoid comparing yourself to others. PhDs are so individual it’s almost useless, and chances are you’re biased against yourself anyway. - You’re here, so you’re smart enough (away, imposter syndrome!). Finishing isn’t about being smart, it’s about luck, hard work, persistence, luck, stubbornness and LUCK. - If you’re sick, stay home and get better. - Although it might not feel like it, this can be an ideal time in your life for things to go pear shaped. You probably have a flexible work schedule and you’re a student so you have access to a lot of student type support, health, counselling, etc, that your 20 something friends with real jobs don’t have access to. - You will feel stupid, lost, overwhelmed, hopeless. It can pass. Just take it one paragraph at a time. - It’s ok if there are times when you want to jack it in and work at McDonald’s. Go home, have a cup of tea , come back tomorrow. It will be a better day. Advice for medics - Ethics takes far, far longer than you ever guessed Advice for engineers - Matlab’s comms toolbox is a big bag of hurt - Matlab is a moderate-sized bag of hurt Finishing off - PhD’s are pass/fail – if it’s good enough, it’s time to stop - At the end of the day, a PhD is pass/fail. - A done paper is better than a good paper. - The finish line may move, but you will get there - The end will be in sight, eventually, even if it keeps going out of view, it will come back! - There ain’t nothing ‘simple’ about a thesis/dissertation! :) This is a PhD, and it’s meant to be hard, otherwise everyone would have one. - Own it: it’s a huge thing and you’re going to get one! Compiled by Susannah Fleming With thanks to all contributors: Gareth, Mike, Steve, Griselda, Gina, Stephany, Carrie, Mary-Helen, Suzi, Esmee, Kim, Jen, Witty, Karen, Sigrid, Ahmar

1 some stuff: 1, $$x_{1} \pm 1$$

A document by michaeljdklein, written on Authorea.

A document by michaeljdklein, written on Authorea.

INTRODUCTION This problem came up in the process of searching for lower bounds on the number of cycles in a graph with certain properties. I chose this proof as my sample because it’s my favorite short proof. The final bit of reasoning is analogous to reducing the problem to a halting problem and solving that halting problem.

Hey, welcome. Double click anywhere on the text to start writing. In addition to simple text you can also add text formatted in BOLDFACE, _italic_, and yes, math too: E = mc²! Add images by drag’n’drop or click on the “Insert Figure” button.