What are really helpful math resources out there on the web?
Please don't only post a link but a short description of what it does and why it is helpful.
Please only one resource per answer and let the votes decide which are the best!
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1$\begingroup$ I edited your question a bit, my main goal was to remove the reference to WolframAlpha; and make the text easier to read. But feel free to revert if you like! (You can see the edit history and revert by clicking on "edited ... ago") $\endgroup$ – Ilya Nikokoshev Nov 4 '09 at 21:23
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3$\begingroup$ The answers below are great. Here's an idea I don't see that may be interesting to think about starting: An online example repository. This would be a place where one could upload and search various "first nontrivial example" notes. I think such a thing would really move mathematics along. $\endgroup$ – Jon Bannon Apr 20 '12 at 13:38
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edit by jc: As of May 11, 2010, the work has been completed!
This is a reference that is not yet complete, but it should be very useful when it finally does arrive:
Digital Library of Mathematical Functions (DLMF)
(book and associated website;
will replace Abramowitz & Stegun's Handbook of Mathematical Functions)
NIST / Cambridge University Press
expected 2009/2010
http://dlmf.nist.gov/
This will contain diagrams, tables, properties of, principal values of, and relationships between many important mathematical functions. For example, the trigonometric and other elementary functions are described, with very many formulae relating them.
The Handbook is very good; the Digital Library will be even better.
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$\begingroup$ The website, hardback and paperback of DLMF are now all available. $\endgroup$ – Rhubbarb Aug 17 '10 at 0:19
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http://maths-magic.ac.uk/index.php
Apparently UK has been building a depository/interactive system for graduate math courses. Click on "courses" to access archives. Many have lecture notes and other materials.
I found this recently. Have not actually personally used it, but potentially very useful.
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2$\begingroup$ A similar "access grid" network comprises the Universities of Bath, Bristol, Oxford, Warwick and Imperial College London. The main site (which includes links to archived course material) can be found at tcc.maths.ox.ac.uk. $\endgroup$ – Nicholas Jackson Oct 24 '09 at 11:04
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$\begingroup$ And there's a similar one for Scotland: smstc.ac.uk $\endgroup$ – Tom Leinster Oct 27 '09 at 17:12
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$\begingroup$ Australia has one also: amsi.org.au/index.php/ice-em/access-grid Unfortunately there is not nearly as much archive material as one would hope for. $\endgroup$ – Terry Tao Oct 27 '09 at 20:46
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$\begingroup$ Now when all of these have unified search capabilities, that would be a great day! $\endgroup$ – Max M Oct 31 '09 at 22:20
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While not as comprehensive as wikipedia, if you find an article on the scholarpedia on a topic, it should be the first place to look:
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Hosts high-level maths discussions, forums have inline LaTeX rendering.
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Alexandre Stefanov keeps an extensive list of free math books / lecture notes. The list is divided according to subject and updated frequently. I have found some very nice books there.
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$\begingroup$ Henning - I'm going to steal this and add it as a comment to the gigapedia answer. (It seems it might be more visible there, and thus help serve to discourage people from (mostly) illegal downloads.) $\endgroup$ – dvitek Oct 27 '10 at 5:10
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It seems this link hasn't appeared above http://www.ams.org/mathweb/index.html The resources there are too rich to describe.
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I recommend archive.org. Books from Fourier, Lagrange, Euler... old stuff.
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All math.[institution].edu/~[professor]/ sites are great with, in of themselves, many links to the favorites of the page's professor.
It is like walking up to the professor at coffee and asking him about the tools he uses (resources) and how it applies to his research . A big cafeteria with the world's professors ressembled and willing to answer any questions... or at least those who keep their site up to date.
An example: http://www.math.ucla.edu/~tao/
Professor Tao's page is mostly blue (links).
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The manifold atlas is pretty cool. I haven't spent enough time on it though... It seems like a different type of mathematical venture. Hopefully, it will inspire other similar projects. http://www.map.mpim-bonn.mpg.de/index.php/Main_Page
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I don't know if this reference is of sufficient generality:
Finite Calculus: A Tutorial for Solving Nasty Sums
http://www.stanford.edu/~dgleich/publications/finite-calculus.pdf
It is only a paper, but it describes the methods of the so-called "umbral calculus": a really useful technique to know.
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1$\begingroup$ It's nice, but I don't think it ever comes to actual umbral calculus. Just working with finite difference operators and falling factorials is not umbral calculus yet. $\endgroup$ – darij grinberg Mar 7 '10 at 23:20
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$\begingroup$ I think Gosper's algorithm is worth mentioning here. $\endgroup$ – teil Apr 25 '10 at 23:44
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The optimization community seems to prefer this specific online repository instead of the more broad one arxiv.
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Since someone mentioned The Digital Library of Mathematical Functions, we better also include The Wolfram function site: http://functions.wolfram.com/
It's really useful for special function identities - especially since they are also available in Mathematica input form that you can copy straight into your code.
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From website: Project Euler is a series of challenging mathematical/computer programming problems that will require more than just mathematical insights to solve. Although mathematics will help you arrive at elegant and efficient methods, the use of a computer and programming skills will be required to solve most problems.
From me: I have personally found it beneficial to go through these to help work at how I think about math problems.
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Quite impressive is this site:
"PlanetMath is a virtual community which aims to help make mathematical knowledge more accessible" - or how they put it: "Math for the people, by the people":
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Resource for books is book.fi - select English from upper right.
Resource for (mostly free) papers is projecteuclid.org
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http://citeseer.ist.psu.edu/ caches a lot of papers that has been posted online. It often comes up within the first few search results in Google. (But you cannot view the cached documents online, since they are directly downloaded.)
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There are some great things here at the small but fine Clay Institute Online Library
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$\begingroup$ According to a little box on the right hand side of the front page claymath.org/index.php "CMI publications are available in PDF form at most six months after they appear in print" but that seems not to be the case for the most recent publications... $\endgroup$ – j.c. May 20 '10 at 1:08
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$\begingroup$ True, it's written there. But for example the outstanding book "Harmonic Analysis, Trace formulas and Shimura Varieties" from 2003 is still there. For me the CMI can stay inconsistent with respect to that :-) $\endgroup$ – Peter Arndt May 21 '10 at 14:03
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People: consider http://www.digizeitschriften.de/ tons of classical papers in english...
I think it is worth to check the 39 journals collection on world class referee-ed mathwork.
One paper on Mathematische Annalen (which is the very amusing): "On the holymorphic flow with an isolated singularity", is the famous GSV, gives you an index formula...
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Jahrbuch Database
A sort of Mathscinet and Zentralblatt for the period 1868-1942. Most of the reviews are in German. It is interesting to read the reviews written by mathematicians like Frobenius, Hilbert, Minkowski, Hasse, E. Noether, Artin, Mittag-Leffler, Landau, Van der Waerden, ...
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LMFDB has been officially launched yesterday (10th May of 2016).
It is an integrated knowledge database of L-functions and modular forms with a nice web interface that helps visualize the connections between these mathematical objects.
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Topology Atlas at York University is a great site with an awesome Q&A board (it of course, was not just restricted to Topology) and has been around for years.
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CiteULike (by Springer), to organize in a library the titles and abstracts of one's preferred papers and books.
(From the FAQ:) CiteULike is a free service to help you to store, organise and share the scholarly papers you are reading. When you see a paper on the web that interests you, you can click one button and have it added to your personal library. CiteULike automatically extracts the citation details, so there's no need to type them in yourself. It all works from within your web browser so there's no need to install any software. Because your library is stored on the server, you can access it from any computer with an Internet connection.
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Very nice Notes and Videos from the Southwest Center for Arithmetic Geometry are available here!
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For students (or even teachers!),the Abdus Salam International Centre for Theoretical Physics has lots of lectures in Advanced Math.Every year the lectures are different.Enjoy! http://www.ictp.tv/diploma/index08-09.php?activityid=MTH
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Mathematics Dictionary & Glossary for students at http://www.tuition.com.hk/mathematics/
This is a very comprehensive source of mathematical definitions.
With over 2000 terms defined, this dictionary is ideal for supporting students who are studying mathematics or related subjects. All terms in our dictionary are cross-referenced and linked for ease of use, making finding information quick and easy.
