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Now hosted on WordPress

The current version of the catalog is at mathindex.wordpress.com

It has been expanded compared to the description below. Still under construction.


Original version of the post

Standard calculus exercises tend to be posted repeatedly, for various reasons. And although one might hope that each new iteration uncovers yet another aspect of the limit $$\lim_{x\to 0}\frac{x-\sin x}{x^3}$$ this is probably not the case.

So, I wonder if thematic catalogs of links to standard exercises could exist somewhere on the site (or elsewhere). If they were sufficiently visible, perhaps the questions would not be repeated as often. If nothing else, a glance at the list can suggest possible duplicates quicker than search.

Using the Data Explorer and a little script I produced a demo version of such a catalog, posted separately at http://meta.math.stackexchange.com/q/16717/. The questions are ordered by "arbitrary hotness points", equal to 20*(total score of answers) plus the number of views. The list is not exhaustive; I took the cream of the crop. Sadly, large number of formulas slows down browsers on less-powerful machines.

After seeing a very preliminary version of the catalog (only trigonometric limits included)... what do you think of its feasibility/usefulness?


Analysis of the attempt

Seeing the painfully slow page loads in the catalog, I think it is preferable to host it elsewhere, reworking it into something like $\pi$-base. (For those who don't know: it's the catalog of topological spaces where one selects the traits that the space should or should not have, and gets a list of such spaces.) E.g., one could check boxes for "limit with $\ln$ and $\cos$, without other trig functions, at $ 0$." (Conceivably, one can deal with integrals, series and products in similar fashion.)

Looks feasible, given a little development time. My approach was to grab the content of tag from Data Explorer and then search through it with regular expressions. For example, "limit involving sine function, at zero":

/\$[^\$=]*\\lim[^}]*0[^\$]*\\sin[^\$]*\$/   

After the likely limits were extracted, I did basic markup checks like comparing the numbers of { and }; broken fragments were rejected.

The encouraging fact is that the catalog does not have to be very good to be an improvement on the current situation.

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    $\begingroup$ One day, perhaps relatively soon, there will come a search engine that doesn't escape and ignore TeX. And one day, OCR will perhaps improve to TeXify math. And one day, a fully functional math search engine will appear. This will be a great day for mathies everywhere (except of course teachers). $\endgroup$ – davidlowryduda Sep 1 '14 at 6:55
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    $\begingroup$ It makes me think we should start deleting bad questions, has this been discussed? $\endgroup$ – user142198 Sep 1 '14 at 7:12
  • $\begingroup$ This is (to some extent) similar: meta.math.stackexchange.com/questions/3967/… $\endgroup$ – Martin Sleziak Sep 1 '14 at 7:12
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    $\begingroup$ I hold little hope for "If there were sufficiently visible, perhaps the questions would not be repeated as often." But such an organizer post would be useful for users with power to vote as duplicate. $\endgroup$ – Willie Wong Sep 1 '14 at 7:44
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    $\begingroup$ @user142198 A little. Not enough. How to repair the close-delete pipeline? $\endgroup$ – user147263 Sep 2 '14 at 5:54
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I feel like the listings you're creating are exactly the opposite of what we want. Your list of limits involving $\sin(x)$ at $x=0$ includes $\sin(x) / x$ literally 12 times, and many more instances of the abstract idea $\sin(x) \approx x$.

The entire merit of a catalog, methinks, is to get rid of the duplication, so that we have a "canonical" posting for each separate idea.

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    $\begingroup$ Well, making duplication visible is the first step toward getting rid of it. I dropped a couple of duplicate votes after seeing the lists (which were automatically generated). Haven't looked at many of them, though. Some instances of $\frac{\sin x}{x}$ were not duplicates: they asked for specific approaches: geometry, particular squeeze, relation to other limits... $\endgroup$ – user147263 Sep 2 '14 at 4:44
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    $\begingroup$ @Saturday: I am trying to keep ahead of your possible names. You do not understand that individual askers are precisely that, individuals. $\endgroup$ – André Nicolas Sep 4 '14 at 5:35
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    $\begingroup$ @AndréNicolas If the 7 billion individuals asked tomorrow "How to compute $\lim \sin x / x$", would the website be a better or a worse place? $\endgroup$ – Najib Idrissi Feb 22 '15 at 8:07
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This will soon be added to the catalog at http://mathindex.wordpress.com/; I'm posting here as a preview and to raise awareness of this addition. Unlike the earlier sections, this one is hand-rolled.

Recurrence relations (related to algorithm analysis)

$$T(n) = aT(n/b) + \dots$$

  1. $T(n) = T(n/2) + n^2$
  2. $T(n) = T(n/2) + 2^n$
  3. $T(n) = T(n/2) + nlog(n)$
  4. $T(n) = T(n/2) + dn\log_2(n)$
  5. $T(n) = T(n/2) + \sqrt{n}$
  6. $T(n) = T (n/5)+ \frac {n}{\log (n)}$

  7. $T(n) = 2 T(n/2) + 2$

  8. $T(n)=2T(n/2) +\log_2(n)$, another one
  9. $T(n)= 2T(n/2)+n \lg (n)$, another one, and more general
  10. $T(n)=2T(n/2)+n/\log n$
  11. $T(n)=2T(n/2) + n\log(\log n)$
  12. $T(n) = 2T(n/2) + \frac{n}{\log(\log n)}$
  13. $T(n) = 2 T(n/2) + 2^n$
  14. $T(n)=2T(n/2)+n-1$
  15. $T(n) = 2T(n/2) + n$
  16. $T(n) = 2T(n/2) + n^3$
  17. $T(n)=2T([n/2]+17)+n$, another one

  18. $T(n) = 4T(n/2)+n^2$

  19. $T(n) = 8T(n/2) + 0.25n^2$
  20. $T(n) = 4T(n/2) + n^2 + n$
  21. $T(n) = 4T(n/2) + n^2 \log n$
  22. $T(n) = 4T(n/2) + \frac{n^2}{\log n}$
  23. $T(n) = 4T({n/2}) + \Theta(n\log{n})$
  24. $T(n)= 7 T (n/2) + 2 \log (n)$
  25. $T(n) = \frac{n}{2}T(\frac{n}{2}) + \log n$
  26. $T(n)=2^nT(n/2)+n$

  27. $T(n) = 4 T(2n/3) + (n^3 )\cdot \log(n)$

  28. $T(n)=3T(n/3)+\log n$
  29. $T(n) = 3T(n/3) + \sqrt n$
  30. $T(n) = 4T(n/3)+n$
  31. $T(n)=2T(n/3)+\log_3(n)$
  32. $T(n)=2T(n/4)+\sqrt{n}$
  33. $T(n) = 3T(n/5) + \sqrt n$
  34. $T(n)=4T(n/5)+n$
  35. $T(n) = 3T(n/4) + n\log n$

$$T(n) = aT(n-b) + \dots$$

  1. $T(n) = T(n-1) + n$
  2. $T(n) = T(n-1) + cn$
  3. $T(n) = T(n-1) + 1/n$
  4. $T(n) = T(n-1) + \log n$
  5. $T(n) = T(n-1) + \sqrt{n}$
  6. $T(n) = T(n-1) + 3^{n-1}$
  7. $T(n) = T(n-1) + 3n-5$
  8. $T(n) = T(n-1) + n(n-1)$
  9. $T(n) = T(n-1) + \Theta(n)$, another one
  10. $T(n) = T(n-1) + O(\log n)$
  11. $T(n) = T(n-1) + \log^2n$

  12. $T(n) = 2T(n-1) + n$, another one

  13. $T(n) = 2T(n-1) + \Theta(n)$
  14. $T(n) = 3T(n-1) + 1$
  15. $T(n) = 3T(n-1) + 2n+2$
  16. $T(n) = 2T(n-1) + 4^n+1$

  17. $T(n) = T(n-2)+\frac{1}{\log(n)}$

  18. $T(n) = T(n-2)+\log(n)$
  19. $T(n) = T(n-2) + n^2$
  20. $T(n) = T(n-2) +3$
  21. $T(n)= 3T(n-2)+9$
  22. $T(n) = T(n-3) + 1/2$
  23. $T(n) = nT(n-1) + n$
  24. $T(n) = \frac{n+1}{n}T(n-1) + c(2n-1)/n$

$$T(n) = aT(\text{nonlinear}) + \dots$$

  1. $T(n) = T(\sqrt{n}) + 1$
  2. $T(n) = 4T(\sqrt{n}) + n$
  3. $T(n) = T(\log n) + O(\log n)$
  4. $T(n)=T(\sqrt n) + \Theta(\log(\log(n))$
  5. $T(n)=2T(\sqrt{n})+\log n$
  6. $T(n)=2T(\sqrt{n})+\lg\lg n$
  7. $T(n) = 2T(\sqrt{n} ) + c$
  8. $T(n) = T(n-\sqrt n) + 1$
  9. $T(n)=3T(\sqrt{n}) +1$
  10. $T(n) = \sqrt{n} T(\sqrt n) + n$
  11. $T(n) = T(n^{2/3}) + 1$
  12. $T(n) = 12T(n^{1/3}) + \log(n)^{2}$

$$T(n) = \text{multiple terms with T}+\dots$$

  1. $T(n) = T(\lceil n/2\rceil) + T(\lfloor n/2\rfloor) + 2$
  2. $T(n) = T(\lceil n/2 \rceil) + T(\lfloor n/2 \rfloor) + \Theta(n)$
  3. $T(n) = T(n-1) + T(\lceil(n/2)\rceil)$
  4. $T(n) = T(n/3) + T(2n/3)$
  5. $T(n)=T(n/4))+T(3n/4)+1$
  6. $T(n)=T(n/4)+T(3n/4)+n$
  7. $T(n)=T(n/4)+T(3n/4)+O(n)$
  8. $T(n) = T(n-1) + T(n/2)+n$
  9. $T(n) = T(n/5) + T(4n/5) + O(1)$
  10. $T(n) = T(n-1) + T(n/2)+1$
  11. $T(n)=T(n−1)+T(n−2)+3n+1$
  12. $T(n) = T(\sqrt{n}) + T(n-\sqrt{n}) + n$
  13. $T(n)=2T(n/2)+T(n/3)+\Theta(n^2)$

  14. $T(n) = T(n/2) + T(2n/3) + T(3n/4) + n$

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