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1:22 AM
Perhaps it might be interesting (or even useful) to try sometimes searching for question about some specific result. And to compare whether we were able to find it using various methods.
Let us try to find questions about $\sum_{n=1}^\infty \frac{n}{2^n}$.
"I saw this question asking for the sum of $\frac n{2^n}$, which we have seen many times before. Typing $sum 2/2^n into the search box did not find any of them. I know I have answered one."
Wei Zhong left this comment about this issue:
For your fourth point, I have not investigated thoroughly. But this post (math.stackexchange.com/questions/1934367/…) is very recent and I am sure it is not indexed (I have not renewed our index for weeks). Also, by searching \sum_{n=0}^{\infty} \frac{n}{4^n}, you will get many relevant results, although I cannot find this one (math.stackexchange.com/questions/50919) too, but I believe the crawler missed that one, I see no reason that such a simple expression cannot be handled correctly by approach0. — Wei Zhong Sep 25 at 3:53
But I guess Ross Milikan's point was not that the linked question is not among the search result. He wanted to find older questions about the same sum (which probably should be indexed).
Probably the fastest way to find this question is to look in the frequent tab of sequences-and-series tag.
Anybody who spent enough time on this site knows that question about sums of the form $\sum nx^n$ appear quite regularly.
Indeed, on the frequent tab we have How can I evaluate $\sum_{n=0}^\infty (n+1)x^n$ and we can find several copies of the specific question about $x=1/2$ among the linked questions.
(The other thing is that I am not sure whether they should have be closed as duplicates. But that is discussion for another time. In this room we should concentrate on searching.)
One thing which quite often works relatively well is to click on ask the question, then type in title similar to the desired question.
I have tried entering "Sum of the series $\sum_{k=1}^{\infty}\frac{k}{2^k}$", "Sum of the series $\sum_{i=1}^{\infty}\frac{i}{2^i}$" and "Sum of the series $\sum_{n=1}^{\infty}\frac{n}{2^n}$'. (The list of similar questions did not appear until I entered also some text, not only the formula.)
I found the question already linked by Ross Milikan in the list: Sum of the series $\sum_{1}^{\infty}\frac{n}{2^{n}}.$. And this one, which is related, but not very close one: The sum of the series $\sum_{n=0}^\infty\left(\frac{4n+3}{5^n}\right)$.
So this did no help.
Another reasonable thing is to check the list of related questions.
This list shows the question Calculate the sum of the infinite series $\sum_{n=0}^{\infty} \frac{n}{4^n}$, but it might be there simply because it was chosen as a duplicate. Another similar question in the sidebar is Compute infinite sum of a arithmetico-geometric series $\sum_{i=0}^{\infty} \frac{i}{2^i}$ (this one is not linked).
However, I am not sure how this list was influenced by the comments and duplicates. So it is not clear whether this would work at the time when the question was posted.
So let us not try what we will be able to find with Google.
Searching for sum "frac n 2^n" site:math.stackexchange.com returns several reasonable hits.
Let us try also built-in search. It seems reasonable to restrict results to the tag .
I have tried this: math.stackexchange.com/…
If I remember a post about the question and I remember who asked or who answered it, I can use this fact and search among posts by a specific user.
Since Ross Milikan mentioned on meta that he remembers answering question like this, we might try searching among his posts.
First I tried: math.stackexchange.com/… There is too many results - I was too lazy to check over 300 hundred posts.
I have tried to restrict the search a bit: math.stackexchange.com/… I did not notice a question about this series there.
Now let us try Approach0.
Searching for $\sum$, $\frac{k}{2^k}$ returns among top results:
Ross Milikan did not link to the search he used in the post on meta. So we do not know, whether it was the same as above. And also the pages which the search returns might have changed because of this (or for other reasons).
Conclusion is that in this case both Google and Approach0 found several relevant results. (And if we find one question, we usually find more copies among linked and related questions.)
This is probably not very surprising, since this question appears very often. (Trying to find a question which does not have that many copies on math.SE will probably be more challenging - ane maybe even more interesting.)
@RossMillikan I have tried whether I can find the question about the series you mentioned using Google and using Approach0. You can see here to which extent this was successful. — Martin Sleziak 9 secs ago
 
2:08 AM
Since I mentioned that I tried searching among Ross Millinkan's posts, I will add that I found at least this (related) question in this way: How do we get the result of the summation $\sum\limits_{k=1}^n k \cdot 2^k$?
 
2:45 AM
@Workaholic Sorry for not being explicit, we want to get something related to \oint \frac{dz}{1-z^2}, which is presumably a high frequent math expression. If we can get \oint there, and \oint \frac{dz}{1-z^2} returns nothing, I think perhaps there is not a \oint \frac{dz}{1-z^2} in approach0 index. To confirm this, I ask people to report if they find any old post containing \oint \frac{dz}{1-z^2}.
 
2:56 AM
@MartinSleziak Ross searched $sum 2/2^n (I do not know if he forget to put a closed dollar sign), but actually the query $\sum_{n=1}^{\infty}\frac{n}{2^n}$ is better because it has subscript under \sum to specify the variable and its initial value.
 
3:13 AM
@MartinSleziak Right, so I have modified that EDIT, I agree that those users who have search problem should stay here and ask, but to push appraoch0 forward and get developer involved, bugs and feature request should be posted on GitHub in my opinion.
 
3:28 AM
@MartinSleziak I think your query is not $\sum$, $\frac{k}{2^k}$ (I opened it, but did not get the top results you listed), instead, I guess it is $\sum_{k=0} \frac{k}{2^k}$? I do not know, I just want to correct it. Since later users come into this chat room and use that link probably will not get expected search results.
 
3:39 AM
@MartinSleziak Thanks for giving a good search example and comparison between different search methods.
I agree that finding most commonly used expression is not that difficult, the difficulty at least I have seen technically (and is the next thing I want to improve on) is let search engine try different equivalent transformations and decomposed expressions (i.e. to decompose a complex and large math expression into small "key expressions").
Because many times, our index does contain a expression, it is the query not being the same form of expression or the query being too complex that we cannot find an expression.
 
 
4 hours later…
When I tried to search for $\sum_{n=1}^\infty \frac{n}{2^n}$ I should have mentioned also possibility to look into various compiled list of questions.
In this case List of Generalizations of Common Questions and Catalog of standard exercises seem like reasonable candidates.
I do not see such question in the List of Generalizations of Common Questions .
And back to Approach0 again.
 
8:03 AM
@WeiZhong The query I used was indeed the one I linked. Maybe I should be more precise. These questions were not the top results, but they were ranked 2, 5 and 6. (Still on the first page, so I was satisfied.)
So let us try some other alternatives.
When I write what was the search query, I always copy the text from "raw query" field.
Searching for $\sum\frac{k}{2^k}$ works approximately the same, the hits number 2, 5 and 6 are about this series.
So I will try to add the lower and upper bound to the sum. There are two alternatives how to write this.
The results seem to be the same (at least on the first page). And almost all results on the first page are questions about this series. (And the rest are posts which contain the series.)
So this returns by far the best results from the things I have tried above.
The fact that I can write the same thing in many different ways in LaTeX begs the question.
Will the result be the same if I write in raw query different TeX-versions of the same expression? For example, $\sum_{k=1}^\infty \frac k{2^k}$, $\sum_{k=1}^\infty \frac{k}{2^k}$ and $\sum^\infty_{k=1} \frac k{2^k}$.
At least if I test this for these three expressions, the results (on the first page) seem to be the same. I have troed the first and the second above. Here is link to the third one.
 
8:55 AM
Oh and still about the above searches. We can only guess what Ross Millikan searched for (since he did not link to the query). But if I enter $\sum k/2^k$ through the raw query, I find no questions about this series.
And with some stretch of imagination, these two are at least related: How to prove $\sum n/3^n$ converges without ratio test? and Finding the Function of a Power Series: $\sum kx^{k+1}/3^k$.
But I guess that it is more natural to write that sum as $\sum\frac{k}{2^k}$ than as $\sum k/2^k$. So the fact that no relevant results were found is probably simply due to the fact that there are no posts where this series is wirte with slash instead of fraction.
 
 
2 hours later…
11:15 AM
@MartinSleziak Yes, the search result is expected to be the same using your examples. This is because Approach0 parses a TeX into "operator tree" in which changing the order of subscript and super script or enclosed a TeX bracket (i.e. "{}") does not affect the "operator tree" structure. So many times when I say "structurally relevant", I mean the operator tree representation of query is the same or subtree of the operator tree representation of another expression.
 
Followup-question (although based on the above I expect the answer to be yes).
@WeiZhong Can Approach0 handle symmetry of equalities and inequalities? For example are results for $x^2+y^2=z^2$ and $z^2=x^2+y^2$ supposed to be the same? Similarly for $\frac{a+b}2 \ge \sqrt{ab}$ and $\sqrt{ab}\le\frac{a+b}2$?
 
@MartinSleziak Try to use "$\sum{k/2^k}$" instead. Link: approach0.xyz/search/?q=%24%5Csum%7Bk%2F2%5Ek%7D%24&p=1
 
This seems a bit "magic" that $\sum{k/2^k}$ and $\sum k/2^k$ is the same expression when rendered by TeX, but we get different results.
If I try the query for $x^2+y^2=z^2$ and query for $z^2=x^2+y^2$, the results seem to be the same.
 
This is because currently Approach0 is really bad at handling ambiguity of math expression, if you are using $\sum k/2^k$, the parser is getting this (I tested locally): \frac{\sum k}{2^k}
This definitely needs improvement, but I guess (technically) improving the grammar rule take some effort.
 
For the queries for the inequalities (approach0.xyz/search/… and approach0.xyz/search/… ) the results seem to be different. But from the highlighted parts, it seems that both queries match both directions of the inequalities.
 
11:27 AM
@MartinSleziak They are the same because the subpath set of their operator tree are identical.
 
@WeiZhong So if I understand it correctly, for any equation I get the same results if I interchange the left hand side and right hand site. Is the correct?
 
I need some time to type and explain to you later. Now there is a soccer match I want to watch (China vs Syria), AFK soon.
 
ok
Is it World Cup qualifying?
 
Yes
 
So good luck to your team!
 
11:29 AM
Thanks
 
Just to add links, so that other users can test for themselves, if I enter $\sum k/2^k$ and if I enter $\frac{\sum k}{2^k}$ (in the raw query field), the results are exactly the same.
And again, to have them side by side (easier to compare) here is your link with $\sum{k/2^k}$ and the link to search for $\sum{k/2^k}$. They return the same results.
 
 
2 hours later…
1:37 PM
@MartinSleziak On symmetry of "equalities and inequalities" (I call it relation token), yes they suppose to get the same results (as long as the relation token is still defined as commutative operator), because their subpath sets (reference) are the same.
@MartinSleziak As for why the two inequalities results are different, I have checked their result scores. (You can also check it by inspecting HTML hidden element near each <li> element, i.e. each search result) Those results in different ranking are actually the same scores.
I need some time to investigate on what makes their ranking different, but since those results have the same score, my guess is the traversing order of document will make a difference on ranking order.
Because results with the same score can be ranked differently if they are pushed into result set in different order.
 
2:39 PM
I have some trouble finding it but now I found something like docid: 179778 and score: 3.96. (After clicking on the number, choosing inspect and then expand all.)
 
2:58 PM
Exactly, if you go to look at the second ranked result (since starting from that point, they are different), their score are both 3.806.
I am investigating right now, from what I can tell, it is one single document (math.stackexchange.com/questions/1319137) that makes the difference. "\frac{a+b}2 \ge \sqrt{ab}" can find it, but "\sqrt{ab} \le \frac{a+b}2" cannot find it. Other than this document, the hit order and score of other documents during search process occurs exact the same. Very interesting, it reminds me the butterfly effect.
To be specific, the TeX "y' = \frac{t+c}{2} = \sqrt{|y|} = \sqrt{y}" in that thread is searchable by \ge query but not by \le query. I assume this is a bug.
Because search engine is using minheap as the result set data structure (it is computer science topics, anyway, not intend to dive into technical details) to hold results during search process, a little difference in the order of pushing results into minheap can make a big difference in terms of overall ranking.
 
 
2 hours later…
4:50 PM
I think I know where is the problem. It is actually an expected behavior, but because of some coincidence, we go into a very rare case. Hard to explain in a few words.
But in general, it is because approach0 sacrifices some degree of strictness for search efficiency, and in this case, neither of the two query should match $\frac{t+c}{2} = \sqrt{y}$, but because approach0 uses a simple rule (number of subpaths) to efficiently eliminate impossible matches, one query passes through this filter in this rare case, and become a match.
I have a workaround to "fix" this (make that two query have same results and reduce the possibility of this case happening in the future again), but to fundamentally solve this problem, I need to add algorithm complexity for strictness, which will be much less efficient. So perhaps I would adopt the ad-hoc solution for better efficiency-strictness trade-off. I probably will patch this issue tomorrow (I am in Beijing time, going to sleep soon).
 

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