I think Scott and I might disagree a fair amount on when you should leave a comment, so I'll try to flesh out my position a bit more. I think I almost always leave a comment when I downvote or vote to close (even if I weren't casting the final vote, though I guess never happens), or I vote up a comment that says what I'd like to say. Here are the main reasons:

1. Negative feedback (in the form of downvotes or votes to close) hurts a little bit, and a comment can mitigate that.
2. Constructive feedback is really useful. It helps people make better posts in the future, it can help them improve the current post, and it feels good to get it. If the question is salvageable, I try to say how you would salvage it.
3. Writing a comment explaining my downvote/vote to close forces me to actually have a good reason (that I can at least partially explain) for downvoting/voting to close. The better I can put my reason into words, the more confident I am that I'm not being an elitist snob, the more likely the person is to improve in some way, and the easier it is for other people to explain to me what they disagree with if they disagree.

Please DO NOT follow the rule "always leave a comment when downvoting/voting to close", but rather follow the reasons. Leaving a nasty comment, or even a more neutral comment like "Doesn't belong on MO" does not accomplish any of the goals of leaving a comment and only exacerbates the negative side effects of not leaving a comment (i.e. it's bad enough when somebody downvotes you, it's worse when they pretend to give you a reason for it).

Writing constructive comments is hard, and I admit that I sometimes don't do it. The FAQ and the How to Ask have lots of specific anchors you can link to, which should make the job a little easier, but please don't make a comment consist entirely of a link. A good constructive comment should go something like this (let's assume you're voting to close because the question is too vague)

I'm voting to close. As stated, it's not clear what you're trying to accomplish and there are too many ways to interpret your question. [examples of different valid interpretations]. I shouldn't have to guess what your question is before answering it. See http://mathoverflow.net/howtoask#specific. If you edit the question to make it clearer what you want, I'll vote to reopen.

Of course, the more specific you can be about what's wrong with the question and how it can be fixed, the better. It's hard to write a good comment for a bad question that doesn't exist.

I was not such a good engineer; had it been so I would not have dropped it after years of effort and taken up math(given that I am not a great mathematician either). In any case I did not provide an answer in the original thread, for the above reasons.

Well, the only thing I used here was that the Fourier transform of a signal is its frequency spectrum. When you have the frequency spectrum in front of you, you can do what ever you want. Beat matching, frequency filtration, etc., and whatever involving frequency are very easy. And when you are done, take the inverse Fourier transform. The only difficulties of implementation are the delays in computing Fourier transform and its inverse. This is most obvious.

Actually here I did not contribute any mathematics at all. This was engineering. All I did was to speak from my memory that FFT can be done efficiently(for this stated problem) on existing microprocessors. It is the "engineering intuition". If you ask me justification how I reached the conclusion, I have no answer to give except that it is the result of fiddling around with this stuff for a while and the resulting experience. When I was an engineering student, I was not so much bothered by mathematical questions, just like any usual engineer. I did not compute using the MIPS ability of the processor and complexity of the problem or whatever; I just knew that whatever I wanted could be done on such-and-such processors, and how approximately speedy they were for the task.

Here's why. I feel that the question first belonged to stackoverflow. Somebody there would have been able to tell the OP that there is no problem in implementing FFT(=fast fourier transform, algorithms for effectively computing the discrete fourier transform) with the existing processors. I suspect that the OP's problem was that he didn't want to get into the "complicated" FFT algorithm and wanted something simpler instead. However that is not an issue since there were tailor-made FFT routines available on the web.

Shannon's sampling theorem is quite intuitive and nobody who are actually beginner programmers in DSP(=digital signal processing, to fix matters) really bothers about getting more into it. Also it is quite evident that you cannot achieve anything more than what Shannon proved. To capture a signal with frequency f, you need a sampling rate of at least 2f. It simply does not make sense to reconstruct an analog signal from digital samples with less information than this. So the sampling rate had to be above double the maximum frequency the OP was dealing with. If the OP needs higher fidelity, he should use higher sampling rates. For instance, the mp3 files you used to play all throughout the 2000's have a sampling rate of 128k as opposed to the 44 or so k he mentions. The professional systems have even higher sampling rates. The motto seems to be, "sample really well, so that we do not lose anything of value", and do the rest in DSP. Computing FFT will not take so much time. For a processor with a clock speed of 100Mhz, all these are just routine work done in a fraction of a second. I programmed on the age-old "ADSP-2181" processor in 2003, and let me assure that even this rather primitive one would have been enough to handle the task. Its speed was something like 40MIPS(Million instructions per second).

Now, if the OP was willing to put time into understanding FFT and the available microprocessors, he could have done it by himself. However he would not have got that information from MO.

Shannon's sampling theorem etc., would be more interesting to the guy who designs the person who reconstructs the analog signal from the analog output, for feeding to the speaker. However again here it is impossible to go beyond the mathematical bound of Shannon and Nyquist, and what can be done to reach it is more or less already done. There are ready-made "Digital to Analog converters" -- DAC ICs-- available from ages ago. Maybe for some cutting-edge military or so application one needs to nitpick further. But it didn't seem like that from the OP's post. If I were the OP, I would just use some ready-made stuff and get the application done. This appears like an elementary DSP question. Nobody who spent a little bit of time on DSP would have asked it.

For programming questions, I make the following proposition. First the questioner must find out from stackoverflow. If there is really a mathematical issue that they cannot address, then possibly it can be asked here.

This could be made into official policy. This can encompass issues like the "long lines of code" problem mentioned in another thread.

The strong reaction against the closing of this question was quite unexpected for me. There was no issue at all in closing this one. However I must complain of a few other closed questions. .

As has been said before by various people several times, there is an argument for having a fairly focused site doing a few things well. Outreach also involves either diagnosis or tailoring to the audience (IMHO). My personal preference is to err on the side of being Stuffy and Uncool and Like Not Chilled Out (but courteous and receptive when the question has been posed helpfully, regardless of its topic).

2) I find this less than convincing. We can make this decision for individual questions, but I am not keen for the site to be swamped with people wondering if we can help get their cat down from the tree, although I don't mind so much being called if there's something strange in the neighbo(u)rhood... Lots of things are popular without being good; lots of things are desired without being healthy.

To recapitulate: your first sentence seems to conflate our responsibility as an academic community with our (purported) responsibility as this online community. I think it's consistent for me to think that in the former role we could and should do more, without thinking MO is the place to do it. Still, I admit that maybe it comes down to my Eeyorish/Benjaminesque turn of personality: this youtube clip (language perhaps NSFW) may perhaps convey something of my underlying prejudices http://www.youtube.com/watch?v=nLb7tOl-pHc

1) Mathematicians have a bad enough perception in the public mind as it is. Anything we can do to convince them that mathematicians are interesting people whose skills have relevance to the real world can only help us out. (We are interesting people whose skills have relevance to the real world, right?)

2) MO is somewhat high up on the public lists of StackExchange sites, so we are going to get a lot of non-mathematician traffic whether we want to or not. The question is whether to view it as a burden or an opportunity.

You once urged me not to bother replying to calculus questions, etc.. Your point was valid. However, I was once an electrical engineer, and I had a desire to do math, and in the days when I explored the idea or just started making the shift, I must have appeared as a complete dumbass to the professional math students, or profs for that matter. I remember those days, and that was why I was feeling kinder to those amateur chaps.

However in this thread the question had nothing much to gain from MO, as FFT algorithms exist already, and are very effective too on the DSP processors of these days. My feeling was that there was nothing inappropriate in closing the question.

It seems it is difficult to convey the problem without actually having taken a course in it. Here's the fundamental dictum of engineering mathematics: "Every function is just its Taylor series around the point you like, and in this Taylor series everything starting from the x^2 term should be ignored".

It is very hard to learn things when presented without any clue of what the hell is going on.

Anyway, I saw your complaints about understanding Fourier transforms, and so I mentioned all that. Here's some 5 cents more:

For example, the Dirac Delta function approximately occurs when you have a sudden high and brief surge of current in your electrical system. Like what happens when you have a lightning. To get the ideal delta function, you start with a smooth function supported on [-a.-a] which when integrated gives 1. And you let 'a' go to 0, and correspondingly scale the value of the function so that the integral is the same. The Fourier transform of a signal is its "frequency spectrum". You watch in that "frequency domain" what is happening, when you do the above approximation process in the "time domain". This is actually something you can do with constructing some circuit and observing on the "CRO" -- the Cathode Ray Oscilloscope, which is indispensable in every electronics lab, however small. However one does not actually need to set it up so and see it; with some experience one knows what will turn up without actually doing it. For example, when a spark or lightning happens, there is a disruption of noise in your radio or tv. . And it happens no matter the frequency you have tuned it to. This is because the delta function has Fourier transform 1, ie it is a constant in the frequency domain, and therefore the noise created by a signal which looks roughly like it will appear uniformly in every frequency.

Similarly, note that the sudden switching on/off of current is like the step function. That also has a very wide Fourier transform, ie, frequency spectrum, and your radio/television experiences a brief noise when this happens. Same is the case with loose contacts touching and going off. When the contact breaks or makes, there is noise even at high frequencies.

Also note how easy it is to prove the Parseval: The energy(rather, power) is the same, whether you look at the time domain or the frequency domain. It's just a different way of computing energy, which is the same wherever you look from, as per a physical law. How very natural!

The Fourier theory is very nice to study from an electrical or communications perspective. I still like it even after seeing the theory of distributions, which is the one making it all rigorous.

You get a certain intuition in Fourier transforms when you use it for electrical or communications systems. Even now I don't know distributions very well, but I can do some Fourier transform calculations in the non-rigorous, but perfectly natural way.. And I find that trying to bring in the theory of distributions destroy all intuition and messes it up.

It is like, it is much easier to decide whether a "usual" function of a real variable is continuous or not by looking at its graph, rather than trying to check with all epsilons and deltas. You do not mention them at the start of your calculus class, do you?

A few of own "pure math" questions have been asked with only a very vague (or half-baked, or only partially coherent) idea in mind. I post such questions in the hope that the vague idea has some sense (or if it's nonsensical, that someone can tell me why it's nonsensical), and that someone can point me to a reference (if one exists) where the vague idea is made more concrete.

Examples:

• http://mathoverflow.net/questions/652/homological-algebra-and-calculus-as-in-newton

• http://mathoverflow.net/questions/9945/analysis-analogue-of-orlovs-theorem

• http://mathoverflow.net/questions/11716/mirror-symmetry-mod-p-physics-mod-p

• http://mathoverflow.net/questions/8772/cohomology-rings-and-2d-tqfts

I would say that all of these questions are vaguer than or as vague as, for instance, the question we had a while back about walking vs. running in the rain. Douglas is right: while vague "pure math" questions seem to be relatively well-received, the applied math questions are often very quickly shut down.

To answer Harry's implicit question, Allen Hatcher got his PhD 39 years ago, and James S. Milne is 67.

Another person that it would be interesting to see here, but who we'll never see, is Don Knuth, who says he stopped using e-mail in 1990.

First of all, Pete, I can almost guarantee we won't see Persi on MO - he doesn't even use email. (I'll spare everyone a little rant here about the implication that probability is applied math.) But I think parts (not all!) of this discussion are making too much of the pure/applied distinction. I've seen a number of excellent questions in fields that aren't well represented on MO disappear quickly because no one answers or even comments on them. Like this one, for example:

http://mathoverflow.net/questions/12420/asymptotic-non-distortion-of-the-separable-hilbert-space

I don't mean to criticize anyone for not trying to answer this question, but I don't think anyone would say it's not up to MO's standards. This just reflects the fact that many areas of mathematics are not well represented here. The fact that there are few applied mathematicians doesn't necessarily have any deeper significance than the fact that there are few functional analysts.

Personally, I think it's (always been) fine to downvote without commenting, and similarly I think it's fine to vote to close without commenting. On the other hand, if you can leave a helpful comment about why you're voting to close, that's better than saying nothing, and at least one of the votes to close should come with an explanation -- which is why I emphasised the role of the final voter.

Moreover, I would add that leaving questions open just because someday someone might come along and answer them is just plain wrong. One of MOs strengths is its speed: it's for finding quick answers to quick questions. Even more for mathematicians than for programmers, it's a "save me a little time here" site. I could trawl through oddles of literature looking for information on whether or not a particular LCTVS is paracompact or not, or I could try to get a head start and ask here first. If I don't get an answer but was truly interested in the question then I'd go off and do the hard slog. Or at least, I could do so. So if I scan back through the unanswered questions, how do I know that the questioner is still interested in the question?

I would love to see lots of applied mathematicians on MO. I'd love to see more algebraic and differential topologists as well. Not to mention functional analysts. But the MO community is not going to grow by forcing it, but by being convincing. And part of that is making sure that it doesn't look like a "Grill a Mathematician" site.

It'd be great if Persi joined MO. I'd love to fire more questions at him and I'm sure he'd love to ask questions of the rest of us. I have a vague memory that he's not all that bothered about computers, though, so that might be a pipe dream.

The basic problem with MO is that it is, at heart, based on a precarious balance. There are "questioners" and "answerers". Left at that, the incentives are all wrong and, what with everything else everyone has to do, the system doesn't work too well. The genius of SO is to realise that this can work if the two groups are the same. With software, there's a large enough community that this is a stable solution. I'm not convinced that we can have the same stability in mathematics so we need to impose it artificially.

(PS My example above is a bad one since I'm very interested in paracompactness of LCTVS but haven't gotten round to doing the hard slog yet; however, if I hadn't said it here no-one else would know that.)

If only the last voter leaves a comment, including a micro-explanation as to why the question is about to be closed, and ideally (but I do not believe there is any duty 'to go out of [our] way to sift questions on MO for the gold that might lie therein', as Yemon puts it) a hint as to how to improve it, then the questioner has to go through the extra bureaucracy of requesting the question be reopened while she might simply have fixed it as soon as the problem was explained.

If you vote to close, I don't have a strong opinion about whether or not you comment on why you're voting to close. However, if you're the vote that actually closes the question, and no one has left a comment explaining why the question is being closed, then you should leave such a comment.

In an ideal world, I think only the last vote to close would leave a comment, but it's a pretty minor matter.

Other reasons mentioned in this thread (like that the mathoverflow population might not be able to help, or that the question is 'applied', say) seem to me quite irrelevant to the issue. Most of the questions that I think are not appropriate for the site, as far as I understand it, would be able to be answered by essentially all people here, and apart from the isolated youngling I doubt anyone seriously thinks applied math is out of place here (there is no need to go all the way to von Neumann or Wiener to show that kind of (sane!) attitude I'd like to think most of us have on this...)

@Pete, Keep in mind that we don't have a real policy on when exactly to leave comments. We're figuring this stuff out as we go along. When the site started, we were concerned about people writing too many HW questions; we weren't concerned about individual users being overzealous about telling people their questions were being inappropriate. The point I was make, at least, is that the distinction should be between constructive and unconstructive comments. Comments that help the user understand what is going on or help them improve their questions are constructive. Comments that say "I, random MO user, don't think your question is appropriate" and nothing else, are not constructive. SE has a mechanism for expressing that sentiment, and it is called "downvoting."

@Andrew, you said, "Of course, any one question will sink without trace, but if people see questions like What does T mean in this vector notation? getting answers."

I absolutely agree that those types of simplistic/homework questions should be closed. My issue is with the more complex ones, like the signal processing question.

About feeling stupid because of academia: I agree. The material is hard, and always feeling stupid is just part of the game. But, like you pointed out, the key is being courteous. If one downvotes or closes a question, he or she should give some constructive feedback; otherwise, it can seem too much like sniping from the sidelines.

Echoing other people, thanks to @Scott for his suggestions, and I too will try to be more constructive and courteous.

@David: The basic problem with this is that they just keep on coming. Of course, any one question will sink without trace, but if people see questions like What does T mean in this vector notation? getting answers, then they'll keep asking them and that will mean that MO becomes effectively useless (for me). There's more of them than of us and by being too welcoming, we risk being overwhelmed.

@Tom: The same goes for your point on that. On your other points, I wouldn't have classified this question as "applied maths". I wouldn't vote to close an "applied maths" question providing I thought that the maths involved would interest an applied mathematician. I'd just ignore such a question. Your scenario about an applied mathematician coming along in 6 months time doesn't work: the person who asked that question wanted an answer now and if they don't get a solution for 6 months then I suspect that they won't be interested in an answer any more. MO is a short-term system.

As for the thick skin comment, I'm sorry but this is academia. To survive here, one has to get used to feeling stupid about a dozen times a day. It's how we progress. I've lost count of the number of times I've felt ignorant, that all my Great Theorems are basically trivial, that even the ones I do think are worth something seem to always get ignored, but then along comes something that I do understand and it all goes away again. It's a piece of advice I remember very clearly from my supervisor: after finding (and then fixing) a hole in my thesis, I remarked "I hope I never feel like that again" whereupon he said something like "Get used to it! You'll feel that again, and again, and again.". MO is a bit different, it's public, and it's over the internet, so we do need to be a little more careful than we would be in daily life, but nonetheless, the voting and closing mechanism is part of the feedback loop that keeps MO working. I wouldn't change that.

But please note that I used the word "courteously" in my original answer. There is, as Scott reminded us, no excuse for leaving a nasty comment - even to someone you know well from outside MO.

If you think a question on mathoverflow is inappropriate: 1) vote to close 2) if you don't have sufficient reputation, flag 3) according to personal preference and politics, downvote.

In particular, there's no 4), leave a nasty comment. Commenting that the question is inappropriate is very often a bad idea. You should always think twice before doing so, and absolutely never do this without first flagging for moderator attention. I think you should only leave such a comment if you think it constitutes constructive communication with the asker of the question. If you just want to get the question closed, either vote to close, or flag, but don't complain in the comment thread. We check the flags often, and I think experience shows that the moderators are slightly less likely to provoke anger than the average leaver of a "this is inappropriate" comment. :-)

And of course, remember Ben's other point: if you're the final closing vote, leave a comment explaining why the question was closed, as nicely as possible!

What I was objecting to was the condescending tone of some commentary, or at least my (mis?)perceptions of it; and the statement of mine which you quote was merely my attempt to explain my POV, and explain why I hadn't offered any constructive comment on the original question. I am not claiming everyone should espouse it; my apologies if it came across that way. FWIW, I did ignore the question that occasioned this comment thread; I'm commenting here on the meta-issue, not so much the question itself.

For what it's worth, I agree with much of what Andrew Stacey said earlier. I have worked on a help-desk centre for non-mathematics students, so I agree that it's important and valuable to interact outside the mathematical academic community. Like Andrew, I personally do not see MO as part of that. On a purely subjective - rather than prescriptive - basis, I'd prefer to see MO remain largely a service by the maths+stats academic community for the maths+stats academic community.

I also admit to getting prematurely cross with questions which merely present confusion or zeal, without context or detail that I can work with. (Again, one reason I didn't downvote the question Adrian originally referred to, was that the detail and context made it clear that the questioner had thought hard about the problem and what he or she wanted to ask.) Part of being a good student is asking questions well, irrespective of the topic.

By the way: let none of this give the impression that I don't respect the efforts and skills of those who can give good answers to such questions. It's not something I can do well without one-to-one contact, which is why I'm not so motivated to do it on MO.

However, I should also clarify that I am of the view that some question are closed too hastily. For instance, I am still remembering my poor closed question here ..

Uninteresting questions naturally fall off the main page as new and more interesting ones take their place.

Right now, the last question on the front page was updated 19h ago. If the MathOverflow population can't help out with a question, then within one day it will be gone and nobody will look at it.

@Pete said, "I don't think that complaning about elitism or bias is the way to get more applied math onto our site. But it's not clear what the answer is: how should we do it?"

First, by not closing questions of an applied math flavor! Consider the following hypothetical situation: suppose that in six months more applied math people join this forum, one searches for the tag signal-analysis, and answers that question. Wouldn't that be a good thing for this community? Instead, she will find that the question is closed and will get the impression that her kind aren't welcome here.

@Andrew said, "In addition, closing a question doesn't mean 'Go away and never darken our doors again.'"

To you, maybe, but not to everybody. People take downvotes and closed questions personally---negative feedback sucks. You can be dismissive and say that people should have thicker skin, or you can try to find a more constructive way to deal with questions you don't find agreeable. If you feel that "the person is going to get a better answer (and quicker) elsewhere," then make a comment to that effect and move on to another question.

@Yemon said, "I don't see it as my duty to go out of my way to sift questions on MO for the gold that might lie therein."

Nobody's twisting your arm to make you look at every question! I count about 50 questions on the main page, and only about 5 that I'm interested in right now. If you don't like a question, ignore it; if enough people feel this way it will be gone soon enough.

@Steve said, "A notional "EEoverflow" would have been a much place for such a question."

Perhaps. But there's not an EEoverflow, and I'm sure that one poster wasn't going to go through all the trouble to build one just to ask that question. MathOverflow is here. It's big, it's popular, and it's robust. I agree that it should be for "research mathematicians," but that's an awfully big tent and we shouldn't be so quick to exclude people from it.

My own view is that MO is primarily an internal tool for professional mathematicians. It is public, but that does not necessarily mean that it is for everyone. Lectures and seminars are often public in that anyone is welcome to come in, but that does not necessarily mean that they are appropriate for everyone to attend. Of course, as it is public we need to think carefully about how to deal with anyone who does find their way in (by mistake) and I hope that we would do so with courtesy.

I think that there probably is a place for an interface between non-mathematicians and mathematicians. MO isn't it, maybe "Art of Problem Solving" is - I don't know, I haven't looked at those sites. Perhaps more of us should hang out on those sites and answer questions there. If there isn't something appropriate, maybe someone should start a sister site to MO - and then there should be the ability to move questions between the two as called for.

So I'm not particularly bothered about providing "mathematical support" for non-mathematicians here, just as I don't expect all the official documents in Norway to be translated into English for my benefit - I do my best to understand and when I find I need help then I go and ask someone who owes me a favour. To continue the analogy, when I go to a class for learning Norwegian, then I expect the instructor to ensure that I understand what's going on - but that requires a lot more work on behalf of the instructor than if he/she were talking to native Norwegians.

When I do encounter borderline questions on MO then I try to be helpful, within the stricture that if it takes more than a few seconds then it's counterproductive - I'll get fed up of MO and leave altogether. So answering a question on the "size" of the skeleton of the category of groups was okay, but trying to figure out what the question actually is in one on the probability of vectors being linearly independent isn't.

In addition, closing a question doesn't mean "Go away and never darken our doors again.". I know that many think it sends that message, but it shouldn't. It says, "This isn't the right place for this question.". That could be because the question isn't well-formulated, but it could just be that the person is going to get a better answer (and quicker) elsewhere. Closing a question also says to others "This isn't the sort of question that's likely to get answered.".

In the current case, I don't understand the audio terminology. But, as I get it, we have two sound files which are believed to be the same noises, distorted by noise and out of synch. How can we line them up with each other? Anyone with ears can roughly tell when the job has been well done; how we quantify this and optimize it algorithmically?

I don't know the answer to either question. But they certainly strike me as the sort of thing that Von Nuemann or Wiener would have been glad to work on. Among living mathematicians, Tao's blog posts certainly suggests that he thinks about applying analysis to functions coming from real word data. Are there no grad. students being trained in this art? Or have none of them found their way to MO?

My wish is that our current users would (1) vote up and encourage questions which involve interesting mathematics, even if they don't know how to solve them (2) when possible, would help nonmathematical posters clarify their terminology and (3) would not close or downvote questions simply because they don't know how to attack them.

Now I bet several of you are saying "But, Ben, you're a huge hypocrite; you've left oodles of comments saying questions weren't appropriate." However, I only left those comments when I had just closed the question. We've had problems on a couple of occasions with users being confused about a question being closed if no comment was left about it, so it's been a convention since the very early days of the site that a moderator closing a question should leave a comment explaining why. It's possible that there's a better way of doing things, but there is good logic behind it.