This is not done by MO or wikipedia. The lean theorem prover might be taking a step this direction.

I would prefer for such a project to reach through from mathematical idea to known engineering applications, or at least to PDE.

]]>– Online, publicly written math books.

Should such things eventually exist and gain popularity, they could interact with the Tricki. For example, usually a textbook will use a well-known and common trick at some stage in a proof, but will rarely have the opportunity to give much insight into the trick itself, its origins, its limits, alternative methods, etc. All too often, the first time you see a trick, it is also much more complicated than the most basic example of its use, which makes it harder to digest. Imagine you learned the Fourier transform before you ever diagonalized a matrix… You might not know why it has certain magical properties when it comes to translations and differentiation — in any event, it’s useful to understand that diagonalizing commuting operators is a truly general trick.

Often the trick appears as an ingenious technical detail in a proof. For example: the proof of Sard’s theorem uses this trick you might call “decomposing the total change into small changes”. For instance, writing

helps you bound the size of the image of critical sets which are “almost connected” (you prove an estimate like ) — a similar technique shows the Lipschitz image of a curve in the plane is not surjective (even has measure 0), a fact which can be used to calculate fundamental groups like that of the 2-sphere. You may recognize this trick from a proof of the Fundamental Theorem of Calculus, but maybe you didn’t see it coming. Or you might find it enlightening to see the same trick in other contexts and in a much more general light. If you can link to the Tricki, then the individual techniques can be explored separately and generally without disturbing the flow of the exposition.

This use of the Tricki would be different from the vision of using the Tricki for your research; instead it would be more like using it for people to better/more easily learn a subject. But I do not think it would get in the way of other purposes.

]]>PS . just kidding 😉

]]>There are features that are missing on both. Vivid style. Motivation. History of mathematical notions and theorems (this subject usually is connected to motivation). Proofs (I suppose that PlanetMath tries to do this).

]]>One thought that occured to me that there is a good reason to expect the number of collaborators to increase in the future, because some of the people who “grew up” mathematically with the tricky will reach the level of mathematical maturity to contribute an article.

Because of this I think that the larger number of article with “undergraduate content”, will be an advantage for the long term growth of the tricky. ]]>

So, I hope that tricki stays alive not for the sake of pure mathematicians only but also for those who are in a bit applied stuff. Cheers!

]]>I (obviously) cannot publish such work in the journals of my discipline, and the techniques are too classical and elementary to constitute modern mathematics research, but it would be nice if I could post them on a well-trafficked public venue and someone else found it useful. I could supply (weak) proofs or justifications for them and have verified them to the best of my abilities, though the proofs would certainly not meet the standards of rigor required by pure mathematicians. If a mathematician could ‘vet’ them, I would be pleased to get a login and contribute.

]]>Your crude ratio of pageviews to unique visitors indicates that a pretty large fraction of visits comes from large depth visits, but this statistic can be measured separately. If you’re using Google Analytics, it allows you to filter and segregate visits by depth. Other analytics tools should offer similar features.

]]>For instance, the psychology wiki here:

http://psychology.wikia.com/wiki/Psychology_Wiki

was written almost completely by one person at least for the first few years — now there are contributions from large numbers of people.

The story is a little different with something like Math Overflow, because it is not primarily an information corpus as a place to ask and answer questions. But even there, it is the dedicated few who monitor the site who made sure that it took off. The need for that “dedicated few” in the case of the Tricki would be substantially more.

]]>Perhaps there might be a group of people who would volunteer to be “editors” for certain pieces of Tricki and when these individuals saw a post to MO, journal article, or web article, with content related to the area they “supervise,” they could contact the author of that item to modify if for posting on Tricki. The fact that there is less work in adapting something nice that has been done already compared with creating something new from scratch may encourage more contributions.

]]>My idea was to make it even more visible.

But the utility of such a feature depends very much, I think, on how much one wants to post something that has not yet the final quality one is used to display in her/his writings.

]]>Well, perhaps that’s not quite true. There’s a label for incomplete articles (roughly corresponding to what Wikipedia would call a “stub”) and there are plenty of paragraphs in articles saying things like “This article needs more examples.” But this doesn’t seem to have encouraged people to supply those examples.

]]>I would suggest to create a label such as “to-be-polished” (or similar) to indicate that an Tricki article not so well written due to lack of time of the contributor is intended to be improved by the same author and/or “polished” by others.

Another possibility would be to create a section for the articles not yet in the final form.

]]>About 770 users have registered on the site since it opened to users about a year and a half ago. Of these, something like 30 users have created articles and something like 70 users have edited articles.

There are about 300 articles on the site in total. Of these, only two were created in 2010.

Over the past six months about 100 users have registered, which averages out at roughly 3.8 registrations per week. These ‘new’ users have created one article and edited eight.

During September the site has had 7651 ‘unique visitors’, though I can’t tell how many of these correspond to genuine users and how many are from spam bots, and about 40000 pages have been served. (I think search engine bots are specifically excluded from these numbers where it has been possible to identify them.) So far September looks to be fairly typical.

Most visitors (by far) arrive at the site from Google.

]]>For all natural numbers m, n satisfying 1<m<n+1 the inequality

sqrt(mxsqrt((m+1)x…xsqrt(n)…)<m+1 holds.

Proof by induction: The statement holds for all natural numbers n, if m=n since sqrt(n)<n+1. Now, suppose sqrt(mxsqrt((m+1)x…xsqrt(n)…)<m+1 for some natural numbers m, n satisfying 2<m<n+1. Then,

sqrt((m-1)xsqrt(mx…xsqrt(n)…)<sqrt((m-1)(m+1))<sqrt(m^2)=m. Now, we have the general statement and get the original one for m=2. ]]>

sqrt(2xsqrt(3x…xsqrt(n)…)<3.

One way to prove this is by proving the following stronger statement:

For all natural numbers m, n satisfying 1<m<n+1 the inequality

sqrt(mxsqrt((m+1)x…xsqrt(n)…)<m+2 holds.

Proof by induction: The statement holds for all natural numbers n, if m=n since sqrt(n)<n+2. Now, suppose sqrt(mxsqrt((m+1)x…xsqrt(n)…)<m+2 for some natural numbers m, n satisfying 2<m<n+1. Then,

sqrt((m-1)xsqrt(mx…xsqrt(n)…)<sqrt((m-1)(m+2))<sqrt((m+1)^2)=m+1. Now, we have the general statement and get the original one for m=2.

I am sorry if somebody elso posted this before or if it is considered a boring example. ]]>

Gil touched on it above, but there should be a critical point where the Tricki can be referenced in MO answers, much like the nLab is. Often there is a bit of a feedback, when someone who is an active nLabber wants to answer an MO question, and to save time puts the answer in the appropriate (or creates an appropriate) nLab page, then links to it in their MO answer with enough padding to explain the first-level understanding of the answer, and a ‘see details at [this page]’. People don’t seem to use MO to ask for techniques, but facts/references/examples, unlike Stackoverflow (at least, as I imagine it from my one or two brief glimpses). A bit of backward and forward, using MO to build up Tricki, then promoting an awareness of the sort of questions that Tricki can help answer (and that Tricki can answer them).

]]>