Wikipedia:Reference desk/Archives/Mathematics/2013 August 22

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August 22

Amateurs Publishing on Complicated Topics, Possible?

This question is specific to myself. As it stands, I have no credentials, degree, etc. However, I really love doing math/physics. At any rate, I have various ideas of my own, mostly spread over various notebooks that fill boxes in my basement; some of those ideas aren't half bad, and if I were to go through my notes and clean them up, could be condensed down into something worth reading (I think, at least.) Given that I am entirely self-taught, if I submitted something, would anyone care? I imagine that whatever I sent would be less likely than if someone with a degree (etc.) submitted the same, but setting that aside, would there be a realistic chance (if it was actually a good idea) of it going somewhere? Most of my stuff is related to algebraic topology, logic, and the theory of computation; all in areas of those subjects that "cranks/crackpots" wouldn't be toiling- I'm not sure if this works for, or against, me. At any rate, if the answer is "you have a snowball's chance in hell", that's fine, I already have a career and what not; I just want to know my chances before spending the time going through my boxes. Thanks for any help on this:-)Phoenixia1177 (talk) 07:13, 22 August 2013 (UTC)

Spending the time is the only way to know your chances. Write a paper and submit it to academia.edu. Bo Jacoby (talk) 07:39, 22 August 2013 (UTC).

Its certainly possible - see List of amateur mathematicians. How easy it is to get taken seriously I don't know. -- Q Chris (talk) 07:40, 22 August 2013 (UTC)

Yes, I think so. Some possibilities:

1) Some publications geared to PhDs will actually publish well-written articles regardless of their credentials. The tricky part might be getting them to actually read it, as the credentials help to narrow down the submissions a lot. That is, your submission might get lost with a bunch of crackpot time cube type submissions. If you could get somebody in the field to endorse it, then they might bother to read it. An old professor, perhaps ?

2) There are also publications aimed more at the amateur author.

3) Then there are publications which will publish anyone who pays them. Of course, such publications have a bad reputation, and deservedly so.

4) Internet publication might be the best option. That way, if your article is really good, people will link to it and others will find it that way. Of course, any of the first 3 categories listed might put things online, too, or you can publish it on any old blog. StuRat (talk) 07:44, 22 August 2013 (UTC)

Think of it as an open source project, anyone can contribute but they need to build up a reputation before anything too complex is taken very seriously. Also writing well is difficult so getting a friend to help after showing something simpler will help. Dmcq (talk) 10:07, 22 August 2013 (UTC)

Also, there are collaborations on the internet where everyone can in principle collaborate in, e.g. Terence Tao's polymath collaboration. We should think about doing similar here on Wikipedia. It would greatly boost the image of the Math Desk if papers would appear regularly in leading math journals written by the "Wikipedia Mathematics Reference Desk Collaboration". Count Iblis (talk) 11:56, 22 August 2013 (UTC)

The injunction against original research in the encyclopaedia is a very sound and sensible one. I think we want to keep a good distance from anything like that here. Dmcq (talk) 12:08, 22 August 2013 (UTC)

But that's only relevant for what can be written in Wikipedia articles. Count Iblis (talk) 12:49, 22 August 2013 (UTC)

I absolutely love this idea. Seriously, if it can't happen here on Wikipedia, I'd be more than happy to collaborate with any interested to get this setup somewhere else.Phoenixia1177 (talk) 07:00, 23 August 2013 (UTC)

I suggest you read the article on Ramanujan and be inspired. Widneymanor (talk) 14:46, 22 August 2013 (UTC)

As a former academic, my one piece of advice would be to read the literature on the subject you're interested in before writing the article, and then position your article so that it contributes to the larger conversation. In fact, the best thing you can do to increase your chances of being published in an academic journal is to read several current issues of the journal and ones similar to it. Ideally, you would be able to find a professional mathematician to use as a sounding board, who can give you advice on how to frame your argument and which journals will be receptive to it. Good luck! OldTimeNESter (talk) 15:24, 22 August 2013 (UTC)

Would it be appropriate for the OP to post a short example here and get opinions? Bubba73 ^{You talkin' to me?} 18:09, 22 August 2013 (UTC)

To be honest, all of my stuff is in "rough form" strewn across a few hundred notebooks, so while most of it is theorems followed by full proofs, none of it is coherently organized. However, there are three specific ideas that I was thinking of writing up in better fashion, in outline:

1.) (Homotopical Stuff) With inspiration from the idea of the etale fundamental group and some stuff from Quillen: for categories of the "right kind" and an associated category of "bases" (not necessarily a subcategory of the first), a functor from their product to the category of quasigroups with identity is setup to function as "homotopy groups". The main interest, of mine, was using this to constrain what groups the homotopy groups, in the normal sense, can be based on how they are constructed from the unit cubes (the Torus being a product of two circles, which in turn are quotients of the interval, etc.). To this end, I start with base categories with nice enough properties, generalize the idea of "construction" to category language, then get a category of "constructions" with the right properties to setup the above homotopy group functor, and that happen to be topological spaces. Then, finally, working backwards from the original topological object, its properties select subcategories out of each base category for each integer (corresponding to the various dimensions of homotopy groups); from this we get a 2dim chain complex whose homology tells us how homotopy in the bases "deviates" from that of the normal notion. The difficulty with all of this isn't that it doesn't work, but that it appears a hell of a lot harder to talk about these quasigroups than the original homotopy groups, so it doesn't shed much light on things (the homotopy groups of spheres was a motivator).

2.) (Abstract Complexity) Looking at the structural (topological, algebraic, etc) properties of recursive sets, generalizing this notion. Then, associating to these spaces, algebraic structures that work something like the logics used in Finite Model Theory. The main goal being to begin with the algebraic associations that would generalize a given logic, then determine the nature of the association back to the recursive structure; in this way, we can, then, attempt to ferret out some of the structure of complexity classes by looking at, essentially, algebra. Unfortunately, the general framework works well enough, but doesn't say much except in the case of unnatural, or uninteresting, classes; also, generalizing quantifiers is horrible, definitely something I would need to spend a while cleaning up.)

3.) (Discrete Time Dynamics) For a function f:X -> X (X a "nice" space) associates a collection of (A, g) [A a space, g a self map], that act as an "approximation" of the f's dynamics under iteration. Specifically, a group is associated to each (A, g), the groups elements are "obstructions" to g and f being conjugate. Then, looking at g's and their groups, we can talk about the dynamics of f. In general, this doesn't allow one to completely specify what f does, but it does allow one to rule out certain behaviours, or demonstrate their existence. It's, spiritually, similar to how one shows that there's no quadratic formula for quintics, just a whole lot messier.Phoenixia1177 (talk) 07:00, 23 August 2013 (UTC)

Just realized all of this could be summed up as, "Take X's of interest, generalize to X-like objects, algebraically classify the deviation from the original, talk about the original using the X-like properties constrained by looking at the algebraic objects." Funny, I never noticed that extremely obvious theme...:-)Phoenixia1177 (talk) 07:10, 23 August 2013 (UTC)

It can happen, though I don't think it's very common. The work George Bergman did on Golden ratio base was published when he was a teenager. In fact Mathematics Magazine published a note from his mom in the same issue thanking them for reviewing the article he submitted even though the author did not have an academic background. The internet is changing the whole landscape of scientific publishing though; the idea that to get your ideas known you have to get someone to put them on flat pieces of ground up tree is rapidly going away. --RDBury (talk) 01:20, 23 August 2013 (UTC)

I think the really hard part is less barriers to publication than it is just doing valuable original work without being in an academic setting. First of all, few people really have time, assuming they have to earn a living some other way. Even if you have the time, you're not going to seminars, you're not talking with colleagues, you don't have a good connection to the state of the art (which is often years ahead of publications). So either you have to have found some niche that no one is working on, or you just have to be really really good. But it is possible, for some very talented individuals. --Trovatore (talk) 01:46, 23 August 2013 (UTC)

This, actually, is what I figured would be a major inhibitor. I don't have massive amounts of free time to do things properly; and being out of the loop, even if I have something interesting to say, it's going to be talking in a dated fashion. I may, nonetheless, give it a shot (just to see what happens). Ultimately, I expect that it would be very hard to get anyone to actually read something submitted, or if so, that it would need to have no imperfections (or, rather, irregularities-not flaws in the logic, but in form).Phoenixia1177 (talk) 07:00, 23 August 2013 (UTC)

Absolutely; I hope you do (give it a shot). My intuition is that you have considerable talent. You may find that your results have been anticipated, but then again, maybe you won't. --Trovatore (talk) 23:44, 23 August 2013 (UTC)

Thank you very much:-) I have a couple weeks of vacation days I need to use, I think I may spend them going through old notebooks:-) As for being anticipated, I like to play the optimist: I figure whatever happens, I either get to contribute something novel or find out where much smarter people took it (so all my nagging questions get answers)- or, I'll find out I've made mistakes I've missed, but that's never a bad thing. -- Genuinely, thank you for your encouragement, that means a lot; I was on the fence about what I was going to do, I now absolutely intend to give things a shot.Phoenixia1177 (talk) 21:53, 24 August 2013 (UTC)

Thank you all for the responses:-) Given that it may be possible, but that there are some definite barriers; I think I'm going to through my old notebooks, improve on what I have as time permits, then try submitting it. Or, if upon completion, it doesn't seem worth publishing, start up a blog and post it there. Probably the latter, in that case I'd be likely to, at least, get some useful criticism. Thank you all again:-)Phoenixia1177 (talk) 07:00, 23 August 2013 (UTC)

I'd like to support the advice of OldTimeNESter: Read! Mathematics by itself isn't all that difficult. So chances are certainly that an amateur finds something that is worth publishing. An amateur is unlikely to just solve a problem that has defeated generations of professionals, but 99.9% of all published mathematics is much more mundane than that. Where amateurs are likely to fail is that they don't read enough. I am not familiar with your areas of interest but chances are that there is other relevant work in those areas. Be familiar with that. Any submission is expected to have a decent literature review that details what is already known about the problem and where your contribution fits in. You should know what journals publish papers in your areas (and be able to tell the serious ones from the ones that publish just about anything). Read papers in the serious journals, know the style. Best by far if you can find a professional mathematician working in your area that you can discuss your ideas with. If I thought I had a really bright idea outside of my area of expertise, I'd go to colleagues in the area, use them as a sounding board and if after that it still seems a bright idea I'd convince them to go for a joint paper. Its pretty difficult to publish in an area you are not completely familiar with even if you are a professional (just one from a related field). I know two people in my field you are basically amateurs. But they go to seminars at the local university, they attend workshops and conferences. They talk to people wherever they can. They know the lingo, they get taken seriously. 31.52.20.70 (talk) 23:46, 23 August 2013 (UTC)

Upper bound on prime gaps?

With respect to the result obtained by Yitang Zhang, does that mean that the maximum gap between prime numbers is 70 million? The wording in the article is somewhat unclear , which is why I ask. Thanks! 174.250.193.70 (talk) 18:07, 22 August 2013 (UTC)

No, there are arbitrarily large gaps between primes. Bubba73 ^{You talkin' to me?} 18:09, 22 August 2013 (UTC)

There is no maximum prime gap. Pick n > 3. Then, n!+2, n!+3, ... , n!+n are all composite. Prime gap >= n. - ¡Ouch! (^{hurt me} / _{more pain}) 06:02, 26 August 2013 (UTC)

See Prime gap. Bubba73 ^{You talkin' to me?} 01:09, 24 August 2013 (UTC)

What it means is that the twin prime conjecture is still too hard to prove, so they invented a related but easier problem to work on. Instead of the gap of 2 between twin primes, you ask if you can prove that there are an infinite number of prime gaps with gap size less than some H. Zhang obtained the proof for H = 70 million, the polymath collaboration improved this to H = 4680. Count Iblis (talk) 18:29, 22 August 2013 (UTC)

How to retrieve the cartesian coordinates of a line of $R^{3}$ from its Plücker coordinates?

Dear contributors to Wikipedia,

amongst my searching on the internet about Plücker coordinates, I have found many clear explanations of how to compute the Plücker coordinates of a line $L$ of $R^{3}$ knowing classical (cartesian) properties of $L$ (such as direction, points belonging to $L$) but nothing clear about the way back. This leads to my question:

knowing the Plücker coordinates $(d:m)$ of a line $L$ (using the same notation as in https://wiki.riteme.site/wiki/Line_geometry#Geometric_intuition ), how one is to calculate the homogeneous coordinates of a point $P$ belonging to $L$?

(Any such point $P$ would suffice to reverse the process as the vector $d$ defines the direction taken by $L$ in the usual 3D space, therefore it will then be easy to compute any classical/cartesian property about $L$.)

Thanks in advance, 193.49.162.4 (talk) 19:29, 22 August 2013 (UTC), Julien F.

$\mathbf {x} \cdot \mathbf {m} =0$ and $(\mathbf {x} \times \mathbf {d} )\cdot \mathbf {m} =|\mathbf {m} |^{2}$. Sławomir Biały (talk) 21:29, 22 August 2013 (UTC)

Thank you for your quick answer! May I ask you how have you found these formulae? Kindly, 193.49.162.4 (talk) 22:53, 22 August 2013 (UTC), Julien F.

I personally got them by inspection, but a more systematic way is as follows. Write d = b − a and m = a × b as 2×2 minors of the matrix

${\begin{bmatrix}1&1\\a_{1}&b_{1}\\a_{2}&b_{2}\\a_{3}&b_{3}\end{bmatrix}}$

The condition for a triple x = (x,y,z) to lie on the line is that all 3×3 minors of

${\begin{bmatrix}1&1&1\\x&a_{1}&b_{1}\\y&a_{2}&b_{2}\\z&a_{3}&b_{3}\end{bmatrix}}$

must vanish. This is four equations: the first works out to be $\mathbf {x} \cdot \mathbf {m} =0$. The remaining three have the form $\mathbf {x} \times \mathbf {d} =\mathbf {m}$. But these three equations have only one independent equation since if x is a vector perpendicular to m, then the cross product $\mathbf {x} \times \mathbf {d}$ is already proportional to m, so we're only interested in the component of this equation in the direction of m, and taking a dot product on both sides with m gives $(\mathbf {x} \times \mathbf {d} )\cdot \mathbf {m} =|\mathbf {m} |^{2}$. Sławomir Biały (talk) 23:42, 22 August 2013 (UTC)

Thanks a lot! Do you mind if I insert your result and demonstration in the wikipedia page about the Plücker coordinates with a reference to you and this section, and is it OK to do so with regards to Wikipedia's chart? I am thinking about a section devoted to this result with a snippet containing your demonstration. 193.49.162.4 (talk) 11:30, 23 August 2013 (UTC) Cheers, Julien F.