Wikipedia talk:WikiProject Mathematics/Archive/2008/Jul

Hilbert space under review

I've now reviewed Hilbert space as promised above. I encourage all editors here to improve the article. Delisting it would cause me a great deal of sadness, but there is so much work that needs to be done to avoid such an outcome, that input from many editors is vital. Geometry guy 22:45, 6 July 2008 (UTC)

Hi Geometry guy! I am willing to do my part to bring the article back up to standards. Is there a deadline after which you plan on delisting it? siℓℓy rabbit (talk) 22:46, 6 July 2008 (UTC)

If there's no activity, then two days. If there's a response, then I will wait at least a week, but I'm willing to follow improvements for much longer than that, in order to achieve the best outcome for WP. Geometry guy 22:55, 6 July 2008 (UTC)

This article seems to be very poorly written. I have very quickly added some obvious missing references. I have not yet added the first two volumes of Dunford & Schwartz, or Courant & Hilbert. It would be quite easy for me to supply citations, although many of the statements seem slightly wrong or inaccurate. I'm not sure whether you really want an expert to help you with this. Mathsci (talk) 23:56, 6 July 2008 (UTC)

Thanks for the input already. There's been enough activity to put this "on hold". I wouldn't say the article is "very poorly written", but it needs work both from experts and non-experts. Citation is not the main problem: as you point out, that is fairly easy to fix in an article like this, and should, at the same time, clarify the statements the article makes. Geometry guy 09:09, 7 July 2008 (UTC)

Yes, I agree not "very", but just a little sloppy. I looked for the first time at Sobolev space and didn't find any reference to Rellich or Weyl's results on elliptic regularity (e.g. contained in F. Warner's book, or Michael Taylor's books, or even Hörmander, or ....). These are stones best left unturned. Often on WP when I have needed to use material on spectral theory, for example on spectral measure / multiplicity theory, I have had to write it from scratch. Since I am possibly over-familiar with this material, I prefer to stay away from it at the moment, although I will chip in occasionally if there's no rush. I wouldn't want to be accused of WP:OWN :-) Mathsci (talk) 10:50, 7 July 2008 (UTC)

(unindent) Many of the changes to this article are good: it is improving nicely. I am not so happy about the definitions of Hardy space and Sobolev spaces. Best to take the theory on a circle (or torus) and use Fourier series. Both are defined using elliptic operators (in general either the Laplacian or the Dirac operator or d-bar operator)- this is a missing ingredient, even if this it the simplest of all examples. The Hardy space projection comes from the Hilbert transform which is essentially the phase of the unbounded operator i d/dθ, the Dirac operator on the circle; and the L^2 Sobolev spaces for a compact manifold are the images of (I L)^s/2 where L is the Laplacian [defined as a positive operator]. One of the most important first geometric applications of Hilbert space theory is in Hodge theory (essentially Hermann Weyl's method of orthogonal projection). Sobolev spaces and Hodge theory are well presented from scratch in Frank Warner's book, Foundations of differentiable manifolds and Lie groups, which is one of the standard references for this subject. Likewise Zimmer's book has a nice treatment of the Sobolev theory. There are plenty of other places to look - e.g. almost any book about the Atiyah-Singer index theorem, e.g. the book of John Roe. It's not clear that books on measure theory or L^P spaces are that relevant asreferences for this article. Avner Friedman's introductory book Foundations of Modern Analysis should also not be forgotten. Similarly the classic "Theory of Linear Operators in Hilbert Space" by Akhiezer and Glazman could be mentioned. Just some random thoughts which might be helpful... Mathsci (talk) 08:01, 10 July 2008 (UTC)

Interesting suggestions. I suggest keeping the definitions of the Hilbert spaces involved as naive as possible, but then later expound on these details in a dedicated section on Hodge theory (this is obviously important enough to warrant a section of its own). siℓℓy rabbit (talk) 14:37, 10 July 2008 (UTC)

I completely agree. Cheers, Mathsci (talk)

Bernoulli numbers

In Bernoulli numbers, after the words

The combinatorics of this representation can be seen from:

There follows a sequence of expressions in which the pattern is clear EXCEPT for the pattern in the plus and minus signs. I left a message on the talk page of the user who wrote that passage inquiring about it, but he's been away from us for six days now, and left no email address. Can anyone explain that pattern? Michael Hardy (talk) 02:20, 8 July 2008 (UTC)

I think that's an error, but I'm not an expert so I won't make the change. I think $B_1$ and $B_3$ should have a negative sign in front of the $|\emptyset|$. Loisel (talk) 03:49, 8 July 2008 (UTC)

I have implemented this suggestion. (Of course, it doesn't change the actual value of the expression, since the cardinality of the emptyset is zero.) siℓℓy rabbit (talk) 04:45, 8 July 2008 (UTC)

That's because there were {} missing from several lines. It should have read |{ \emptyset }|, which is 1, not zero. Loisel (talk) 14:38, 9 July 2008 (UTC)

No. Now it's wrong, since the empty word is not a word of positive length. (E.g., there are exactly zero ways to make a word of length two using zero letters.) I am reverting. siℓℓy rabbit (talk) 14:59, 9 July 2008 (UTC)

The equations as shown give the wrong sign for B₁ and the wrong value for B₃. And I don't understand why in the equation for B₂ the string bb is missing, in the equation for B₃ the strings bbb, ccc, and eight other strings that use two out of the three symbols a,b,c are missing. —David Eppstein (talk) 05:31, 8 July 2008 (UTC)

The words on the strings have to use all of the letters, I believe. You were right about the value of B₃: two of the words bab and abb had been left out. As for the sign on B₁, it is positive, which agrees with some conventions for the Bernoulli numbers and not others (according to the article, it's news to me). siℓℓy rabbit (talk) 05:42, 8 July 2008 (UTC)

I'm used to B₁ being -1/2 myself; but the quote from Bernnouilli in the article shows clearly that he would have used 1/2. Septentrionalis PMAnderson 00:42, 9 July 2008 (UTC)

Sum of two squares

Sum of two squares currently redirects to Brahmagupta–Fibonacci identity. That is an identity in algebra that says the product of two sums of two squares is itself a sum of two squares. Since the idea of a sum of two squares seems relevant to number theory, I wonder if maybe making it a disambiguation page might be appropriate? Michael Hardy (talk) 18:00, 10 July 2008 (UTC)

I agree. Until I read your note, I would have assumed that Sum of two squares discussed the theorem that says that an integer is the sum of two squares if and only if every prime of the form 4k 3 appears an in its prime factorization an even number of times. --Dominus (talk) 20:12, 10 July 2008 (UTC)

Wikipedia discusses the latter theorem at Fermat's theorem on sums of two squares. -- Dominus (talk) 20:14, 10 July 2008 (UTC)

OK, I've made it a disambiguation page, currently listing ONLY those two items. Michael Hardy (talk) 20:25, 10 July 2008 (UTC)

Adjectives as article titles

If this is not the right place to ask, please accept my apologies and tell me where to go (ooer). The adjectives "isgonal", "isohedral" and (to a lesser extent) "isotoxal" are commonly used to describe certain polyhedra and tilings. There are also related ideas such as isohedral numbers. Wikipedia would normally expect article titles to be nouns as in "isogonality" and so on, but these are used far less often - indeed there seems only one documented example of "isotoxality" in existence. In this unusual circumstance, would it be acceptable to use the adjectives as the article titles? -- Cheers, Steelpillow (Talk) 19:42, 10 July 2008 (UTC)

Is "Isotaxal polyhedron" ok? For instance, instead of "solubility", we have an article on soluble group. You can make "isotaxal" and "isotaxal tiling" redirects. JackSchmidt (talk) 19:57, 10 July 2008 (UTC)

The problem is, "Isohedral" for example references at least three articles; "Isohedron" (isohedral polyhedron), "Isohedral tiling" and "Isohedral number". Would it be sensible to set "Isohedral" as a redirect to say the "Isohedron" article, and then provide disambiguation links to the others from there? -- Cheers, Steelpillow (Talk) 20:09, 10 July 2008 (UTC)

That is definitely sensible. If there is later a need for a disambiguation page (I think their page names are also supposed to favor nouns), then this will not interfere. In the meantime, it follows the naming conventions and quickly directs readers to the right page. Good idea! JackSchmidt (talk) 20:18, 10 July 2008 (UTC)

Many thanks. I'll see what I can do. -- Cheers, Steelpillow (Talk) 21:27, 10 July 2008 (UTC)

Applications of symbolic dynamics

There is a long section "Hematopoietic Stem Cell Kinetics" at Symbolic dynamics that clearly belongs elsewhere. The question is (to people who understand what it is), where? Arcfrk (talk) 02:38, 11 July 2008 (UTC)

Have you tried moving it to the article on Hematopoietic stem cells (a cell type which is the precursor to all kinds of blood cells)? JRSpriggs (talk) 07:34, 11 July 2008 (UTC)

OK, moved it there, hopefully, the'd know what to do with it (other than deleting). Arcfrk (talk) 10:44, 13 July 2008 (UTC)

Integral of secant cubed

I sometimes think of myself as the foremost advocate of the need to remember that some people who read Wikipedia articles are not mathematicians. Whenever (fairly often) someone starts an article by saying "Let {T_n} be a sequence of bounded linear operators on a separable Banach space", I point out that the lay reader could read that whole sentence and maybe the long paragraph that it initiates without finding out that mathematics is what the article is about. But now some people have raised questions at talk:Integral of secant cubed whose answers I hadn't thought to put into the article, of which I am at this point the primary, but not the only, author. Can people here help explain to the broad public that this particular integral has reasons for being singled out? Michael Hardy (talk) 14:21, 11 July 2008 (UTC)

I think I agree with Kusma on the talk page: it is not inherently clear to the mathematicians what's so special about this integral. Silly rabbit gave some nice comments as to how the integral comes up (and might be avoided!), but surely similar comments could be made for most of the other integrals that one might do in a calculus class: they all compute something and often something reasonably natural. I wonder what you think is so special about the integral? (Not a rhetorical question; I'm interested to know.) I do agree with you that for some reason this integral is trickier than most other integrals one meets in a course on this level: it is one of the few integrals that I try to remember to work out in advance, because the typical calculus class doesn't want to watch you try something reasonable to evaluate an integral, have it fail, scratch your head for several seconds and then do something else, eventually arriving at the solution. (This is a shame, because such a presentation would be more honest and arguably more instructive, but on the other hand I see where they're coming from.) It might be interesting -- and difficult -- to try to explain why the integral is so tricky to compute: note that giving the derivation is not at all helpful in doing this: it doesn't look hard! Plclark (talk) 15:30, 11 July 2008 (UTC)Plclark

Part of it is just that it's one of the ones to which others get reduced. Most integrals in exercises lack the sort of naturalness that user:silly rabbit refers to, and among those to which others get reduced by routine transformations, most are either really easy or become so when some other topic (e.g. differentiation of polynomials or exponential or trigonometric function, integration by parts, etc.) gets treated, but this one doesn't. And unlike all those artificial routine exercises in integral calculus, this one does arise naturally when thinking about some other things. Michael Hardy (talk) 15:53, 11 July 2008 (UTC)

OK, maybe I'd better rephrase that. The reduction technique used to get the recursion formula for integrals of powers of secant is all there in this one integral. It's the simplest one of those. And it arises naturally in several places. Other integrals of which that can be said are usually relatively trivial, but this one has difficulties that those don't have. Michael Hardy (talk) 15:57, 11 July 2008 (UTC)

PPS: I'm not sure I agree that it's more instructive to show the class how you really solve math problems. Doing that fails to give them bite-sized pieces and confuses them. Michael Hardy (talk) 16:04, 11 July 2008 (UTC)

The "arguably", "on the other hand..." and my admission that I do try to disingenuously work out the integral beforehand were all meant to indicate that I am not sure either. Plclark (talk) 16:23, 11 July 2008 (UTC)Plclark

If I'm going to "scratch my head", figuratively speaking, in class, then I'm going to make a show of it. If you appear uncertain about something before a class of naive freshmen, then many of them lose confidence in you and stop paying attention. And then they start complaining. Not at all reasonable, given that you knew a hundred times as much back when you finished the course than they will when they do, but they're naive freshmen and they don't know that. Michael Hardy (talk) 18:54, 11 July 2008 (UTC)

This is so true. Ozob (talk) 15:29, 12 July 2008 (UTC)

BTW, the most difficult antiderivative presented in first-year calculus is

$\int \sec \theta \,d\theta .$

But they don't need to do that because it's handed to them. It's easy to check the answer, and the derivations usually presented are easy to check, but the derivations I've seen involve steps that, although easy for the students to check, would be impossible for them to anticipate, barring an archangel coming down from Heaven and revealing it, or a standard textbook or the like. Is there any reasonable way to prepare students so that one could then make an exercise of that integral? Michael Hardy (talk) 19:01, 11 July 2008 (UTC)

Somehow I got

$\int \sec \theta \,d\theta =\operatorname {arccosh} (\sec \theta ) C$

instead of the usual (and correct)

$\int \sec \theta \,d\theta =\ln |\sec \theta \tan \theta | C$

as seen at List of integrals of trigonometric functions. Funny, eh? JRSpriggs (talk) 04:36, 12 July 2008 (UTC)

If you were in my class, I'd give you an A on the assignment, but you'd have to work harder to get one in the class :-) --C S (talk) 07:17, 12 July 2008 (UTC)

Thanks. I should have said

$\int \sec \theta \,d\theta =\operatorname {arcsinh} (\tan \theta )\cdot \operatorname {sgn}(\sec \theta ) C$

so that it would get the sign correct. JRSpriggs (talk) 16:25, 12 July 2008 (UTC)

graph problem

Can somebody with practical experience in graph search algorithms help me out with the following (this request is related to my work on zeteo, a database of books): I want to measure the "distance" of two arbitrary WP articles. The idea is to create a graph, whose vertices are the WP articles, and edges between articles A1 and A2, whenever A1 contains a wikilink to A2 or vice versa. The question now is, what algorithm is best to solve the graph search algorithm? The data are approx: 2,500,000 vertices (=articles), the graph is very, very sparse (every vertex is connected to some 10-100 other vertices). I think I don't need weighted edges, just constant weights. Also, I don't have to measure distances between all possible couples of articles, only a few. Also, if the distance is known to exceed a certain threshold, the algorithm may stop without telling the exact distance. Thank you so much, any help is appreciated. Jakob.scholbach (talk) 12:56, 13 July 2008 (UTC)

This has already been implemented at [1], which gives a form to give the distance between any 2 wikipedia pages and the code used.R.e.b. (talk) 14:04, 13 July 2008 (UTC)

Example of a non-associative algebra

Is there any way that this article is not WP:OR? I've helped patch it up, but there needs to be a reference that someone else has found this notable. (Besides, the octonians form a more interesting non-associative division algebra, being also an alternative algebra.) — Arthur Rubin (talk) 14:24, 12 July 2008 (UTC)

In general, I don't think "Example of" articles fall into our mission; any example important enough to have its own article would probably have a name of its own. This particular example is also mentioned at Division_algebra#Not_necessarily_associative_division_algebras. I think that the example article should just be redirected to that section. — Carl (CBM · talk) 14:47, 12 July 2008 (UTC)

I think the example is valuable but should be a section, not an article. CRGreathouse (t | c) 16:36, 12 July 2008 (UTC)

Also, in general, the longer and more creative the example, the more helpful it would be to find sources that mention the same example in order to avoid charges of original research. I mean, if an article requires an example of a prime number and you pick 29, there's no point in sourcing that, but this example is more complicated than that and it would be of interest to know where one could go to read more about it. —David Eppstein (talk) 18:16, 12 July 2008 (UTC)

Use textbooks as references. This "Example of" article should be merged into another one, like Non-associative algebra. Gary King (talk) 18:17, 12 July 2008 (UTC)

This arbitrary example is already mentioned in Division algebra by the same editor as CBM has already said. I have a problem with that article, as almost always books on division algebras are about "associative algebras" (Artin, Nesbitt & Thrall, Jacobson, Herstein, Cohen, Schofield, etc, etc). Has this article been distorted by the same editor? (Division ring does not seem to have been distorted in this way.) A more serious imbalance is that the fundamental role of division algebras in the theory of central simple algebras and the Brauer group does not seem to have been mentioned. Nor has their important relationship with universal enveloping algebras, Goldie's theorem and the Ore condition. Just my ten centimes worth. :) Mathsci (talk) 10:19, 13 July 2008 (UTC)

That's the only article that links to it, in fact; presumably it exists to get this demonstration out of division algebra. How about renaming it something like Proof that an non-associative algebra can be a division algebra or can satisfy the other criteria for a division algebra? Trivial analytic statements don't need sources. Septentrionalis PMAnderson 20:45, 15 July 2008 (UTC)

We also have Example of a commutative non-associative magma, which suffers from a similar lack of references. It's only really a problem, though, because it's a separate page and because editors are conditioned to expect at least one reference per page — if it was a section in a longer article, no-one would bat an eye, since it's immediately obvious to anyone who knows what "commutative", "associative" and "magma" mean (or takes the time to check the linked articles) that the example given is indeed a commutative non-associative magma, and, as Septentrionalis notes above, "trivial analytic statements don't need sources." Of course, we could sidestep the issue here by replacing the example with one that does have references, but I'd kind of hate to lose the current example: OR or not, it's much simpler than something involving non-zero octonions or whatever. —Ilmari Karonen (talk) 14:21, 17 July 2008 (UTC)

Spectral layout

Hi. I often tag articles for Wikiprojects, but in this particluar case, I don't know if this article comes under your purview - I draw the line at categorising any article with more than three words I don't understand in the same sentence :-) Perhaps someone from the project can take a look and classify/expand if necessary ? Thanks :-) CultureDrone (talk) 09:46, 13 July 2008 (UTC)

Searcges suggest that the Laplace matrix, not the Laplacian, is meant; does this help? Septentrionalis PMAnderson 21:18, 15 July 2008 (UTC)

Jose Luis Massera

Jose Luis Massera is now a totally stubby new article. Can anyone add more to it? Michael Hardy (talk) 16:00, 16 July 2008 (UTC)

I gave a link to a mathematical obituary. My French is pretty marginal, but it sounds like world renown mathematicians are calling him world renowned. I filled in his birth, release, and death dates, as well as added the basic math bio categories. I didn't add any categories for his 1975 years. JackSchmidt (talk) 18:32, 16 July 2008 (UTC)

I added a bit more. He was apparently a pretty big deal, mathematically. I linked to some of his most heavily cited papers (but not all by any means), and included a red link for his work on periodic differential equations, which again gets over a 100 hits on math reviews. JackSchmidt (talk) 19:12, 17 July 2008 (UTC)

Point-set conditions for manifolds?

Originally a discussion on the RefDesk

There seems to be some uncertainty on Wikipedia as to the precise definition of a (topological) manifold. The definition in manifold includes second-countability and Hausdorffness, but hedges this with a footnote stating this is 'the narrow sense' of the term and includes a broader definition. Other articles adopt different approaches: to quote the examples I've looked at, differentiable manifold requires the point-set conditions, and doesn't mention any other conventions while Riemannian manifold doesn't include a definition of manifolds, but includes the line 'Let M be a second countable Hausdorff differentiable manifold of dimension n', perhaps implying that the point-set conditions are extraneous to the manifold concept. Long line (topology) describes the long line as a manifold (sometimes with the modifier 'non-separable'), but mentions that some authors require second-countability/separability in their manifolds.

Is this situation undesirable, and if so, what should be done about it? Algebraist 22:06, 13 July 2008 (UTC)

I don't think the situation is horrible; the most important thing is that each article separately make clear what it's talking about. Just the same it wouldn't necessarily be a bad thing somehow I left out the predicate of this sentence the first time to agree on a baseline set of terminology; you could bring it up at Wikipedia talk:WikiProject Mathematics/Conventions. But again, the most important point is that a convention is not a substitute for making ourselves clear in each article—it is never acceptable to assume that readers know which convention we've chosen --Trovatore (talk) 22:43, 13 July 2008 (UTC)

I ignored the conventions talk page as de facto abandoned. Algebraist 22:47, 13 July 2008 (UTC)

The problem with that is that whenever we mention manifolds in an article, we link it to manifold, which is going to cause confusion if we mean something different by "manifold" than that which the linked article describes. --Tango (talk) 23:13, 13 July 2008 (UTC)

That's why the linking article always has to make clear what it's talking about, without relying on the link. The text in the manifold article could change, after all, and it's unlikely that all the links to it will be correspondingly updated. --Trovatore (talk) 23:16, 13 July 2008 (UTC)

Oh, I should say, that's one reason why. Another important reason is that you don't want to confuse readers who know a different definition -- you can't rely on them checking to see what the manifold article says, whether our articles are consistent among themselves or not. --Trovatore (talk) 23:17, 13 July 2008 (UTC)

True, we need both consistency between articles (for the benefit of those that do click the link) and clarity within articles (for those that don't). --Tango (talk) 23:35, 13 July 2008 (UTC)

(unindent) Trovatore's advice to be clear in each article about what we mean is spot on. Having said that, we should also remember that references rule. I know of no modern text that doesn't require a manifold to be second countable and Hausdorff. (Otherwise, any argument using a partition of unity—something that is certainly desirable—would be ruined.) In other words, it behooves us to make sure the article Manifold doesn't get changed too much from this standard so that when people link to it, they will see the "correct" definition, which is much more than just a convention, IMHO. But I am open to other interpretations provided that solid references can be provided. VectorPosse (talk) 23:37, 13 July 2008 (UTC)

Allen Hatcher's algebraic topology book does not require second countable although it requires Hausdorff. My impression is that there are large groups of mathematicians that don't require their manifolds to be paracompact. I would be surprised if Hatcher's was the only "modern text" that defined manifold like this. But you know, I'm sure this exact discussion has happened before. Are we doomed to repeat history? --C S (talk) 01:37, 14 July 2008 (UTC)

I stand corrected, then. At some point I had the impression that the second countable thing was not well-established in "older" texts, but had become so. But if Hatcher doesn't mention it, I'm sure others don't either. (I should have known that about Hatcher. I've spent a lot of time in that particular book.) Just for the record, I agree with everyone that this isn't a really big deal, so I didn't mean to kick up too much dust.  :) VectorPosse (talk) 08:34, 14 July 2008 (UTC)

I also agree with Trovatore. I don't think this is a big deal. Much of manifold theory is independent of second countability and Hausdorffness: it is pretty much only when partitions of unity are needed that these conditions are used in anger. They are nowhere near as important as "locally isomorphic to Rⁿ" (or whatever) when it comes to the meaning of what a manifold is. Geometry guy 23:45, 13 July 2008 (UTC)

Locally homeomorphic. --Tango (talk) 23:50, 13 July 2008 (UTC)

Homeomorphisms are isomorphisms in TOP, assuming that's what G Guy meant. He could just be doing that snooty thing where he is speaking about every type of manifold modeled on some property of R^n. --C S (talk) 01:37, 14 July 2008 (UTC)

Hausdorffness may be pretty universal, but I don't remember second-countability being required when I first learnt about manifolds. Unfortunately, I'm away from Uni for the summer, so don't have any textbooks to hand. --Tango (talk) 23:50, 13 July 2008 (UTC)

I have seen "non-Hausdorff manifolds" named "premanifolds". This might be something French—the same distinction used to be made between preschemes and schemes (the latter had a Hausdorff condition, the former not).

My impression is that when people who aren't geometric topologists talk about manifolds, they always want second countability and even smoothness. Geometry Guy says above that "much of manifold theory is independent of second countability and Hausdorffness"; but that depends on your perspective of what "manifold theory" is. Certainly a differential geometer will want second countability so that he can (angrily?) use his partitions of unity. But since we're writing an encyclopedia, I think we ought to adopt the point of view that "manifold" means only "the really essential conditions for being a manifold and nothing more." So for us, "manifold" would mean "not necessarily Hausdorff or paracompact, but locally homeomorphic to Rⁿ" (or locally isomorphic in your favorite category). And while that may not be a common definition, it is the only one that will work simultaneously in, say, articles that need to discuss π_k(PL/TOP) and articles that need to discuss Dolbeault cohomology. Ozob (talk) 22:04, 14 July 2008 (UTC)

Wait, how does smoothness depend on second countability? Can't you put a smooth atlas on the long line? I've never really thought about it, maybe there's an obvious reason you can't, but intuitively I'd ahve guessed you could. --Trovatore (talk) 22:09, 14 July 2008 (UTC)

According to our article, you can even give it an analytic structure. Algebraist 22:40, 14 July 2008 (UTC)

This is a miscommunication on my part: I don't mean to imply that smoothness requires second countability, I mean to suggest that to most people, smoothness feels more restrictive than second countability. (Which is an opinion about other people's value judgements, so take it with a grain of salt.) Ozob (talk) 21:28, 15 July 2008 (UTC)

As a minor aside and counterexample to Ozob's claim, when computer scientists talk about manifolds, they are much more likely to assume piecewise linearity than smoothness. And finiteness of the piecewise linear structure, which implies second countability, but if you mention second countability to them you're likely to get a blank stare. —David Eppstein (talk) 23:01, 14 July 2008 (UTC)

My experience, being in geometric topology, is totally different. Geometric topologists are the most likely to require paracompact manifolds. Algebraic topologists (see my Hatcher reference above) often do not care. --C S (talk) 00:37, 15 July 2008 (UTC)

Then it seems that I should have said "algebraic topologists". My impression is still that most non-topologists are likely to require second countability. Ozob (talk) 21:28, 15 July 2008 (UTC)

Can't say if that's true or not, but if true it's a bit strange, as it's the only non-local part of the definition. To me being a manifold seems like it should be a local property. (I guess technically being Hausdorff is not quite local, because every point of the bug-eyed line has a Hausdorff neighborhood, but somehow it seems "local enough", whereas second countable is not local at all.) --Trovatore (talk) 03:53, 16 July 2008 (UTC)

According to Local property, there are two definitions of local (well, actually it's talking about a property being satisfied locally, rather than a property being local, but I think you can turn them into the definitions we need), I think Hausdorff would qualify under the second definition. --Tango (talk) 04:29, 16 July 2008 (UTC)

It doesn't. Take the bug-eyed line: Let X and Y both denote R, let X' and Y' denote the subspaces R\{0}, and consider the bug-eyed line obtained by gluing X' and Y'. The two bugeyes, call them x and y, each have a neighborhood basis consisting of open sets in one of the copies of R: E.g., x has a neighborhood basis consisting of all open intervals containing the origin in X, and similarly for y. But each member of each of these bases is Hausdorff. Ozob (talk) 22:05, 16 July 2008 (UTC)

Isn't one of those open intervals, union the extra origin, also an open neighbourhood? It's the union of an interval in X and an interval in Y (all modulo the equivalence), the union of two open sets is open. --Tango (talk) 22:22, 16 July 2008 (UTC)

So? Algebraist 16:55, 17 July 2008 (UTC)

So, a neighbourhood base would have to include a set containing both origins, which would not be Hausdorff. --Tango (talk) 19:03, 17 July 2008 (UTC) Ignore me, I had the definition of a neighbourhood basis confused... --Tango (talk) 19:05, 17 July 2008 (UTC)

(unindent) I do not see any article on the bug-eyed line. Is there one? If not, should we not make one? JRSpriggs (talk) 02:47, 17 July 2008 (UTC)

Probably. Do you have a source in mind? If it's in Counterexamples, then I'm too stupid to find it. Algebraist 16:55, 17 July 2008 (UTC)

For now, I've created it as a redirect to a section discussing it in Non-Hausdorff manifold. I do feel it might be worth expanding into a separate article at Line with two origins, if someone has the time and inclination to do it. There's an article on it at the Topospaces wiki, but it's under CC-BY-SA, not GFDL, so we can't just borg it. (Incidentally, while googling for it, I found Baillif M, Gabard A (March 2008). "Manifolds: Hausdorffness versus homogeneity" (PDF). Proc. of the AMS. 136 (3): 1105–1111., which might somewhat be relevant to the discussion at hand.) —Ilmari Karonen (talk) 18:59, 17 July 2008 (UTC)

What should we do?

Seeing as we can't even agree about what point-set assumptions are the most common, I'd like to propose that we adopt the convention that for a plain "manifold" with no adjectives, no point-set conditions are necessary. That is, I propose that:

A manifold is a topological space X that is locally homeomorphic to a finite dimensional Euclidean space of a fixed dimension.

Manifolds are not assumed to be smooth, PL, Hausdorff, second countable, or paracompact. Unless otherwise specified, they are assumed to be equidimensional, finite dimensional, and without boundary or corners. (E.g., calling something a "Banach manifold" counts as "otherwise specified".) Homology manifolds and (if I recall my counterexamples correctly) simplicial manifolds don't count. While more restrictive assumptions may be appropriate for certain classes of manifolds (smooth, complex, etc.) these seem to be the best for the encyclopedia as a whole. The manifold article will of course retain its discussion of the various possible definitions, though it would require updating to reflect the new consensus.

Thoughts? Agreement or disagreement? Ozob (talk) 22:31, 16 July 2008 (UTC)

Hmm -- you're not requiring "connected", I guess? So you'd allow the dijoint union of two spheres, but not the disjoint union of a sphere and a line? Why exactly is that? --Trovatore (talk) 22:37, 16 July 2008 (UTC)

You're right, I'm not requiring connected, and I'd allow the disjoint union of two spheres but not a sphere and a line. This is a convention and is totally arbitrary. Not that I want to start this again, but: It reflects my impression of what people usually mean when they say "manifold". We could drop this requirement, too, if it seems overly restrictive. I don't know where one would find a non-equidimensional manifold in nature, but if there are such situations then we should certainly drop equidimensionality. Ozob (talk) 23:17, 16 July 2008 (UTC)

It is not a question of finding one in nature. It is a question about what the established terminology is. I am not sure regarding the point set topology requirement, but I am sure that most topologists would agree that a disconnected union of two spheres of the same dimension is a manifold (for example, the boundary of a manifold with boundary should be a manifold), but that the disconnected union of a 2-sphere with a line is not a manifold. Oded (talk) 23:22, 16 July 2008 (UTC)

In my experience, fixed dimension is a standard requirement for a manifold. I've never seen someone talk about something with variable dimension being a manifold (I stand ready to be corrected, of course). --Tango (talk) 23:47, 16 July 2008 (UTC)

This isn't going to work, I would have thought for obvious reasons. The biggest one I can see is that you are proposing we use an extremely unusual definition that is only going to confuse everybody and cause OR problems. --C S (talk) 01:40, 17 July 2008 (UTC)

It does no real harm to use Hausdorff with a link routinely. That would be the major effect of this proposal. Septentrionalis PMAnderson 02:34, 17 July 2008 (UTC)

Likewise, it does no real harm to use the prefix "non-Hausdorff". In fact, there is a separate article non-Hausdorff manifold which treats specifically this case. Since all manifolds are Hausdorff for me, I doubt I could say when something required the manifold to be Hausdorff, but prefixing it by saying "Hausdorff" may create the false impression that it is not true in the non-Hausdorff case. Unless someone is willing to audit carefully all of the pages linking to manifold, I suggest that we keep the more restrictive (and also more common) definition which most naive contributors to related articles have been using. Incidentally, all manifolds are also paracompact for me, although I doubt very many of my contributions have relied on this property. siℓℓy rabbit (talk) 02:46, 17 July 2008 (UTC)

Actually that would not be the total effect. We would also need to routinely link to second countable. We would also need to routinely verify if new submissions that omit these terms really mean to omit them. That's what happens when you use a nonstandard definition that is not used off of Wikipedia. --C S (talk) 03:23, 17 July 2008 (UTC)

Well, some of that may be unavoidable in any case. I do think we should go with the most standard definition (at least if there's a clear gap between it and the next-most-standard) but we still have to make clear in each article what we're talking about. So even if we agree that our manifolds are second countable, when there's a result that depends on that assumption, we need to say something like if M is a (second countable) manifold, or put a footnote or something. --Trovatore (talk) 07:44, 17 July 2008 (UTC)

And we should link to second countable once per article, when it does matter; not doing so puts us in the unrealistic world in which our readership can be assumed to know exactly what it means. Septentrionalis PMAnderson 23:17, 17 July 2008 (UTC)

Nobody disputes that, so it's rather pointless to mention we can add links. Of course we can add links. We can always do that. But why? You still have not answered why we should use a definition not in the literature, something that will undoubtedly confuse people who actually know the standard definitions of manifold , and would also violate WP:NOR. --C S (talk) 03:10, 18 July 2008 (UTC)

A few somewhat unordered thoughts here:

• It's not quite "not in the literature"; it's the definition used in the paper Ilimari references above. (Well, strictly speaking that paper says locally homeomorphic to Euclidean Rⁿ without assuming n is finite, but I think that can be taken as implicit, given that I don't know what "Euclidean" might mean in an infinite-dimensional space.)
• It's also the definition from our article on topological manifolds -- possibly a problematic article; should this be merged with manifold? Is this a standard usage of the term topological manifold? What's a non-topological manifold?
• Even if we say that this proposed definition is OR, that doesn't resolve the "second countable" issue; you yourself pointed to Hatcher's book showing that the no-second-countability-assumed definition is current in the literature, so we certainly have to address that issue in any article where the assumption is important.
• Are we agreed that failure of Hausdorfness is more pathological, does more violence to the notion of "manifold", is less likely to be permitted in the literature, than failure of second countability? This seems natural to me; non-Hausdorff spaces are usually either pathological counterexamples or else structures that come from some non-topological field (the Stone and Zariski topologies being examples of the latter), whereas the long line has a very clear geometric intuition to it. --Trovatore (talk) 07:45, 18 July 2008 (UTC)

I would say the only time you would find that definition is in precisely the papers that investigate the relationship of these point set theoretic conditions. For example, the topological manifold article clearly states at the beginning the space is Hausdorff (it says it again later in the section called "formal definition" in a sloppier way). The typical use of "topological manifold" always includes "Hausdorff" and most often "second countable". I would agree the latter condition being dropped is a well known phenomenon, but the first never happens except for very special papers where the manifold concept is being explored in a very basic way. The homogeneity condition mentioned in the linked paper is probably the main reason. It is even more basic than partitions of unity. --C S (talk) 05:22, 19 July 2008 (UTC)

It's not stated very clearly in the online version of his lecture notes, but I'm happy to go with what I believe is the definition Thurston follows: a manifold is a Hausdorff space M together with an atlas consisting of a pseudogroup on Rⁿ and a maximal family of homeomorphisms ƒ_i from open subsets of M to Rⁿ such that all the compositions ƒ_i^-1ƒ_j belong to the pseudogroup. So different pseudogroups mean different manifolds, even if the underlying space is the same: a piecewise linear structure on a sphere is a different manifold than a smooth sphere, etc. That said, this level of technicality is likely to be too much for most other articles that use the concept of a manifold. The idea that a manifold is "just" a space locally homeomorphic to some Rⁿ may be good enough in those contexts. —David Eppstein (talk) 03:00, 17 July 2008 (UTC)

I've never heard that definition before. As manifolds, they are the same, they only differ when considered as differentiable manifolds. As far as I'm concerned, a manifold is just a type of topological space, it doesn't have any additional structure. --Tango (talk) 16:23, 17 July 2008 (UTC)

This definition encodes the type of manifold in the pseudogroup: a differentiable manifold is just a manifold whose pseudogroup is the pseudogroup of differentiable homeomorphisms between open subsets of Rⁿ. A manifold "without any additional structure" is a manifold whose pseudogroup is the pseudogroup of all homeomorphisms between open subsets of Rⁿ. Etc. —David Eppstein (talk) 16:34, 17 July 2008 (UTC)

List of mathematical examples

I just added examples of generating functions to the list of mathematical examples. The latter article seems neglected. Michael Hardy (talk) 17:42, 16 July 2008 (UTC)

Examples of generating functions should probably be merged into another article as "Examples" isn't really encouraged in the title. Gary King (talk) 07:46, 18 July 2008 (UTC)

Why shouldn't "Examples" be used in a title if the article is in fact about examples? Michael Hardy (talk) 18:14, 18 July 2008 (UTC)

Reassessment of GA status of Hilbert space

Currently a reassessment is taking place of the "Good Article" status of Hilbert space; see Talk:Hilbert space/GA1.  --Lambiam 12:16, 11 July 2008 (UTC) added by User:C S on 08:34, 25 July 2008 (UTC) as original comment was misplaced on a subpage

Milü

Could someone else check Talk:Milü? I checked the fraction 103993⁄33102 (I must be an incredible nerd) and it seems wrong to me. I'm hesitating though, maybe I made some stupid mistake. Piet | Talk 11:09, 22 July 2008 (UTC)

No, you are correct - the next "best rational approximation" to π after 355⁄133 is 52163⁄16604, not 103993⁄33102 (although that is the next continued fraction convergent).

I have given a longer explanation at Talk:Milü. Gandalf61 (talk) 15:46, 23 July 2008 (UTC)

Wikipedia:Articles for deletion/Duality (mathematics)

Please see Wikipedia:Articles for deletion/Duality (mathematics). — Carl (CBM · talk) 13:46, 24 July 2008 (UTC)

Eigenvalue perturbation

Eigenvalue perturbation needs some help to make it clear. Please help. Tony (talk) 17:14, 24 July 2008 (UTC)

Hmm, quite unsavoury :( What about the classic book of Tosio Kato on Perturbation theory for linear operators? Mathsci (talk) 17:43, 24 July 2008 (UTC)

I just returned a book to the (my unnamed employer's) technical library on Perturbation theory. Grumble. — Arthur Rubin (talk) 18:25, 24 July 2008 (UTC)

Optimal classification

Optimal classification has been nominated for deletion. Comment at Wikipedia:Articles for deletion/Optimal classification. It looks as if some of the people saying it should be deleted have no interest in or knowledge of the subject matter. Michael Hardy (talk) 15:24, 25 July 2008 (UTC)

Eigenvector slew

This article was a speedy deletion candidate, which it survived, although its existence remains borderline. Some helpful input or a contribution to the ensuing discussion at Talk:Eigenvector slew would be appreciated. Ben MacDui 19:38, 25 July 2008 (UTC)

thanks for the heads-up. article, notable subject or not, is bad and of elementary nature. the reaction to the speedy deletion tag on talk is, well, not credible. makes one wanna apologize to the person who introduced the tag. Mct mht (talk)

Thanks for your prompt attention folks. I'll drop a note to said user. Ben MacDui 20:44, 25 July 2008 (UTC) PS Do I get an Erdős number out of this?:D

Mainpage experiment

Someone ran a short (7-day) experiment on the main page, and posted the results here. I just thought it was interesting that so much traffic hits the maths portal. Cheers, Ben (talk) 06:25, 26 July 2008 (UTC)

Cantor's second diagonal argument

Today I added a little section to the article on Liouville numbers on how these numbers constitute an uncountable set (of measure 0, dense in the set of real numbers).

I referred to Cantor's second diagonal argument, which I have not found anywhere in Wikipedia, but made a provisional (red) link.

One place an explanation could go is in the article on uncountable sets. I would prefer it not clutter the article on Liouville numbers. There may be other articles where links thereto would be nice. Scott Tillinghast, Houston TX (talk) 04:53, 19 July 2008 (UTC)

OK, I found Cantor's diagonal argument. Scott Tillinghast, Houston TX (talk) 04:59, 19 July 2008 (UTC)

What do you mean by the "second" diagonal argument? It's clear enough that the methodology of the diagonal argument gets you the result you want, but I'm not familiar with any division into a "first" and "second" (or subsequent) diagonal argument. Do you have a reference where this nomenclature is used? --Trovatore (talk) 05:00, 19 July 2008 (UTC)

Some people call Cantor's enumeration of the rational numbers his first diagonal argument, and his proof that any sequence of binary sequences is missing some binary sequence his second diagonal argument. JackSchmidt (talk) 05:05, 19 July 2008 (UTC)

Oh. I don't think the enumeration of the rationals is usually called a "diagonal argument"; that would be confusing, given that it has nothing to do with diagonalization in the more general sense in which it's come to be understood. --Trovatore (talk) 05:07, 19 July 2008 (UTC)

You can kind of see why from the picture in Cantor pairing function. Cauchy has lots of theorems and Weyl has lots of theorems, so its not too weird that Cantor has more than one diagonal argument. JackSchmidt (talk) 05:12, 19 July 2008 (UTC)

I understand that the enumeration is often presented in a way that has diagonal lines. I don't think it's very often called a diagonal argument; that term is more or less reserved for arguments where you assume something exists and then feed it back to itself to get a contradiction. But there are a few hits for it on the web; I'm not saying it doesn't exist at all, just that it's not standard mathematical terminology. --Trovatore (talk) 09:04, 19 July 2008 (UTC)

I have to agree with Trovatore that Cantor's proof the rationals are countable is not a "diagonal argument" as the term is commonly used today by mathematicians. When I read the section title here, I thought "second diagonal argument" would probably refer to Cantor's second proof that the reals are uncountable. This may explain why Cantor's second diagonal argument didn't already exist. I'm going to create it as a redirect. — Carl (CBM · talk) 12:48, 19 July 2008 (UTC)

It may not fit the usual definition of the term, but I have certainly heard it described as a diagonal argument. --Tango (talk) 17:45, 19 July 2008 (UTC)

I meant the proof that the rational numbers are countable when I implied a first diagonal argument. Actually the problem is all solved now that I have linked to the page 'Cantor's diagonal argument.'

I think the Liouville numbers are a nice example of an uncountable set of measure zero. They are dense in the reals.

I may have seen the arguments called first and second in a textbook or in a 'Scientific American' article about 1965. 2008 (UTC)

An alternative definition of Liouvile number can be made in terms of the values at a number x of linear integral polynomials p-xq. Let the height h of p-xq be the greater of |p| or |q|. Define omega₁(x) as the lim sup of all numbers -log|p-xq|/log h over all positive integers h. x is a Liouville number iff omega₁(x) is infinite. The converging numbers equal the logarithm to base h of 1/|p-xq|.

The concept of Liouville number can be extended. Let the height h of a polynomial be the maximum absolute value of its coefficients. Define omega_n(x) equal to lim sup -log|P(x)|/log h taken for all integral polynomials of degree n and height h, for all positive integers h. A number x is a U number in case there is a positive integer n such that omega_n(x) is infinite.

A number x for which all the omega_n(x) are bounded is an S number. The complement of the S numbers has measure 0. e is an S number. The nature of pi is unknown but it is not a U number.

If all the omega_n(x) are finite but unbounded, x is called a T number. It was difficult to prove that any T numbers existed.

The definition of U number could be iterated on the page about Liouville numbers, with a link to a fuller definition of S, T, and U numbers, probably in the page on transcendental numbers. Scott Tillinghast, Houston TX (talk) 01:12, 28 July 2008 (UTC)

Another place could be the page on transcendence theory. Scott Tillinghast, Houston TX (talk) 01:27, 28 July 2008 (UTC)

Inverse function

An insistent anon keeps on adding what I'd call his pet formula to inverse function. Comments on the article talk page about its value would be welcome. Oleg Alexandrov (talk) 02:41, 29 July 2008 (UTC)

Michael Atiyah

Could more eyes please look at the talk page here? It seems to have been hijacked by extremists determined to include a long section in his BLP proving that he is a plagiarist. Thanks, Mathsci (talk) 23:50, 24 July 2008 (UTC)

I've read over the cited sources and agree they fall into two classes: those that are not reliable sources and those that do not support the claims they are being used to support. Trying to put such things in the article is a gross violation of our BLP policies. JackSchmidt (talk) 15:52, 25 July 2008 (UTC)

An attack article created by User:Bharatveer entitled Raju - Atiyah Case has been speedily deleted, following my request. However no administrators have warned this user or his fellow editors. Could a mathematical administrator please try to sort this out? Thanks, Mathsci (talk) 22:40, 26 July 2008 (UTC)

I'm not sure why an admin hasn't already deleted the talk page and warned those editors not to keep pushing their arguments. The length of the "discussion" on the talk page is at a preposterous level. I see BLP and RS noticeboards have been notified. You might get a more immediate and stronger response if you posted to AN/I. --C S (talk) 01:33, 27 July 2008 (UTC)

Since things seem to be dying down there, it may be best to do nothing. Paying too much attention to hissy fits by posting them on ANI sometimes just extends them. R.e.b. (talk) 02:34, 27 July 2008 (UTC)

I agree with User:R.e.b.. Enough mathematics administrators seem to know about it. (User:David Eppstein reverted another unnoticed Atiyah BLP violation by User:Bharatveer.) This user is actually on ArbCom editing restrictions - there was a whole case about him. If anything flares up again, we can rethink putting something on WP:AN/I. It is premature at present. Mathsci (talk) 03:06, 27 July 2008 (UTC)

Agree with R.e.b. and Mathsci. Fowler&fowler«Talk» 14:02, 27 July 2008 (UTC)

And there's a BLP noticeboard if things do get out of hand. Septentrionalis PMAnderson 18:42, 27 July 2008 (UTC)

It's been posted there for some time. :-) Mathsci (talk) 18:45, 27 July 2008 (UTC)

User:Bharatveer has been indefintely blocked by User:Nishkid64 until he explains his use of VPN. Meanwhile, during his absence, User:Perusnarpk is currently appealing to Elonka to take actions against User:Fowler&fowler and me. This seems strange, because, unlike Perusnarpk, User:Fowler&fowler and myself are good faith editors of long standing, not SPAs whose only purpose is to insert unsourced attack material on Atiyah into a BLP on this encyclopedia. Since his only purpose here has been to insert unverifiable libellous material on WP and since he now seems intent on causing even more disruption, the obvious action to take at this stage against him and his fellow editor User:Abhimars would be an indefinite block. Perusnparpk seems too familiar with editing policies on WP for a newly arrived editor. Mathsci (talk) 07:29, 28 July 2008 (UTC)

Perusnarpk has now set up an RfC against Fowler&fowler. Mathsci (talk) 00:28, 29 July 2008 (UTC)

Wikipedia:Requests_for_comment/Fowler&fowler. --C S (talk) 01:24, 29 July 2008 (UTC)

Since several people went and left a comment, i did likewise. But the RFC has yet to be signed by two editors who have tried and failed to resolve the dispute. So probably it will be shut down unless someone (e.g. Bharatveer) comes along. --C S (talk) 04:26, 29 July 2008 (UTC)

I dont mean to move the discussion on the talk page here, but given the existence of this thread, I felt I should contribute a small paragraph explaining myself. There is a lot of information on the talk page and faced with this avalanche, people tend to react by saying: "I can't make out what is happening in this confusion, but Atiyah is a well known man so let me give him the benefit of doubt". This is a reasonable point of view. However, in this case, it should not take more than 5 minutes to examine the evidence by following the links in the paragraph below.

Atiyah gave a research seminar at KITP and delivered a large public lecture at Lincoln discussing a possible link between functional differential equations and quantum mechanics. He was informed and acknowledged the similarity of these ideas with previously published work. So far so good. The dispute arose because subsequent to this, he personally approved the publication of a large article in the Notices of the AMS, reiterating his priority.

To emphasize, the controversy is not about the first oversight, at Lincoln and KITP, which can be understood in good faith. The question is whether it was justified for Atiyah to permit the publication of this prominent article, reiterating his priority, despite having been informed of prior published work.

I understand that this cannot be included in the page on Atiyah without mainstream media attention. However, in the meantime, I would like to request the undoubtedly perspicacious editors at this forum to follow the links above and form their own conclusions about whether this constitutes ethical academic conduct. Perhaps this will explain why I have followed this issue persistently. Thanks, Perusnarpk (talk) 09:47, 29 July 2008 (UTC)

Perspicacious editors please note [2]. Mathsci (talk) 10:41, 29 July 2008 (UTC)

Careless attribution

Hi,

I expect the case of Raju vs. Atiyah is one of careless attribution rather than the more serious charges mentioned. At the same time, one cannot fail to notice a circling the wagons phenomenon around Sir Michael on the part of the mathematicians. If the Perelman-Yau controversy has merited an article of its own, I don't see why the biographical entry cannot contain a subsection on a controversy that seems to be publicly documented and therefore beyond the stage of OR. I imagine Sir Michael has only to gain from any media publicity likely to be brought on the issue. Katzmik (talk) 12:03, 29 July 2008 (UTC)

I think your observation is colored by a misconception. Perelman-Yau only got its own article after it was covered in the New Yorker and newspapers like the New York Times. Yau's lawsuit was a subject of media coverage (including a press release and webcast by Yau). Saying this controversy is "beyond the stage of OR" is extremely puzzling. I think you need to think upon the differences further. When you say "media publicity likely to be brought...", you seem to recognize there hasn't been any thus far. Is there really a circling of the wagons or a complete misunderstanding of Wikipedia policy on your part? --C S (talk) 21:14, 29 July 2008 (UTC)

The Raju-Atiyah has only been publicly documented by the SSV (an ethics board in India) who said there was a prima facie case here, and by Raju and his supporters. The SSV source fails to elaborate on the nature of the case, as it only claims the possibility for plagiarism on Atiyah's part. The issue remains unresolved, and it seems libelous to include this material in a BLP when the matter has not even been addressed at the slightest or received significant attention. Nishkid64 _{(Make articles, not wikidrama)} 12:50, 29 July 2008 (UTC)

If the SSV picked up the case merely out of solidarity with Raju, then clearly there is no sufficient basis for inclusion. Those interested in including the controversy should focus on the issue of whether there is any substance in the case as documented by the SSV, and avoid loaded language beyond careless attribution. Katzmik (talk) 12:59, 29 July 2008 (UTC)

The SSV hasn't documented anything yet. That's the issue. The editors on Talk:Michael Atiyah are relying on a petition signed by Raju's supporters as a reference for the case details. Nishkid64 _{(Make articles, not wikidrama)} 13:03, 29 July 2008 (UTC)

...should focus on...whether there is any substance in the case.... Actually, that is a misunderstanding of policy. We aren't suppose to focus on any such thing. We are supposed to focus on whether reliable sources have decided there is substance, just like we didn't determine whether there was any substance to the Perelman-Yau thing. It's no different than when a politician gets accused of some scandal. We just include what the reliable sources say in a matter of fact way. Those seeking to include the material must show the SSV counts as a reliable source for the alleged claims in this matter. --C S (talk) 21:19, 29 July 2008 (UTC)

For what it's worth, after placing a request for better sources, apparently the best that User:Abhimars could come up with were entries in a blog. This story has been taken up by zero reliable sources, as far as I can tell, and the editors in question have been unable to produce reliable sources on request. Until an independent reliable source (such as a major newspaper) runs the story, it has no place in a biography of a living person, and I fully support the removal of this potentially libelous material under Wikipedia policy. In my opinion, any more BLP violations committed by this cabal of editors should be grounds for an immediate 24-hour block (at least) by an uninvolved admin, just to drive the point home that this is really against one of our most fundamental core policies. siℓℓy rabbit (talk) 15:02, 29 July 2008 (UTC)

Of course, I agree with this. In addition, I still think the talk page should be blanked of the material, per the usual BLP procedure. At the moment, it's working as a magnet for further people with agendas to come by. No better sourcing is forthcoming, but that doesn't stop these people from making protracted comments to the talk page. I know from experience that BLP enforcement can be extremely inconsistent. Unless one BLP-concerned admin makes this article his/her project (or Atiyah sends an email to Jimbo Wales), it's unlikely that appropriate actions will be taken in a timely fashion. --C S (talk) 21:42, 29 July 2008 (UTC)

Primarily, in my view, this has simply been a case of weak journalism by the AMS, and they have paid the price for that already. Attempts to extend the issue into Wikipedia articles smack of recentism. Geometry guy 23:39, 29 July 2008 (UTC)

Looking for better ways to silence what seems to be a significant portion of the physics community is not a good way of mediating the conflict. One of the greatest mathematicians living today certainly deserves the benefit of the doubt as does anyone else, as do his would-be critics, who should instead be encouraged to await the outcome of the procedure started at the ethics organisation in India. As far as the narrow issue of inclusion or non-inclusion in wiki, this is certainly the most reasonable course of action. Katzmik (talk) 12:21, 30 July 2008 (UTC)

Again, your response seems to me typical of those who see this issue of Wikipedia policy as an extension of some conflict in the real world. No community is being silenced here. Those people are perfectly free to continue on with their lives and fight whatever battles they choose. Whether their accusations and arguments remain on a Wikipedia talk page has very little to do with "silenc[ing] a significant portion of the physics community". The "would-be critics" have been encouraged countless times to await the outcome, to wait until something substantial in terms of coverage has occurred. They are not satisfied with that. They are sure there is some conspiracy, based on euro-centric bias and involving protecting Einstein's reputation no less, to keep this material off Wikipedia. It seems to me you're buying into this theory, albeit in a much smaller way. But there is no conspiracy here. I've never met Atiyah, never heard any stories about him (good or bad), or studied his work (although I did sit in a very introductory course on topological K-theory). I have no desire to silence some community of people in order to protect Atiyah. Nonetheless it's clear as day their comments have passed the stage of being helpful remarks on how to add useful information to a biography to the stage of being advocacy for their cause, with little understanding of the relevant policies. Rather than understanding the policies and then seeing if the material is worth including, they are approaching policy from the perspective of "how can we interpret the policies to include the material we want to include?" This is never a good idea. --C S (talk) 12:39, 30 July 2008 (UTC)

I fully agree with you. Incidentally, I misused the word "likely" in my first comment above, which was meant to be a kind of a subjunctive, as in "whatever is likely to come of this", not that I think media attention is actually likely. My understanding is that SSV is thought to be a serious ethical panel but has not lodged any substantive charges until now. There seems to be a general agreement among the people who commented above that if and when SSV does move further on this controversy, there may be room to include a subsection in the wiki article. What I think exacerbates the situation is the fact that as mathematicians it may be very hard for us to evaluate Raju's actual contribution, it being in a different field, and very speculative even by the standards of that field (everybody seems to agree on that). Katzmik (talk) 12:51, 30 July 2008 (UTC)

How long do we have to continue these pointless discussions? If and when Atiyah publishes any paper on this topic, and it has been reviewed, it might merit a comment. However at the moment editors are actually adding details about the main body of his work to the BLP, rather than his post-retirement musings. Can we possibly regain some sense of proportion here? It seems that elsewhere more SPAs are continuing to hatch, but Nishkid64 is fortunately monitoring them - many thanks to him! Mathsci (talk) 16:16, 30 July 2008 (UTC)

Ultralimit

I'd like to request initial assessment of a newly created article Ultralimit (the topic was on WP:Articles requested for more than a year). Thanks, Nsk92 (talk) 22:45, 27 July 2008 (UTC)

I moved the request here from a WP 1.0 page that is not highly watched. I haven't looked at the article. — Carl (CBM · talk) 00:18, 28 July 2008 (UTC)

Having glanced at but not read the article, an obvious deficiency is that it only covers one use (to do with metric spaces). We still need (at least) an article on the model-theoretic meaning (direct limit of a sequence of ultrapowers). Algebraist 00:31, 28 July 2008 (UTC)

Being basically a geometer, I am/was not aware of any standard usage of the term "ultralimit" other than for a limit of a sequence of metric spaces. In geometric group theory, geometric topology and metric geometry the term "ultralimit" does have a standard meaning and it always refers to limits of metric spaces. If there is indeed some other standard model-theoretic usage of the term, some model theorist has to write a WP article about it. I am certainly not the right person to do that. Nsk92 (talk) 05:31, 1 August 2008 (UTC)

I'm not really familiar with Algebraist's meaning. In set theory such direct limits come up all the time, but they're usually just called "ultrapowers". For example you can take the ultrapower of the universe by some extender, which is an article we really ought to have. --Trovatore (talk) 06:31, 1 August 2008 (UTC)

We do have an article ultraproduct which discusses ultrpowers in a fairly general algebraic context. Maybe it should be extended... Nsk92 (talk) 11:47, 1 August 2008 (UTC)

OK, I talked to one of the model theorists in our department and he explained to me what Algebraist meant above. Apparently there is a notion in model theory (somewhat obsolete and rarely used according to the colleague I consulted) that is also called "ultralimit" or "limiting ultrapower" and it does refer to a direct limit of ultrapowers. Basically, as I understood it, one can take a nice algebraic structure (say a ring) A₀, and an ultrafilter D on $\mathbb {N}$ and form an ultrapower $A_{1}=A_{0}^{\mathbb {N} }/D$. Then we do the same to $A_{1}$ to form $A_{2}=A_{1}^{\mathbb {N} }/D$. And so on, inductively, define $A_{n}$ for every $n\in \mathbb {N}$. For each n there is a canonical diagonal embedding $A_{n}\to A_{n 1}$. Then one can take the direct limit $A_{\infty }$ of the sequence $A_{0}\to A_{1}\to A_{2}\to \dots$ and apparently this direct limit is also called "ultralimit". I must confess that I had not seen this notion before ,and it is, of course, a completely different notion from the notion of an ultralimit of a sequence of metric spaces that the article ultralimit is currently about. This other algebraic notion deserves either a separate article or maybe a separate section in the ultraproduct article. But somebody more well versed in model theory than me will have to do it. Nsk92 (talk) 22:23, 1 August 2008 (UTC)

It may for all I know be "obsolte" in model theory, but definitely not in set theory, although as I say I don't recall hearing the word ultralimit used for the concept. In general you won't have just a single D but a sequence of them, or even a tree, and you take direct limits along branches of the tree. This is a fundamental technique in inner model theory, which despite the name is more part of set theory than it is of model theory. --Trovatore (talk) 23:48, 1 August 2008 (UTC)

Arc length

The article titled Arc length could probably benefit from the attention of mathematicians. Michael Hardy (talk) 16:24, 28 July 2008 (UTC)

There haven't been any recent comments on this article's talk page. At first glance it looks like a fairly typical example of a math article at this level: i.e., adequate but not great. The one thing which seemed fishy to me is the global choice of $\pm$ in the arc length of a curve in a pseudo-Riemannian manifold. I am far from an expert on such things, but I see no reason why the function $t\mapsto g(\gamma '(t),\gamma '(t))$ should have constant sign, and thus the definition makes no sense to me. Is this a case of excessive generalization, or what? Plclark (talk) 22:47, 31 July 2008 (UTC)

Sorry, I meant to say: so what possible improvements do you have in mind? But I got distracted by the mathematics. Plclark (talk) 22:50, 31 July 2008 (UTC)

atiyah

User:Fowler&fowler suggested getting Michael Atiyah to FA status so it can go on the main page for his 80th birthday next april.

Most of the WP:SPAs on the talk page have just been blocked, but at least one is coming back as an IP, so it would be helpful if a few people could keep an eye on talk:Michael Atiyah for a few days. My understanding of WP:BLP policy is that all additions by these people should be reverted on sight with no discussion. R.e.b. (talk) 19:05, 1 August 2008 (UTC)

IP blocked. Nishkid64 _{(Make articles, not wikidrama)} 20:11, 1 August 2008 (UTC)