Wikipedia talk:WikiProject Mathematics/Archive/2010/Aug

Self-study and this project

Wikipedia:WikiProject Mathematics/Advice on using Wikipedia for mathematics self-study seems to be, from a thread above, a project page that should exist. I'm going to get this started. Please come and help. Charles Matthews (talk) 11:43, 13 July 2010 (UTC)

Nice idea, but how will people find it? -- Radagast3 (talk) 12:28, 13 July 2010 (UTC)

Please read the thread above (naturally skipping most of it) where Portal:Mathematics is discussed. Given that there are real issues, the proposal I made (that we add to the pages accessible from the portal in order to have some FAQ-like material only two clicks from the Main Page) seems to have gone down well. Charles Matthews (talk) 12:38, 13 July 2010 (UTC)

What I meant was, as yet it's not linked on the portal. -- Radagast3 (talk) 13:46, 13 July 2010 (UTC)

Well, to state the obvious, it seemed premature and presumptuous to link my first draft without any input from others. Charles Matthews (talk) 13:50, 13 July 2010 (UTC)

Another approach that might be helpful is creating more "articles with a separate introduction". Currently there is only a handful of math pages in that category. A considerable amount of ink (and sarcasm) has been expended in this space to discuss the problem of accessibility. What I would suggest is choosing one of the pages mentioned here, and investing the effort instead in writing an introduction. Tkuvho (talk) 14:27, 13 July 2010 (UTC)

Well, it might. But it is hard to get ahead with anything in threaded discussions if everyone introduces their own tangential topics. Charles Matthews (talk) 15:13, 13 July 2010 (UTC)

I think that is brilliant. I have made a couple of minor changes. Yaris678 (talk) 15:34, 13 July 2010 (UTC)

Yes, I like it, too — it's engaging and pithy. In addition to posting it at the Portal, I'd suggest that you add a prominent link to it on this wikiproject, e.g., at the top of this page. That might forestall some people from posting here in the future. I have a few other ideas, but out of kindness and respect, I'll restrain myself from introducing tangential topics now. ;) Thank you, Charles! Willow (talk) 13:30, 15 July 2010 (UTC)

Looking reasonably complete now (assuming we do want a "gist and pith", as Ezra said). Please go ahead and link to it on the basis that this is the work of half-a-dozen hands and probably represents a collective view. Charles Matthews (talk) 19:36, 17 July 2010 (UTC)

Well done. --P64 (talk) 03:43, 27 July 2010 (UTC)

History of calculus

I'm all for avoiding a western bias, but History of calculus has almost nothing about the history of calculus as it is conventionally understood in reliable sources, and instead presents a highly misleading whiggish narrative connecting everything from Archimedes to Lui Hui to Aryabhata and others, as though they were as much a part of the story as Newton, Leibniz, Bernoulli, and Cauchy (who is barely even mentioned!) It's clear that some emergency intervention is called for here. I've started a thread at Talk:History of calculus. Sławomir Biały (talk) 13:54, 21 July 2010 (UTC)

It's not very good. It would be improved by conventional copy-editing, first. There is some old encyclopedia material there (where it starts talking about Abel), which is about foundations of differential algebra and algorithmic integration, topics that should be linked to but certainly elsewhere. Questions about the key decades in the 17th century (when what we know as calculus emerged) have come up on Talk:John Wallis and elsewhere. I'd certainly like to see all that material reconsidered. Charles Matthews (talk) 19:58, 21 July 2010 (UTC)

I have replaced the lead and formulated a proposed solution at the talk page. Arcfrk (talk) 06:13, 22 July 2010 (UTC)

Concerning Sławomir Biały's comment on Lui Hui to Aryabhata: why, ancient remains of the Aztec civilisation provide strong evidence that their children used to slide down staircase railings, thereby exploiting slope, proving calculus to be an anti-eurocentrist, decidedly multicultural thing. Tkuvho (talk) 09:09, 22 July 2010 (UTC)

I performed some drastic surgery at History of calculus, it would be good if a few people could look it over for bugs. Tkuvho (talk) 19:17, 25 July 2010 (UTC)

obligatory archiving?

At http://wiki.riteme.site/w/index.phptitle=Talk:Criticism_of_nonstandard_analysis&curid=19173182&diff=375360445&oldid=375295630 an editor who has no record of contribution to mathematics articles seeks to impose an archiving scheme. Is this required by some wiki regulations, or can I override him? Tkuvho (talk) 12:17, 25 July 2010 (UTC)

You can override him. If discussion is not frequent, many people think it's better to leave up older discussion so that other people have a chance of seeing it. I know I have seen useful comments from months earlier when I randomly happened upon a rarely-visited talk page. — Carl (CBM · talk) 12:26, 25 July 2010 (UTC)

Please read WP:TPNO#When_to_condense_pages. The talk page in question shows up in Wikipedia:Database reports/Long pages, so it is too long. Let's take this discussion to the affected talk page.--Oneiros (talk) 13:37, 25 July 2010 (UTC)

I agree the page was too long. In general, I prefer manual archiving for pages unless the receive nearly constant comments, in order to try to avoid losing comments to lightly-watched pages. — Carl (CBM · talk) 13:55, 25 July 2010 (UTC)

Indeed not related to this case, but in general some people "abuse" archiving to remove directly visible criticism from the discussion page and to interrupt slow discussions that they dislike. Archiving where it is not really needed is somewhat of a disservice to readers, since less experienced readers may not realize that they have to check the archive as well.--Kmhkmh (talk) 15:01, 25 July 2010 (UTC)

Exactly the way I feel about it. Just because it appears on some list, does it obligate the wikiproject to archive it? It is hard to believe the TPNO is binding. Tkuvho (talk) 19:07, 25 July 2010 (UTC)

When a talk page is over about 100k, it loads very slowly for many editors, particularly those without a high-speed internet connection. So if there are older threads, archiving is helpful. But that can be done manually; nothing obligates automated archiving. — Carl (CBM · talk) 04:22, 27 July 2010 (UTC)

Conical hat vs Pointy hat

Hello maths people. We need your help. There is an article called Pointy hat, and one called Conical hat. A merge has been suggested. So, can you get your thinking caps on and tell us what is best? We might debate this endlessly, but you, being maths people, have large hat sizes, and therefore can solve this with logic and mathematical definitions. The thread is here. Thank you. Anna Frodesiak (talk) 15:13, 26 July 2010 (UTC)

I think an prod should be placed on both as not having any source talking about the general topic of either pointy or conical hats. Dmcq (talk) 16:48, 26 July 2010 (UTC)

Prod seems a bit drastic. There is, after all, some useful disambiguation that should be provided as a service to readers typing in "pointy hat" or "conical hat" into the search box, so these articles clearly serve a useful purpose consistent with our mission as an encyclopedia. Furthermore, I don't think assembling a list of pointy hats constitutes original research, despite the lack of a consolidating source. The lead to the pointy hat article and the History section of the conical hat article should perhaps be removed or rewritten based on this concern, but deletion of the entire article seems unwarranted. Sławomir Biały (talk) 17:40, 26 July 2010 (UTC)

I suppose they could be turned into disambiguation pages. A disambiguation page is not an article on a topic. The prod would be of the topic as such. People do describe some hats as conical, that doesn't mean that anyone outside Wikipedia has decided to write about it as a topic in any notable way. The individual hats though are probably notable and list of hats is fine and I can't see that a section on hat shapes that listed conical hats would raise any objection. Dmcq (talk) 18:08, 26 July 2010 (UTC)

Prod? Like cattle prod? I don't know what that means. Anna Frodesiak (talk) 18:11, 26 July 2010 (UTC)

Prod is short for "proposed deletion" (a process that is in-between "speedy deletion" (CSD) and regular AfD deletion), and the template is named {{prod}}. (we Wikipedians love our terminology, yes we do :)

I agree that prod is overly drastic, especially whilst a discussion to merge/fix/ref is already ongoing. -- Quiddity (talk) 18:23, 26 July 2010 (UTC)

This doesn't really seem like an issue for WPMATH. Both articles are claimed by Wikipedia:WikiProject Fashion so I'd suggest raising the issue there.--RDBury (talk) 19:28, 26 July 2010 (UTC)

Oh that prod. I see. The reason I posted here is because, in the past, the debate had been over whether or not all conical hats are pointy, and can a really obtuse angle mean that it has no point, if a cone is a bit floppy, is it still conical, etc. You know, maths stuff. I thought you could provide an incontrovertible argument. Maybe you're right about the fashion folks. Thanks all! Anna Frodesiak (talk) 02:09, 27 July 2010 (UTC)

Well if that's all you want: no, if the tip is rounded then it's not a mathematical cone. We'd call those folks from Remulac "Paraboloidheads". —Tamfang (talk) 02:24, 27 July 2010 (UTC)

Good. I see. What about straight sides? Can a cone bend a bit? Anna Frodesiak (talk) 02:38, 27 July 2010 (UTC)

Cones are pointy. (Trying to think of a Coneheads clip where a hat or something gets torn but am unable to...) SharkD  Talk  03:32, 27 July 2010 (UTC)

Coneheads, eh? :). So, all conic hats are pointy, but not all pointy hats are conic. Is that right? If so, then Conical hat can be merged into Pointed hat. Sorry to bother you all with this stuff. But, it can really help decide the merge. The folks at Fashion might cause a dramafest. Thanks. Anna Frodesiak (talk) 04:05, 27 July 2010 (UTC)

Neither article seems to mention the type of hat worn by Fujiwara no Sai which was popular during the Heian period of Japanese history. It is a very tall hat, similar in shape to File:Cône d'Avanton, musée des Antiquités Nationales.jpg, but smooth and black in color. JRSpriggs (talk) 06:52, 27 July 2010 (UTC)

Of course, I know the hat. You can get them on eBay. I kid, I kid. I will see if I can find it and add it to the article. Thanks. Anna Frodesiak (talk) 07:06, 27 July 2010 (UTC)

Axiom of choice, "definable", "explicit"

There are quite a few articles involving objects whose existence is proved with the help of the axiom of choice, and often they make an assertion that it is impossible to find an "explicit" or "definable" example. Examples: non-measurable sets, paradoxical decompositions (Banach-Tarski), ultrafilters, Hamel bases of R/Q, probably lots more.

There is generally some sense in which this claim is true, but unfortunately such a sense is almost never specified. There is at least one sense in which it may be false; namely, if there's a definable wellorder of the set-theoretic universe, or equivalently if V=HOD, then there are definable examples of all these things. Not definable in any very useful way, but definable nonetheless.

This is a problem that's been on my radar screen for years, and I've never figured out what to do about it. I invite suggestions, and I'd also like people to keep it in mind and avoid making the problem worse. --Trovatore (talk) 05:05, 25 July 2010 (UTC)

How about 'constructible' instead? Dmcq (talk) 11:23, 25 July 2010 (UTC)

Another important "counterexample" to such use of "definable" is the result of Kanovei and Shelah, who described a definable model for the hyperreals. As far as use of AC and such, perhaps more appropriate adjectives are "explicit" or "constructible". Tkuvho (talk) 11:30, 25 July 2010 (UTC)

Yes, I felt this problem when writing into Borel set#Non-Borel sets the phrase (at the end) "However, this is a proof of existence (via the choice axiom), not an explicit example." Advice from logic experts is welcome, which word to use in this situation. No, I do not mean here a theory that stipulates a definable wellorder of the set-theoretic universe. For me, the boundary between "good" and "bad" in this aspect lies between the dependent countable choice and any stronger choice. Does it make sense? Boris Tsirelson (talk) 12:32, 25 July 2010 (UTC)

A useful model of the hyperreals can be obtained in the context of Skolem's non-standard models of arithmetic, by forming the field of fractions. Since there exist countable such models, one does not use anything beyond countable choice. This was explained in a paper of Avigad's. Tkuvho (talk) 13:32, 25 July 2010 (UTC)

The most for which one could hope is "If ZF is consistent, then there is no formula of set theory which can be proved from ZF alone to well order [such and such a set].". JRSpriggs (talk) 15:20, 25 July 2010 (UTC)

OK, responding to various suggestions above:

• constructible doesn't work for a couple of reasons: first, it's likely to be taken to mean "in L", which is not what's intended here. Second, even if we make it clear that that's not what we mean, it's still not clear what we do mean.
• explicit has the same problem as the second sentence above — no demarcation is offered for what constitutes an "explicit" example.
• The problem with talking about the axiomatics (Boris's suggestion and JR's) is that it perhaps says too little. We can usually say something (absolutely, not in terms of provability) about the example not being definable in some especially useful way; we can say it's not in some simple definability class (e.g. Borel, for examples that can be coded as sets of reals, or even projective, assuming enough large cardinals). Question is, do we want to, and how do we avoid drowning the statement in technicalities? --Trovatore (talk) 19:24, 25 July 2010 (UTC)

May constructible work if qualified by some adjective or proper name? (Such as classically constructible or Brouwer constructible, but I don't know the history to suggest any particular coinage.) Maybe the disambiguation Constructible needs a fifth mathematical sense which will then serve the purpose in many discussions that "turn on AC" (whichever particular axiom of choice is most useful in this context). Of course an article devoted to constructibility in the fifth sense would be necessary. --P64 (talk) 04:37, 27 July 2010 (UTC)

To P64: See the definition of "constructive" in Constructible universe#The difference between constructible and constructive.

To Trovator: Saying that a set cannot be contructive in some specified variation of this sense (essentially saying that it is above a certain level in the hierarchy of complexity) might be more like what you want. JRSpriggs (talk) 07:09, 27 July 2010 (UTC)

Well, if R is not contained in L (which of course I'm confident it isn't) then it seems kind of trivial to say that, say, a paradoxical decomposition of the sphere is not in any L_α. To P64: Constructivism as I understand it is more a restriction on methodology than ontology; I don't think there's any clear demarcation of what constitutes a constructively acceptable object. Or to the extent that such demarcations have been offered (maybe by Errett Bishop?) then you may find that they rule out things you would intuitively accept as "explicit". For example, in Bishop's mathematics, there is no such thing as a discontinuous function from R to R, which naively to me would seem to imply that there is no such thing as a nontrivial partition of R into two sets at all. --Trovatore (talk) 10:13, 27 July 2010 (UTC)

We have an article indecomposability on your last comment. Algebraist 16:16, 27 July 2010 (UTC)

For example, you can't constructively subdivide a circle as a union of a pair of disjoint antipodal sets. Tkuvho (talk) 11:21, 27 July 2010 (UTC)

OK, so you see the problem, then. The issue is not so much "if you constructively cut the ball into five pieces, you can't reassemble them into two balls of the same size" as it is "you can't constructively cut the ball into two or more pieces at all". In this sense of constructively, at least.

That is surely not what is meant ordinarily by the claim that there is no explicit paradoxical decomposition. --Trovatore (talk) 18:41, 27 July 2010 (UTC)

Actually I don't see the problem exactly yet. Constructivists are worried about eliminating the law of excluded middle, which is what accounts for indecomposability. I take it you like the law of excluded middle as much as I do. The other source of nonconstructiveness in classical mathematics is AC. General results such as existence of maximal ideals, Hahn-Banach, Banach-Tarski depend on AC. Why can't we define (classical) nonconstructiveness as reliance on AC? Here one can make finer distinctions in terms of countable AC, dependent AC, general AC, as Tsirelson suggested above. What's wrong with that? Tkuvho (talk) 19:16, 27 July 2010 (UTC)

"Reliance on AC" doesn't, on the face of it, tell you anything about the decomposition itself, but only how you prove things about such a decomposition. Don't read too much into the fact that I'm sticking to Banach–Tarski here — I just think it's good to talk about one concrete example, and I think this is a good one because it's involved enough to pick up a lot of the relevant issues.

If we say "there is no explicit paradoxical decomposition", it seems to say more than "we cannot prove the existence of a paradoxical decomposition without using AC". What if there's an explicit paradoxical decomposition, but we just can't prove (in ZF, or perhaps even in ZFC) that it is a paradoxical decomposition? That would still falsify the claim as stated, even if we couldn't prove it.

One solution, as Boris and JR have noted, is simply to reword all the claims in terms of provability. But that weakens the statement more than is perhaps necessary. It is certainly the case, for example, that there is no paradoxical decomposition consisting of analytic sets, and that rules out pretty much anything that a novice is likely to try as a decomposition. Given sufficient large cardinals we can show that even much more complicated sets can't possibly work (for example they can't be in L(R)). Conceivably this last point could be taken to argue that no definition of a paradoxical decomposition can be predicative, because you would have to divide reals into sets based on the behavior of objects more complicated than reals. --Trovatore (talk) 19:32, 27 July 2010 (UTC)

Speaking about choice axiom I did not really mean provability (sorry for wrong use of terms, I am not logic expert). Rather I mean the following. We have the universe of sets (yes, so naive); a number of predicates and operations act on it; they all are canonical (wrong term again? I mean there is no free choice of them); but this is ZF, not ZFC. Now we introduce (postulate) another operation, highly non-unique. We use this bad operation when constructing some objects; these are badly constructed (no one can know which element of the universe is really meant). But still, some are nicely constructed, using only canonical operations. This is what I meant in "Non-Borel set". Boris Tsirelson (talk) 19:57, 27 July 2010 (UTC)

But this is probably too restricting. Now I like to modify the above "definition" as to allow dependent choice of a sequence (not axiom, again, but operation). Boris Tsirelson (talk) 20:01, 27 July 2010 (UTC)

I think before I can say anything sensible I'm going to have to ask you to explain better what you mean by "operations" as opposed to "axioms". For example, consider the powerset axiom; certainly for any given set X, the axiom asserts the existence of a precise object P(X). But then what about the elements of P(X), are you allowing them, or only P(X) as a whole? Are you allowing iteration into the transfinite, and what do you do at limit ordinals? And what is the "dependent choice operation"? --Trovatore (talk) 08:21, 28 July 2010 (UTC)

Can your objection to claims that, for example, "paradoxical decompositions cannot be constructed without AC" be formulated in the spirit of reverse mathematics: is the existence of such decompositions consistent with ZF together with negation of AC? If it is consistent, then apparently claims such as the above are not merely vague but actually inaccurate. Tkuvho (talk) 08:57, 28 July 2010 (UTC)

Oh, certainly they're consistent with the negation of AC. The negation of AC by itself rules out very very little, because the smallest rank of a set that can't be wellordered might be very large, and so AC might as well be true as far as it concerns sets of small rank.

More concretely, it's not too hard to show that there's a wellfounded model of ZF in which R can be wellordered, but P(R) cannot. The wellordering on R is enough to get Banach-Tarski. --Trovatore (talk) 09:07, 28 July 2010 (UTC)

Can one establish a tighter relationship between, say, paradoxical decompositions of R^3 and well-ordering of R? Tkuvho (talk) 10:01, 28 July 2010 (UTC)

(unindent) Surely I am inventing something well-known, but let me try anyway.

Let A(x) be a (well-formed) formula of ZF(C) with no free variables other than x, such that there exists one and only one x satisfying A(x). Then this x will be called famous.

Of course, this is not a definition within ZFC. Of course, the set of famous sets is countable, but this is not the countability within ZFC. I intentionally use truth rather than provability; it is naive, but I do mean "the right" model of ZFC.

For example, the real line is famous, and the first ordinal of cardinality continuum is famous. We may hope that every bijection between these two is non-famous, and every non-measurable subset of R is non-famous.

The first example of non-Borel set given in Borel set#Non-Borel sets is famous, the second is hopefully not.

The dependent choice is a harder matter; for now I leave it aside. Boris Tsirelson (talk) 16:47, 28 July 2010 (UTC)

OK, right. So your famous is what would ordinarily be called "first-order definable, without parameters, over V". I'll keep your terminology because it's shorter.

"Famous" is a bit hard to work with because it's not definable in the first-order language of set theory, but we can look at a larger class, the ordinal-definable sets. Every famous set is ordinal definable, and ordinal definability is definable in the language of set theory. Then a set is hereditarily ordinal definable if it's ordinal definable, all its elements are ordinal definable, all elements of its elements are ordinal definable, etc. The class of all hereditarily ordinal-definable sets is called HOD, and is a model of ZFC.

There is a canonical definable wellorder of HOD.

Now, suppose V=HOD, which is consistent with ZFC. (This would be implied by V=L, which is also formally consistent with ZFC, but which is much less plausible for various reasons.) Then there is a famous paradoxical decomposition of the ball. Specifically, define it as "the least paradoxical decomposition of the ball, in the canonical wellorder of HOD".

On the other hand, suppose HOD(R)-determinacy holds (also consistent with ZFC, though you need large cardinals to prove that). Then there is no famous paradoxical decomposition, because if there were, it would be ordinal definable, and each of its finite number of pieces would also be ordinal definable. They wouldn't necessarily be hereditarily so, because there might be individual elements of R^3 that aren't ordinal-definable, but that's taken care of by going to HOD(R), so the pieces would be HOD(R). But given HOD(R)-determinacy, every subset of R^3 contained in HOD(R) is Lebesgue measurable, so this is a contradiction. --Trovatore (talk) 22:50, 28 July 2010 (UTC)

Thank you, I see: first-order definable, without parameters, over V! I abbreviate it to FODwP(V) and waive the temporary term "famous".

Is FODwP(V) described somewhere in Wikipedia?

Returning to your initial problem, "There is generally some sense in which this claim is true, but unfortunately such a sense is almost never specified", I proclaim Boris thesis :-) (like Church thesis):

The informal notion of an explicit example is formalized adequately by the formal notion of FODwP(V).

Other notions mentioned by you above are probably more interesting for set theory/logic, but address different questions, not this one. Do you agree?

Boris Tsirelson (talk) 06:07, 29 July 2010 (UTC)

Well, so if that's the notion that the terms like "explicit" are meant to capture, then I would argue that we do not know whether there is an explicit paradoxical decomposition, an explicit nonprincipal ultrafilter on the naturals, an explicit Vitali set, etc etc etc. Because, while it's reasonable to say that V=L has been settled negatively, it's a lot harder to claim that about V=HOD. And that's why I brought up the matter in the first place. --Trovatore (talk) 06:31, 29 July 2010 (UTC)

Oh, really? I had the (wrong) impression that it is proved that every FODwP(V) set of real numbers is Lebesgue measurable. Or maybe the point was that the opposite is not provable? Anyway, I'd be rather satisfied with phrases like "this is an explicit example" and "this is not", the second being interpreted as "no one made it explicit" rather than "it is proved that it cannot be made explicit".

And, by the way: the notion of FODwP(V) is a mix of theory and meta-theory, but can be internalized at the expense of a large cardinal. Boris Tsirelson (talk) 07:38, 29 July 2010 (UTC)

Right, the second thing you said: The opposite is not provable.

Your suggestion as to how to fix the articles might work. Would require some careful choice of words. I'll ponder it.

Here's the thing, though. Suppose you make the following definition:

X is the HOD-least non-Lebesgue-measurable set of reals in HOD, if there is a non-Lebesgue-measurable set of reals in HOD; otherwise, X is the empty set

The question of whether X is Lebesgue measurable is independent of ZFC.

Now, this is a definition; it fits the criterion you gave. But it's a singularly unhelpful one. Good luck figuring out anything at all about X from this definition.

Partly that's because it's so terribly impredicative. Suppose you want to know whether π is an element of X — a question about whether a real is an element of a set of reals. The answer may depend on subtle questions about sets of sets of sets of sets of reals, or even much higher types than that.

But suppose you ask a bit more of your definitions. Suppose you say, "if I want to know if a real is in a 'nicely definable' set of reals, at the very most I should have to quantify over reals, not anything more complicated". Then (granted large cardinals) it is true that every "nicely definable" set of reals is Lebesgue measurable.

So that's my quandary — whether to try to work in something about that. I'm getting closer to the idea that that's too ambitious, and it's probably better to reword so as to say that the constructions "do not give" us an explicit example, rather than that there is none. --Trovatore (talk) 08:06, 29 July 2010 (UTC)

"Suppose you want to know whether π is an element of X..." — but this may be a problem even for a Borel set X. Say, "X = if ZFC is consistent then R else empty".

"So that's my quandary — whether to try to work in something about that." — as for me, the matter is interesting enough for using both approaches.

"it is true that every "nicely definable" set of reals is Lebesgue measurable" — I guess it is about the projective hierarchy; right?

"V=L has been settled negatively, it's a lot harder to claim that about V=HOD" — probably you have a reason to believe in V=HOD≠L, but for now I am inclined to believe that V≠HOD.

Boris Tsirelson (talk) 08:32, 29 July 2010 (UTC)

Scope of project, use of banner

I am not a member of the project so I would not post its banner, which seems to imply taking some responsibility. Right?

Anyway, I don't know where in the world of probability, statistics, and mathematical modeling you all would draw the line. For example, here are some articles claimed by no Wikiproject: Chinese restaurant process, Urn model, Multivariate Polya distribution, Generalized Dirichlet distribution. Meanwhile both the Math and Stat banners deck Dirichlet process,Dirichlet distribution, and Stochastic process.

Afterthought ... Markov process, none. Markov chain, both. --P64 (talk) 04:07, 27 July 2010 (UTC)

I think all those fall in the scope of our project. They are listed on the list of mathematics articles, which is very complete. That list is updated by a bot based on article categories.

Our use of talk page tags is not very complete. The main limiting factor in adding talk page tags is human effort – it takes time to assess the quality and field of each article. However, because we have the list of articles already, it is not a high priority. I have been slowly working on assessing articles, and there are several tools on my user page that can make the process less labor intensive. — Carl (CBM · talk) 04:20, 27 July 2010 (UTC)

It looks like there are 104 articles in Category:Stochastic processes that do not have a {{maths rating}} tag [1]. I will work on those over the next few days, if nobody else gets to them first. As always, to assess an article, you should to fill in all three of the class (quality), priority, and field parameters, as explained on Template:maths rating. — Carl (CBM · talk) 04:26, 27 July 2010 (UTC)

Ah, so, Math is well organized in this respect too, with an automated list of articles. (Elsewhere some of us use banners without any assessment at all, simply to notify each other that articles exist.)

Thanks for the whole explanation. It fills a few gaps for me, beside the point of my question --answers what I didn't know I was asking ;-) P64 (talk) 01:30, 29 July 2010 (UTC)

Inter-wiki links to nowhere

81.214.68.90 (talk · contribs) has been adding many inter-wiki links to the Turkish Wikipedia to various articles including many mathematics articles. So far as I have checked, all these links lead to nowhere — there are no corresponding target articles on the Turkish Wikipedia. I have no idea how to deal with this except to tell you-all and hope someone among you knows what to do about it. JRSpriggs (talk) 15:49, 29 July 2010 (UTC)

Now I have found one which does have a target, namely Infinite set. However, from the formulas, the target appears to be the equivalent of our Infinite series article. JRSpriggs (talk) 16:03, 29 July 2010 (UTC)

Anyone have a bot to check these, and delete those which point to an empty article? — Arthur Rubin (talk) 07:25, 30 July 2010 (UTC)

Homology theory

The article has had a radical edit and update, essentially a new article pasted in. People had better look to see what has gone on. Charles Matthews (talk) 07:46, 28 July 2010 (UTC)

It appears that all the historical material has been removed, which is a pity. Tkuvho (talk) 08:01, 28 July 2010 (UTC)

The person who made the change has only done some rv's and similar maintenance up to now. So for a first edit this surprisingly good, but I'd especially look out for newbie issues such as encyclopedic tone etc. I don't think the previous version was all that well written that the change should just be reverted, but someone familiar with the subject should take a swipe at merging the two versions. This is a top priority subject so it wouldn't hurt to have a few more editors looking at it anyway. I'll put an FYI on the person's talk page.--RDBury (talk) 14:46, 28 July 2010 (UTC)

Not surprising at all in fact, as he is a first rate topologist (joint papers with Hopkins, book, etc.). But the historical material should be restored. Tkuvho (talk) 15:10, 28 July 2010 (UTC)

There would be a case for putting back some historical stuff, but if so then it really ought to be a bit more systematic rather than just a random selection of observations. There is currently an external link to an article by Peter Hilton which would be a good source for that, and there is the voluminous book by Dieudonne if anyone wanted to dig deeper. There is also plenty of non-historical stuff that could usefully be added, which I might do later, but I will be off the web for the next three weeks. Neil Strickland (talk) 16:58, 28 July 2010 (UTC)

Dieudonne's histories are not reliable sources. They have serious errors. Ozob (talk) 23:46, 28 July 2010 (UTC)

I don't have my own opinion on this question (mathematical history is not really my thing) but there are very complimentary reviews by Saunders MacLane and Peter Landweber (in MathSciNet and Zentralblatt respectively) and both of those people deserve to be taken seriously. Neil Strickland (talk) 06:27, 29 July 2010 (UTC)

My own experience trying to trace the history of a particular result was that Dieudonne referenced only the first work on the problem, declared that this one page sketch solved it, and ignored all the later work. But it took decades of work by some very good people to turn that sketch into a correct proof! Someone else I talked to had a similar experience. This is why I say his histories are not reliable sources. Perhaps they are better for matters which he experienced first-hand; I don't know. Ozob (talk) 23:45, 29 July 2010 (UTC)

Dieudonne was well known for having strong opinions. While I didn't read his history of algebraic topology, his article "The Historical Development of Algebraic Geometry" in American Math Monthly is very doctrinal and cannot be used as an encyclopaedic source for history of algebraic geometry, unless you view all of algebraic geometry as a prelude to Grothendieck's scheme-theoretic approach. By the way, my reading of MacLane's and Landweber's reviews is that Dieudonne's book is grandiose, but they make it clear that it's far from perfect either mathematically or historically. Arcfrk (talk) 08:21, 30 July 2010 (UTC)

I don't mean to question the mathematical competence of the newest contributor, but I am concerned that after the latest edit the article has become visibly more technical: verbal explanations have been replaced with text saturated with formulas; I don't consider that an improvement. On the subject of removing and putting back historical material: while it would be nice to have a systematic treatment written by a professional historian of mathematics, it's also somewhat unrealistic to expect this to materialize any time soon. Let us also keep in mind that it's unwise to rely for history on an article by one of the principals, especially if it's entitled "A subjective view"! If the quality of the historical section was acceptable, it should be restored in the meantime. Arcfrk (talk) 08:21, 30 July 2010 (UTC)

The lengthy first section "the general idea" is mostly new. It discusses the 1-dimensional case in detail. I think it is off the mark somewhat. The focus on the 1-dimensional case is misleading. Since homology in this case is most easily described as the abelianisation of the fundamental group, the whole discussion may give an inaccurate idea of where the interest of homology theory lies. The applications given later on include fixed point theorems, whose proof involves relative homology which was not even discussed. Tkuvho (talk) 09:21, 30 July 2010 (UTC)

Detailed comments should go to Talk:Homology theory. Charles Matthews (talk) 09:59, 30 July 2010 (UTC)

Another point that is an essential feature of the theory but is not discussed early enough is the dependence on the coefficients. Tkuvho (talk) 11:13, 30 July 2010 (UTC)

IP edits of group theory articles

In the past day or so, Special:Contributions/98.81.17.64 has made quite a few edits to group theory articles, including Finite group, Abelian group, and Burnside problem. Many of them, frankly, strike me as sophomoric: for example, standard mathematical language is replaced with something that sounds better as an English phrase, but does not carry the same meaning, or even makes little sense mathematically. In addition, he doesn't seem to be sufficiently familiar with editorial practices of Wikipedia: some red links to names of mathematicians have been removed, while in front of links to names of others, nationalities have been added (selectively, and not always correctly); changes to See also sections and categories don't seem to be improvements. Can people, please, take a look at these articles/IP's contributions and weigh in on whether the changes should be summarily reverted, or are there linguistic improvements there that are worth retaining? Arcfrk (talk) 07:55, 5 August 2010 (UTC)

I've tidied Finite group, retaining a table we can probably live without. In hindsight, I should probably just have reverted all the edits. I haven't looked at Abelian group or Burnside problem. -- Radagast3 (talk) 09:56, 5 August 2010 (UTC)

Irrational rotation

Irrational rotation has been moved. I suggest that the move be reverted. IIRC, the original content was about C*-algebras arising from irrational rotations of the circle. Mct mht (talk) 16:17, 5 August 2010 (UTC)

I have reverted the latest additions and moved back to the old title. Arcfrk (talk) 18:40, 5 August 2010 (UTC)

Reverted (i.e. deleted) all mentioning of plane rotations, pigeon-hole principle, units of angle... just because you dislike something? Where do you propose to write about these things?! Incnis Mrsi (talk) 19:24, 5 August 2010 (UTC)

I didn't mean to offend, Incnis Mrsi. Perhaps the article doesn't convey this well yet, but, as it was first written, it's meant to be about the dynamics of a irrational rotation on the circle and a corresponding family of C*-algebras. Surely there's a more suitable home elsewhere for the contents you mention above? Mct mht (talk) 23:10, 5 August 2010 (UTC)

Note that the link Irrational rotation algebra is red yet, and the article does not give any definition of an algebra, only states that irrational rotations have much use in C* algebras and dynamical systems. It seems that the article covers nothing more than an irrational rotation of a circle, a thing trivial for mathematicians, and presented in a quite unintelligible way used in some narrow researches. Incnis Mrsi (talk) 08:11, 6 August 2010 (UTC)

Looking at this version I can't see anything on "irrational angle" in the sources. The closest is "irrational screen angles" mentioned here except that's about printing not maths, and the link is a redlink (or the PrintWiki equivalent). So the additional content looks like OR.--JohnBlackburne^words_deeds 23:30, 5 August 2010 (UTC)

OR is that irrational angles is those that are not rational? Incnis Mrsi (talk) 08:11, 6 August 2010 (UTC)

Indefinite logarithm

I came across indefinite logarithm. It appears to be a neologism which I only found in this arxiv paper. Should the article be deleted? Jakob.scholbach (talk) 16:57, 7 August 2010 (UTC)

illustration of logarithm vs. hyperbolic geometry?

The logarithm article currently says

“ In geometry the logarithm is used to form the metric for the half-plane model of hyperbolic geometry. ”

Can anyone think of an illustration underlining this? Thanks, Jakob.scholbach (talk) 13:30, 7 August 2010 (UTC)

See Beltrami–Klein model#Distance metric for a disk model metric that uses log. I assume there is a half-plane model that uses a similar one but I couldn't find it in WP. It's a bit of an overstatement in the article though since there are several models of hyperbolic geometry.--RDBury (talk) 14:24, 7 August 2010 (UTC)

PS. The original edit has a link to Hyperbolic motion#Elementary half-plane geometry and if you read that the statement makes more sense. The link was lost in subsequent edits though. In any case, the statement is unsourced and to me it doesn't seem that helpful as an example. I'll add a fact tag for now.--RDBury (talk) 14:40, 7 August 2010 (UTC)

Our article on the Poincaré half-plane model of hyperbolic geometry seems to focus (at present) on the symmetry group to the exclusion of everything else. This is unfortunate. I may try to add some other stuff. JRSpriggs (talk) 23:36, 7 August 2010 (UTC)

Just replace "half-plane model" with "Cayley–Klein model" (which some people insist on calling "Beltrami–Klein model"). The formula also involves cross-ratio, but the connection to logarithmic function is simpler than in the conformal models (disk or half-plane). Arcfrk (talk) 01:43, 8 August 2010 (UTC)

Hm. Actually I'm wondering whether this is worth mentioning in the logarithm article. What do you think?

I already polled here some time ago, but since I'm still struggling with balancing the material, let me ask again: what are the most important applications/occurrences of logarithms in maths and beyond? (so far, we have [with varying quality]: Richter scale, slide rule, pH, decibel, Weber-Fechner law, mental representation of numbers, entropy in physics and information theory, fractal dimension, Hausdorff dimension, prime number theorem, Benford's law, musical intervals).

I know the collaboration of the month is dead, but I'd be most happy with a few more minds working on that topic! Jakob.scholbach (talk) 21:45, 8 August 2010 (UTC)

Perhaps this is already there? The floating point representation of real numbers used by computers has an exponent which is the characteristic of the logarithm of the number. By the way, does our article on logarithms mention the traditional division of the logarithm into a characteristic (integer part) and mantissa (decimal fractional part)? JRSpriggs (talk) 00:04, 9 August 2010 (UTC)

This is mentionned but only in a crappy way. I'll work on that. Thanks for pointing it out. Jakob.scholbach (talk) 18:28, 9 August 2010 (UTC)

Logarithmic plots are very important, but haven't been explicitly named. I stand by hyperbolic metric. Lyapunov exponents are a possibility. Arcfrk (talk) 10:02, 9 August 2010 (UTC)

Thanks. Actually I forgot to mention logarithmic plots; they are already covered. Anything else? Jakob.scholbach (talk) 17:54, 9 August 2010 (UTC)

Ka Pat number

Comments welcome on whether this is a hoax or where confirmation might be found - the only references offered are Arabic, and English-language searches return nothing. JohnCD (talk) 20:43, 7 August 2010 (UTC)

Author has not replied to questions; article now at AfD here. JohnCD (talk) 19:30, 12 August 2010 (UTC)

Eyes on P versus NP problem

In light of Vinay Deolalikar's proposed proof of the P ≠ NP problem, the article P versus NP problem could use some eyes.

CRGreathouse (t | c) 12:57, 10 August 2010 (UTC)

FYI, the article Vinay Deolalikar is on AfD. Sławomir Biały (talk) 11:16, 12 August 2010 (UTC)

Hehe I was waiting for that to show up here:).

On a related note - do you consider the Polymath project notable enough to get its own article? So far there's a redirect to Timothy Gowers article, where it is shortly mentioned.--Kmhkmh (talk) 13:31, 12 August 2010 (UTC)

Coverage at level of, say, an article in the American Math. Monthly might justify an article of its own. Charles Matthews (talk) 14:11, 12 August 2010 (UTC)

Listing numbers and topological invariants

I just created the stub titled Listing number, and I was going to add it to the list of topological invariants, but that doesn't exist, and neither does Category:Topological invariants. The list of topology topics doesn't even have a section for those, and seems possibly deficient otherwise. Hence some tasks:

• Create the list of topological invariants.
• Create the category.
• Improve the present list of topology topics.
• Improve the now severely stubby article titled Listing number.

Michael Hardy (talk) 19:24, 12 August 2010 (UTC)

See also Topological property (and my remark Talk:Topological property#What about homology?). Boris Tsirelson (talk) 19:35, 12 August 2010 (UTC)

And see also the "See also" section at the bottom of the page of Topological property as well as the other pages

• topology glossary
• List of general topology topics
• List of geometric topology topics
• List of algebraic topology topics
• Publications in topology 78.146.39.137 (talk) 22:07, 12 August 2010 (UTC)

An old notation for factorial

I mean n! written as n resting in a sans serif L (more or less) as early in [2]. I'm interested because I have just come across this in connection with Thomas Jarrett, who introduced the notation around 1830. Any TeXperts able to put this on the page, at least? Charles Matthews (talk) 17:54, 12 August 2010 (UTC)

How about ${\begin{array}{|c}n\\\hline \end{array}}$ but maybe too big. --Salix (talk): 22:32, 12 August 2010 (UTC)

Not too bad. Here's an image from one of Jarrett's books for comparison. CRGreathouse (t | c) 00:20, 13 August 2010 (UTC)

(ec) You could try ${}^{\displaystyle {|}}\!{\overset {\displaystyle {n}}{\displaystyle {\text{---}}}}$, but you might consider making an image file. RobHar (talk) 00:33, 13 August 2010 (UTC)

Or a pure html/css solution n--Salix (talk): 07:04, 13 August 2010 (UTC)

Neat! RobHar (talk) 00:55, 14 August 2010 (UTC)

Thanks, I've put that in the article. Charles Matthews (talk) 07:37, 13 August 2010 (UTC)

Merge discussion

Okay, we have Angular distance and Angular diameter distance and Angular diameter. I am proposing the first two be merged into the third. Discuss at:

Talk:Angular_diameter#Merge_discussion Casliber (talk · contribs) 20:58, 13 August 2010 (UTC)

The second two are more astronomy than math. Also, with several million articles, it shouldn't be surprising that there are some with similar names, though it's definitely worth checking.--RDBury (talk) 14:18, 14 August 2010 (UTC)

pseudo-Riemannian geometry

I just noticed that we have no page titled pseudo-Riemannian geometry. Should we? Michael Hardy (talk) 02:55, 17 August 2010 (UTC)

Is that different from semi-Riemannian geometry? --Trovatore (talk) 02:56, 17 August 2010 (UTC)

I don't know. I'm definitely not a differential geometer. Michael Hardy (talk) 03:11, 17 August 2010 (UTC)

Moot point since we don't really have an article on semi-Riemannian geometry either.--RDBury (talk) 03:28, 17 August 2010 (UTC)

A Pseudo-Riemannian manifold is the same thing as semi-Riemannian. I have the impression that the latter term is used more frequently than the first, but I might be wrong. Here one of the main objects of interest is Lorentzian geometry because of the connection to general relativity. The redirect from semi-Riemannian geometry to Riemannian geometry is not helpful. There is also a sub-Riemannian geometry, related to Carnot-Caratheodory metrics. Tkuvho (talk) 04:07, 17 August 2010 (UTC)

I think the frequency of use of the terms "Pseudo-Riemannian" vs. "Semi-Riemannian" is heavily field dependent. The former is common with mathematical physicists with a connection to GR. The latter seems more common with pure "Geometers". I also think it makes more sense to redirect semi-Riemannian geometry to semi-Riemannian manifold.TimothyRias (talk) 07:30, 17 August 2010 (UTC)

A quick note: in differential geometry a manifold sometimes comes equipped with a quadratic form in each tangent plane. If the form is degenerate, we may be in sub-Riemannian geometry. If the form is definite, we are in the Riemannian case (positive definite form), in the semi-Riemannian case (eigenvalues of different sign), or more specifically Lorentzian case (all but one eigenvalues are the same sign). Tkuvho (talk) 10:03, 17 August 2010 (UTC)

Pre-emptively redirecting pseudo-Riemannian geometry to semi-Riemannian geometry would make sense at this point; otherwise people might create and work on separate articles without being aware of each others' work. But unfortunately that's not allowed. Michael Hardy (talk) 23:04, 17 August 2010 (UTC)

I have made them both redirects to pseudo-Riemannian manifold. --Trovatore (talk) 08:48, 18 August 2010 (UTC)

ALMOST GA-class: Dirac Delta Function

Dirac Delta Function GA-class assessment pending only on more references! Please help find and cite references! Adavis444 (talk) 03:17, 17 August 2010 (UTC)

Here's a better link: Dirac delta function (lower-case initials except for "Dirac"; the other one of course redirects). Michael Hardy (talk) 23:02, 17 August 2010 (UTC)

IMU prizewinners

Some articles to expand/keep an eye on: Fields medal: Elon Lindenstrauss, Ngô Bảo Châu, Stanislav Smirnov, Cédric Villani.

Nevanlinna prize: Daniel Spielman.

Carl Friedrich Gauss Prize: Yves Meyer.

Chern Medal Award: Louis Nirenberg Fm2010ax (talk) 07:42, 19 August 2010 (UTC)

Edited Clenshaw Algorithm, have some questions

When I'm looking for some specific information and come across a WP math article that's poorly written, I often just quickly move on to another source (PlanetMath, DLMF, etc.) and forget about it. Today I overcame that lazy tendency and edited an article. In fact, completely rewrote it: the Clenshaw algorithm. I happened to have a copy of the book Fox and Parker "Chebyshev Polynomials in N.A" lying on my desk, so it was easy.

I still have some questions about how WP works. The original article was lacking references, which I added. When will the "This article does not cite any references" flag disappear? Another request: might someone have a quick look at the article and fix any formatting blunders I may have introduced? I'm competent with TeX, but I've noticed some of my formulas rendered strange. Thanks! Gagelman (talk) 22:21, 17 August 2010 (UTC)

If you think that you have added enough references to support the claims in the article, then remove the "{{Unreferenced|date=December 2009}}" template yourself.

Tex can be rendered in two different ways here. The thin-line version (or ugly version to my way of thinking) is the default. To get the pretty version, the formula must contain any of several things which cannot be rendered in the default version (including "\frac") or "\!" or "\," (which also adds a blank space except when it appears last). I usually put "\," just before the "</math>". If I intend to show a comma or period following the formula, I put that punctuation between the "\," and the "</math>". JRSpriggs (talk) 23:56, 17 August 2010 (UTC)

Thanks! I'm getting the hang of things slowly. Your comments about TeXing formulae are also really helpful. Gagelman (talk) 11:23, 19 August 2010 (UTC)

FAR of 0.999...

I have nominated 0.999... for a featured article review here. Please join the discussion on whether this article meets featured article criteria. Articles are typically reviewed for two weeks. If substantial concerns are not addressed during the review period, the article will be moved to the Featured Article Removal Candidates list for a further period, where editors may declare "Keep" or "Delist" the article's featured status. The instructions for the review process are here. — Preceding unsigned comment added by TeleComNasSprVen (talk • contribs)

To clarify: the article was marked as FA, so the "review" is to see if it should stay that way. Anyone can unilaterally start one of these reviews. — Carl (CBM · talk) 00:06, 21 August 2010 (UTC)

n-ball

A few months ago, someone replaced the redirect to the mathematical concept of n-ball at n-ball with an article on a computer game with that name. Later, someone else inserted a dab link at the top. Question to those familiar with naming and dab policies: since the mathematical meaning of the term appears to be more commonly used than the computer game meaning, shouldn't the redirect be restored and the present article moved to something like "n-ball (computer game)"? Arcfrk (talk) 23:30, 22 August 2010 (UTC)

Even simpler would be for the computer game article to be deleted. The article currently contains no evidence that the game is at all notable (I have yet to look for such evidence myself). Algebraist 23:36, 22 August 2010 (UTC)

I've moved it to n-ball (game) and then changed the new n-ball redirect page to point to ball (mathematics), to which I've added a suitable dablink in a hatnote. I also be a "main" tag on the section titled "n-ball" in the article called n-sphere. Only two articles link to n-ball, and neither appears to intend the video game; they both appear to intend the mathematical concept. Finally, I put an "orphan" tag on n-ball (game). Michael Hardy (talk) 01:26, 23 August 2010 (UTC)

Abel, sums

These are currently two different things:

• Abel's sum formula
• Abel's summation formula

How should we organize them? Michael Hardy (talk) 03:43, 23 August 2010 (UTC)

Just out of curiosity, what is the relationship (if any) between this and the Euler–Maclaurin formula? Grammatically, I would think that an X formula would be formula that helps you to do X. So a summation formula would be a formula that helps with summation, but it sounds wrong to have a formula that helps with sum. Maybe 'Abel's sums formula' is more grammatical, depending on which side of the Atlantic you were born and how old you are. Or maybe 'Abel's sum' since a sum is a formula. Also, 'Abel's sum formula' gets 0 hits on Google books so I'd say just PROD it.--RDBury (talk) 16:45, 23 August 2010 (UTC)

Completeness

The completeness disambiguation page lists a number of mathematical senses of the word, and I wonder if we need another one. See this discussion. The Chebyshev polynomials are asserted to be a "complete" orthonormal basis of the set of functions quadratically integrable with respect to the measure

${dx \over {\sqrt {1-x^{2}}}}$

on the interval [−1, 1]. I take that to mean there are "enough" Chebyshev polynomials, in that no nonzero function on that space is orthogonal to all of them. That is quite a different thing from asserting that this L² space is complete in the sense of convergence of Cauchy sequences. After all, if you take all but one of the Chebyshev polynomials, they still span a complete inner product space; that space is just not all of the L² space, and that set of polynomials is not "complete" in the relevant sense. Thus complete metric space is not an appropriate link. Does some other link exist that is appropriate? Michael Hardy (talk) 19:57, 23 August 2010 (UTC)

Something relevant: Orthonormal basis#Incomplete orthogonal sets; Defective matrix ("that does not have a complete basis of eigenvectors"); Generalized eigenvector ("full set of linearly independent eigenvectors that form a complete basis"); Two-state quantum system ("complete basis spanning the space..."). Does "basis" already imply "complete"? Boris Tsirelson (talk) 20:16, 23 August 2010 (UTC)

Strangely enough, Banach space does not mention "basis" (except for this stupid Hamel basis). Ah, we have Schauder basis; this is better. Boris Tsirelson (talk) 20:29, 23 August 2010 (UTC)

To Michael Hardy: You said "... no nonzero function on that space [quadratically integrable] is orthogonal to all of them [Chebyshev polynomials].". Suppose A is a non-Lebesgue measurable subset of [-1,0]. Let f(x)=2·1_A(x)-1 using the indicator function for A. Then (f(x))²=1. So f is quadratically integrable. But it cannot be orthogonal to any Chebyshev polynomial because the integral cannot even be performed. JRSpriggs (talk) 13:46, 24 August 2010 (UTC)

It's quite obvious that he meant "quadratically integrable" to be a synonym for L², which in particular requires the functions to be measurable.—Emil J. 13:57, 24 August 2010 (UTC)

Moderately cute example, but Emil is right. Michael Hardy (talk) 17:02, 24 August 2010 (UTC)

....also, JRSpriggs' comment doesn't really bear on the issue that this is about. Michael Hardy (talk) 17:02, 24 August 2010 (UTC)

Merge proposal: List of geometric shapes

I have proposed to merge List of geometric shapes, List of mathematical shapes, and List of surfaces into one consolidated article. The discussion is here.  --Lambiam 14:16, 24 August 2010 (UTC)

Let me guess. It's going to be called Outline of geometric shapes. —Preceding unsigned comment added by 92.29.76.81 (talk) 21:03, 24 August 2010 (UTC)

Closed form

Closed-form formula and Closed-form expression seem to be about the same material, with the latter being the main article and the former an orphan written by financial types. Should these be merged, or are they actually different?

CRGreathouse (t | c) 21:36, 24 August 2010 (UTC)

Bleh. Just redirect the former. There's nothing worth keeping here. Sławomir Biały (talk) 22:58, 24 August 2010 (UTC)

Agreed, make it a redirect, just like Closed formula and Closed expression. Move on, folks, there's nothing worth seeing here.  --Lambiam 00:05, 25 August 2010 (UTC)

Move proposal: move the "Domain (complex analysis)" entry to a new entry "Domain (mathematical analysis)"

Hi to everyone: in the following section I have reported a more detailed version of a move proposal concerning the Domain (complex analysis) stub which I submitted to Oleg Alexandrov here yesterday: my proposal is exactly to move it to a new entry Domain (mathematical analysis) and to create redirection pages Domain (complex analysis) and Domain (real analysis), in order to not miss any link to its present content and to redirect there new links to related concepts in real analysis there. I'll edit the new entry according the scheme described below: I also describe below the reasons that inspired my proposal.

1. The concept of domain is ubiquitous in all parts of mathematical analysis, not only in complex analysis: think of partial differential equations, the theory of Sobolev spaces, theory of functions of several variables and the like, just to say a few: a description of these concept presently lacks in Wikipedia. Therefore it would be interesting to have an entry that explain the basic concept of what a domain in mathematical analysis is (an open connected set), why it is so important in analysis, and redirecting the interested reader to many Wikipedia entries related, already existing or to be created by wikipedians.
2. I think the best way to start such a entry is a section on domain in real vector spaces: every complex domain is isomorphic to a real domain, and the first classification of such manifolds is done according to the the smoothness caracteristic of their boundary. Main classes should be
   • Caccioppoli sets
   • Sets whose boundary is $C^{k}$-smooth
   • Sets whose boundary is Holder or Lipschitz continuous
   • Lyapunov manifolds (which where the first domains on which potential theory was developed by Aleksandr Lyapunov)
   • Piecewise smoot domains
   • Domains with the cone property or the flexible cone property
   • And a subsection about the common notations for such domains
3. Domains can also have convexity or concavity properties which have important applications in the fields of optimization and partial differential equations
4. Domains in complex spaces/manifolds have the same topological structure as domains in real vector spaces, but have also other distinctive characteristics: this would hopefully be another important section of the entry.
   • Analytic polyhedrons (and Weil domains)
   • Classical domains
   • Hartogs domains
   • Holomorphically convex domains
   • Lineally convex domains: this should be merged with the complex convexity entry
   • Pseudoconvex domains
   • Polynomially convex domains
   • Reinhardt domains
   • Runge domains
   • And a subsection about the common notations for such domains

It would also be nice to create an historical section with two subsections, one for real domains, the other for the complex ones, even if in this last case the complexity could be serious. And last, but not least, in the discussion page it would fit well a dedication to Oded Schramm, who was the creator of the stub. Well, this is the project exposed in the most detailed way I was able to. What do you think about it? Thank you in advance for your attention and suggestions, and Best Regards. Daniele.tampieri (talk) 21:25, 24 August 2010 (UTC)

As for me, it should be excellent. Boris Tsirelson (talk) 19:48, 26 August 2010 (UTC)

A better solution would be to first create a new article Domain (mathematical analysis) which will treat some of the topics outlined above. Presently, the article Domain (complex analysis) consists only of a few sentences, but more can certainly be said about complex domains in the case of one complex variable, whereas the more general article exists only as an outline, and it will require a lot of work to flesh out this outline in an article, if that ever happens. I just don't see any merit in deciding at this point. Arcfrk (talk) 21:00, 26 August 2010 (UTC)

Integrable function

We have a dreadful article Integrable function that has gone pretty much untouched since 2004. It's almost worth redirecting to the much more informative article Integral for now. Sławomir Biały (talk) 21:08, 26 August 2010 (UTC)

"In mathematics..."

In seems that one of our unwritten rules of style is that every math article should begin with "In mathematics..." or something almost as vague such as "In geometry...". This phrasing appears as an example in WP:MOSMATH, but it doesn't say it must be used, only that the lead should indicate the field of mathematics to which the subject belongs. From what I've seen, it's rare for articles outside this project to begin an article this way so I thought it would be a good idea to question this convention. WP:LEAD gives as it's example of a lead sentence "The electron is a subatomic particle that carries a negative electric charge." It seems to me that we could use a similar style in more of the articles in this project. For example, drop the "In mathematics," in the lead of Prime number, "In mathematics, a prime number (or a prime) is a natural number that has exactly two distinct natural number divisors: 1 and itself."--RDBury (talk) 14:29, 21 August 2010 (UTC)

It is however important to make the context of the article clear in the lead. The typical "in mathematics" phrase does just that, albeit in a stylistically poor way. In the case of prime number it is completely superfluous, since the use of the word number already clearly establishes this context. In some cases it may be harder to get rid of though.TimothyRias (talk) 15:40, 21 August 2010 (UTC)

What about the article local Euler characteristic formula. I created it with "In Galois cohomology, ...", but Michael Hardy changed it to "In the mathematical field of Galois cohomology, ...". I think this is a much more relevant example than that of prime number, since most mathematics articles refer to things that most people have never heard of. I, for one, have no preference either way, but I know that Michael Hardy is a strong proponent of the "In mathematics, ..." beginning. A stylistically better way to start this article might be "The local Euler characteristic formula, due to John Tate, is a result that computes Euler characteristics in the Galois cohomology of non-archimedean local fields.". However, most people would have no idea what any of this is about. Putting "In mathematics, ..." in front at least singles out which of the many words in the first sentence actually tell you what the article is about. RobHar (talk) 16:10, 21 August 2010 (UTC)

I don't think it's generally a problem, or even particularly bad writing. But there's no requirement that the context has to be provided in any particular way, so you can always rephrase it.

The motivation is just that we need to establish context somehow, particularly for esoteric topics, because an untrained reader might honestly have no idea what subject an article is about For example:

• In commutative algebra, the Krull dimension of a ring R, named after Wolfgang Krull (1899–1971), is the number of strict inclusions in a maximal chain of prime ideals.
• In constraint satisfaction, constraint inference is a relationship between constraints are their consequences.
• In applied mathematics and the calculus of variations, the first variation of a functional J(y) is defined as the linear functional $\delta J(y)$ mapping the function h to
  $\delta J(y)(h)=\lim _{\varepsilon \to 0}{\frac {J(y+\varepsilon h)-J(y)}{\varepsilon }}=\left.{\frac {d}{d\varepsilon }}J(y+\varepsilon h)\right|_{\varepsilon =0},$
  where y and h are functions, and ε is a scalar.

These are all tagged as stubs. For a long and well-established article, the lede may already provide enough context that an explicit "in" phrase is unneeded. — Carl (CBM · talk) 16:15, 21 August 2010 (UTC)

I think it's generally harmless and in two ways useful. First it can alert readers that an article is mathematical - important as much of the language of mathematics is different from normal English language, often in subtle ways. An article's lede is often non-technical but it can still be very mathematical, and use language in a way readers are not familiar with. Prime number is a good example as (at a stretch) you can imagine a reader looking for the name of a head of state, confusing "Prime minister" and "Number One" finding it - "Prime" redirects there also.

Second the link mathematics acts almost as a FAQ link, to a page which summarizes the topic and has links to many mostly less-advanced topics, for e.g. background reading, something that came up only a month ago. It's not possible to write a maths article that suits everyone, but I think a mathematics link at the start strikes a good balance between making an article accessible and presenting the topic straightforwardly. --JohnBlackburne^words_deeds 16:50, 21 August 2010 (UTC)

Actually in many cases the link to the mathematics article is a bad thing. My view on this (not sure if anyone has written it up anywhere) is that links from specialized articles to general articles should never make huge jumps in generality. So for example it would be reasonable to start the topology article with in mathematics, topology is..., because topology is sufficiently general that mathematics as a whole is arguably only one step above it.

But it would be very very bad to have a link to mathematics in the first line of Stone–Čech compactification. Instead that one should start with something like in the mathematical field of general topology, the Stone–Čech compactification is.... And the word mathematical should absolutely not be a wikilink. That would be overlinking, because if you're looking at the level of detail of the Stone–Čech compactification, a general article on mathematics would just be noise. --Trovatore (talk) 19:35, 21 August 2010 (UTC)

Please tell me you're not suggesting to regulate with which phrase the lead of math article has to begin. Suggesting any such thing in a style guide seems to be utterly ridiculous to me.--Kmhkmh (talk) 17:39, 21 August 2010 (UTC)

Thanks for the comments. What I'm getting is that the "In ..." phrase performs a few functions that shouldn't be lost, and I think the functions themselves should be documented in MOSMATH but without a prescription of how to achieve them. Certainly if the lead in article already does everything it's supposed to do then we shouldn't be adding "In ..." to the front just to make the article conform to some standardized phraseology. What I'd suggest to to have more than one example in MOSMATH so editors can have more options to choose from. In the Prime number example, I'd have to agree with TimothyRias in that readers will know it's an article about mathematics as soon as as they hit the words "natural number". I'm not sure how a link to the Mathematics article will clarify anything in this case but perhaps there should be a link to Number or Number theory so the reader can get an an idea of where prime numbers play the most significant role. So perhaps the rule which should be documented in MOSMATH, if it isn't already there, is that the lead sentence should always contain a link to a more general subject. I think this may already be implied in the rule that the lead should establish context but there's no harm in spelling it out. The reason I brought this up is that I do (occasionally) look at non-math articles and noticed it's not used elsewhere, so it seems a bit odd that we're using this sort of catch phrase that no else seems to think is necessary. An example is Main sequence, a featured astronomy article which gets as technical as FA generally can; the lead there is "The main sequence is a continuous and distinctive band of stars that appear on plots of stellar color versus brightness." No one saw the need to add "In astronomy..." to the start of that article even in a FA review.--RDBury (talk) 18:45, 21 August 2010 (UTC)

I think rather than merely linking to "a more general subject", the key function of "In mathematics" is to link to a sufficiently general subject that we can reasonably expect all readers to have heard from it, so that even the most mathematics-illiterate reader can get some idea of the context of the article. In your example, the link on star performs a similar function. —David Eppstein (talk) 19:03, 21 August 2010 (UTC)

See above, though, on my point on huge jumps in generality. Mathematics should never be linked from the first sentence of a very specialized article. --Trovatore (talk) 19:36, 21 August 2010 (UTC)

I agree with both David and Trovatore. There's a balance, and we don't want the context link to be too narrow (e.g. group cohomology) or too broad (mathematics). Certainly the context we need to give is broader here than in a journal, because the journal name already gives some context that we don't have as a general encyclopedia. So I wouldn't start a paper of my own with "in mathematical logic" or "in computability theory" but I think it's fine for a general purpose encyclopedia. Hypothetically a reader can follow these links until she arrives at a familiar topic, even if it takes a few steps. — Carl (CBM · talk) 21:14, 21 August 2010 (UTC)

To clarify my position, I have no objection to making "mathematics" or "mathematical" part of the context of even very highly specialized articles. I just don't want the words wikilinked in that case. Everyone knows (more or less) what "mathematics" means; common words should not be wikilinked unless there's a reasonable probability that the typical visitor would actually want to follow the link. --Trovatore (talk) 05:56, 22 August 2010 (UTC)

As long as we're on the topic, it is amusing to recall this edit and its summary. Michael Hardy (talk) 23:30, 25 August 2010 (UTC)

From what I've seen, it's rare for articles outside this project to begin an article this way ...

For what it's worth, articles on technical terms commonly begin "In baseball" (home run, passed ball) and "In the partnership card game contract bridge" (Blackwood convention). Takeout double is a hilarious bad counterexample to the latter. Routinely "baseball" or "card game" and "contract bridge" are linked.

The leads of particular baseball articles never identify baseball as a sport or game, nor do they specify modern or American baseball. The leads of many contract bridge articles do identify bridge as a card game or even as a partnership or trick-taking card game. The articles on particular players, teams, or institutions typically begin with the proper noun (name) and identify the context later in the first sentence.

Here we sometimes specify "mathematical probability theory" (Bayes Law) rather than merely "probability theory" (Chinese restaurant process). Compare "card game contract bridge" versus "contract bridge" or even "bridge". The longer is commendable, in my opinion, because I guess "probability theory" and "contract bridge" fall short of the near-universal reader familiarity that I presume for "geometry" and "baseball" --as well as "mathematics" and "sports".

To me it looks bad and feels wrong to begin a wikipedia article with the indefinite article "A". The preposition "in" is twice long and it feels four times better! (Pardon the mathematics.) --19:34, 30 August 2010 (UTC)

Continuing the last paragraph, note that this section begins with another alternative, from WP:Lead, "The electron is a subatomic particle ...". I'm not sure why I accept that where I would object to "The divisor is ..." or "The theorem is ...". --P64 (talk) 19:41, 30 August 2010 (UTC)

I would be fine with the card game of contract bridge but I would absolutely object to the card game of contract bridge. I hope the analogy is clear. --Trovatore (talk) 19:47, 30 August 2010 (UTC)

I understand the point of the analogy. To specify "card game of contract bridge" (with only one link) fulfills the need that I sense in that case.

I'm not sure that "probability theory" and the one-word term "statistics" work well, as I believe that "mathematical logic" works well. If am right to doubt, the solution is neither "mathematics of probability" nor "mathematical probability" with two links, which is your point here.

P.S. Bayes Law has been revised since I wrote last hour. --P64 (talk) 20:53, 30 August 2010 (UTC)

MOS:MATH

@RDBury: I don't think WP:MOSMATH prescribes any particular way to establish context, it just says "Name the field(s) of mathematics this concept belongs to and describe the mathematical context in which the term appears." There are many ways to achieve that in English, e.g.

• Reverse mathematics is a program in mathematical logic that seeks to determine which axioms are required to prove theorems of mathematics.
• Mathematical logic is a subfield of mathematics with close connections to computer science and philosophical logic.

The main benefit of the "In XXX" method is that it's easy to use for poor writers and ESL speakers, and it can be added by people who don't really know the topic in detail. Fancier writing requires more skill and knowledge. — Carl (CBM · talk) 21:23, 21 August 2010 (UTC)

I'm going to take a stab at tweaking MOSMATH to capture the main points in this discussion. I've just noticed that the example we're using from Continuous function defines Open functions instead, probably something that should be fixed anyway.--RDBury (talk) 14:56, 22 August 2010 (UTC)

Per WP:BEBOLD I did a bit more than tweak, and I removed all the continuous function text, but I don't think the changes are controversial.--RDBury (talk) 17:57, 22 August 2010 (UTC)

I'll look it at a bit more later. I do have one strong objection at first glance, which is to the line about a high-school student or first-year undergraduate. I do not believe anything can be said in general about the level of reader to whom it is possible to give even an informal introduction. That has to be topic-by-topic. There is no particular "intended audience" for Wikipedia as a whole. --Trovatore (talk) 18:05, 22 August 2010 (UTC)

Strongly agree with Trovatore. Paul August ☎ 18:47, 22 August 2010 (UTC)

I edited MOS:MATH to slightly address these concerns. Please improve what I have written. CRGreathouse (t | c) 19:04, 22 August 2010 (UTC)

In my defense, the "high-school student or first-year undergraduate" phrase was in there before I started, I moved it a bit so maybe it was more noticeable. CRGreathouse' changes seem reasonable.--RDBury (talk) 03:09, 23 August 2010 (UTC)

I don't like CRGreathouse's change, but it's a step in the right direction. CRGreathouse (t | c) 19:33, 23 August 2010 (UTC)

I haven't yet read most of this discussion. For now, my position would be this:

• One need not always say "In mathematics,..." but one should always immediately make the lay reader aware that mathematics is what it's about. "In geometry,..." or "In algebra,..." or "In number theory,..." does that. "In Galois cohomology,..." does not.
• I've just deleted the phrase "In mathematics,..." from the article titled prime number. The point here is that what was already there does make it clear to the lay reader that it's a math article. Hence one need not always say "In....". Common sense should guide us on this point.
• In some cases, the very title of the article may be sufficient to alert the lay reader.

Michael Hardy (talk) 17:11, 24 August 2010 (UTC)

I agree--Kmhkmh (talk) 00:17, 26 August 2010 (UTC)

Are open access publications suitable as references?

Hi, at Wikipedia_talk:Verifiability#WP:PAYWALL_and_freely_accessible_sources there is currently a debate about a proposal to encourage providing citations to freely accessible sources in addition to paywalled ones (if they are of equivalent reputation). One user who opposes this argues that "encouraging the use of open access scholarly journals is ... at this point in time, problematic" because:

In my own field, mathematics, almost all open access journals at this point are very low quality journals - basically paper mills with extremely perfunctory/pro forma peer review that make a profit by charging rather exorbitant per page publication fees to the authors.

Is this a widely held view?

Regards, HaeB (talk) 12:49, 26 August 2010 (UTC)

Quite strange. I am proud to publish good articles in The New York Journal of Mathematics and Probability Surveys. (And I did not pay for that.) Boris Tsirelson (talk) 13:18, 26 August 2010 (UTC)

I'm actually not familiar with open access journals in Mathematics; however, there is at least one open access journal(family) frequently quoted in 9/11 articles which has basically no credible reputation. I would have to say that it depends on the journal, and whether the reviewers or editorial board are seen as credible. — Arthur Rubin (talk) 15:18, 26 August 2010 (UTC)

I agree. I don't think we can make hard-and-fast rules of this type. For instance, I think the Electronic Journal of Combinatorics is quite reputable despite its open access status. —David Eppstein (talk) 16:02, 26 August 2010 (UTC)

Nowadays researchers in mathematics and related fields often have freely accessible preprint versions of their papers on their (or their institution's) website. Then it is possible to use a paid journal article as a reference, and include a link to the preprint to satisfy accessibility demands. Sources should be judged according to their quality, not access restrictions.—Emil J. 16:25, 26 August 2010 (UTC)

"Sources should be judged according to their quality, not access restrictions." I agree completely. Almost every source we might consider using is freely available to the public in dozens of university libraries. The availability of online versions is not a relevant consideration. — Carl (CBM · talk) 16:58, 26 August 2010 (UTC)

I don't quite agree. There is no issue that quality/reputability comes first, but nevertheless does a free access have its own merit, because it simplifies verification and opens it up to a broader base. Or to put it this way, in the (mostly theoretical) situation that you have 2 sources of equal quality/reputability with one being freely accessible online and the other one not, then you should definitely go with the freely accessible one.--Kmhkmh (talk) 17:51, 26 August 2010 (UTC)

In most cases when there are two sources of equal reputability, I think the better option is to go with both. —David Eppstein (talk) 18:06, 26 August 2010 (UTC)

Even better.--Kmhkmh (talk) 18:31, 26 August 2010 (UTC)

I'd just like to comment that for many math articles, and particularly for most highly viewed articles, most sources come from textbooks rather than contemporary published journals. This could be different from a field such as medicine where current research almost always has an immediate effect on the lay person's life and quoting cutting edge research is common. For the most part, math textbooks are not available in open access. Certainly, certain well-established mathematicians have reliable high quality course notes/textbooks available (e.g. James Milne, or Hatcher's Algebraic topology) and they are, in my view, reliable sources, but I view the collection of these to be a rather small set. Most mathematical breakthroughs significant enough to pop up in a wikipedia article would probably be published in the annals or inventiones, and not available in any open-access journal anyways. RobHar (talk) 18:17, 26 August 2010 (UTC)

For journals the issue isn't open access or not but whether it's peer reviewed. There is probably a correlation between the two but seems plausible that an open access journal could be peer reviewed. As RobHar says, most of the math material on WP comes from books, but recent developments that are relevant to a subject might not be in textbooks yet so that material would usually be sourced from journals.--RDBury (talk) 00:30, 27 August 2010 (UTC)

Correct, although even with peer-reviewed journals the quality of the peer-review process, the composition of the editorial board and the journal's reputation matter a great deal. There are quite a few "peer-reviewed" journals that publish low quality and simply incorrect stuff. For example, Santilli's journal "Algebras, groups and geometries" a few years back published several "short proofs" of Fermat's Last Theorem, see[3], VOL. 15, SPECIAL ISSUE NO. 3, 1998. Nsk92 (talk) 14:32, 28 August 2010 (UTC)

I have not seen this thread until today and I want to clarify the meaning of my original statement. I was talking about open access journals where reading access is free but where there are substantial publication per-page fees changed to the authors. I am not aware of any good math journals of this kind. Quite a few of them have poped-up in the last couple of years and I receive e-mails with announcements of such journals being launched a few times a month. I have even been invited to join the editorial boards of several of them (which I politely declined). Every single math journal of this kind that I have looked at appeared to be basically a for-profit venture/paper mill where the goal is to make money off the authors desperate to publish their work anywhere. These types of journals usually say that they are peer-reviewed but they also promise a fast refereeing process - something like 2-3 weeks. In math such fast refereeing is simply not practicable as it takes a lot longer to go over a math paper in detail and typical refereeing periods in good journals are on the order of 6 months or so, and longer. So any math journal that promises refereeing in 2-3 weeks is not going to be doing serious peer review but rather a pretend one. Nsk92 (talk) 07:00, 28 August 2010 (UTC)

I would appreciate some examples, if you could give them. CRGreathouse (t | c) 07:17, 28 August 2010 (UTC)

I don't particularly like the idea of denigrating specific journals, but if you insist, here are a few examples, from searching my mailfolder[4][5][6][7]. Nsk92 (talk) 14:03, 28 August 2010 (UTC)

That wasn't my intent, of course. Thanks, that does clear up your position. CRGreathouse (t | c) 20:43, 28 August 2010 (UTC)

Nsk92, you said that this applies to almost all open access journals in mathematics. You also said about two counterexamples that they are exceptional in the sense that they don't charge for either publication or access and are completely free to both authors and readers. Typical open access journals charge significant publication fees to the authors. Since you didn't answer it at WT:V, let me ask you the following question again: Is your statement based on personal impressions, or on actual statistics? This study found that "most Full Open Access journals (52%) do not in fact charge any sort of author-side fees".

Regards, HaeB (talk) 10:54, 28 August 2010 (UTC)

My statement was based on a combination of personal experience and fairly extensive discussions that I had with my colleagues regarding this very issue. It is possible that in some other disciplines the situation is different (which may account for the results of the study that you mention), but in mathematics, as far as I know, totally free journals are fairly rare; most open access math journals seem to have popped up in the last few years and they do charge substantial author publication fees. I also have a little bit of direct experience here. I am on an editorial board of one journal, and I am currently a guest editor for a special issue (dedicated to a particular mathematician) of another journal. So I have to deal with the secretarial staff and some amount of logistics involved in publishing a journal. While the costs of publishing a math journals have certainly gone down, particularly with advent of LaTeX, they are still non-negligible and require having at least one secretarial staff support person, plus printing and typesetting expenses (which still exist, even with LaTeX), plus some misc expenses (space, phone, paper, copying, etc). If one is pretty frugal, that's still on the order of 50K-100K per year and that money has to come from somewhere. For a typical math department, even at a good university, it is unlikely that the department would be willing to fork out that kind of money out of the goodness of its heart, and I can't imagine too many university administrators (at the Dean or Provost level) who would be willing to do that either. Some big math societies (like AMS and LMS) and institutes can afford to produce some totally free journals, on a fairly limited basis because they can rely on membership fees and income from their other, non-free, publications. But as a large-scale phenomenon I just don't see how totally free journals in math could displace subscription-based journals in the near future. Nsk92 (talk) 13:47, 28 August 2010 (UTC)

Ok this starts to sound more like an argument about and profit in science publishing rather than about reputability/reliability of journals. Also there is no need for the "goodness of heart" of math departments required to shoulder the unavoidable costs you've mentioned in collaboration projects, because ultimately for the science/math community at large it is cheaper to cut an "unneeded" profit generating middleman. In some regards commercial science publishing is facing a similar problem in the long run, that commercial encyclopedias have with wikipedia (or alike projects). Math departments and universities have a financial motivation to move away from subscription based journals as a cost cutting measure. How quickly that migration occurs remains to be seen, but it is fair to assume that in the near future the subscription based journal will still play an important role and represent some of the most prestigious journals.--Kmhkmh (talk) 15:16, 28 August 2010 (UTC)

Basically I agree. The title of this section is misleading. I never argued for a blanket denial of open access math journals as reliable sources. As with everything else, with open access journals one needs to look at the particulars and evaluate them on a case-by-case basis. I was simply saying that for the time being most open access math journals charge significant author fees and that that the open access math journals that do that mostly appear to be rather low quality journals. I expect that there will be significant evolution in the prevailing publishing model in math in the next 10-15 years, but how exactly it will work out, remains to be seen. I personally hope that the author-pays open access model will not become dominant. On the other hand, I very much hope for proliferation of low price math journals by non-commercial publishers where the copyright terms for their papers expire relatively quickly after the publication date. Nsk92 (talk) 15:43, 28 August 2010 (UTC)

Triangular array

I've created a new article titled triangular array, concerning such things as Pascal's triangle, Stirling numbers, Narayana numbers, Bell polynomials, etc. It is severely stubby. Currently 17 articles (not counting redirects) link to it.

So:

• Expand it.
• Add any appropriate new links to it from other articles.

Michael Hardy (talk) 21:47, 30 August 2010 (UTC)