Talk:Butcher group/Archive 1

Archive 1

Editing and clean-up

This article is a real mess. The first few sentences are poorly written run-ons. I've tried to help, but there are ownership issues at work unfortunately. ChildofMidnight (talk) 17:03, 24 June 2009 (UTC)

Definition of the group

Should this be "non-zero" homomorphisms, or should it be explicit that homomorphisms are unital? The identity is not stated explicitly either. A.K.Nole (talk) 06:37, 25 June 2009 (UTC)

In this case the issue is the spectrum of a polynomial ring, albeit in infinitely many variables. What's the problem? Mathsci (talk) 09:09, 25 June 2009 (UTC)

I am told that unless you explicitly or implicitly assume that the homomorphisms are unital, then the zero map would be a non-invertible. I'm also told that the definition of a group requires an identity and an inverse and that neither of these are specified here. 92.233.48.153 (talk) 17:39, 25 June 2009 (UTC)

You can't have read the article. The identity is defined there using the counit and the inverse is defined using the antipode. (In the case of the spectrum, all homomorphisms are unital.) It's true that defining a group is sometimes done at high school or in a first year undergraduate course in mathematics. Mathsci (talk) 17:56, 25 June 2009 (UTC)

I see that you added the missing definitions after I first raised the issue: thank you for your prompt response. As you say, it is important to write an article so that even the school or university student can get some value from it. A.K.Nole (talk) 18:29, 25 June 2009 (UTC)

Yes it's true that I hadn't yet added all the definitions for the character group, so your question was useful. There are indeed many mathematics articles written with accessibility as one of the main guiding principles. Differential geometry of curves is one example. It's normally a second year or third year undergraduate course; sometimes a graduate course. In this particular article the later sections are definitely at a graduate level: Lie algebras, Hopf algebras, renormalization are usually only taught at a graduate level. The part I'm writing now on renormalization involves Birkhoff factorization, which is a special case of the Bruhat decomposition for loop groups. I'm not sure that the article "Bruhat decomposition" is that accessible; nor for that matter the essentially equivalent Riemann-Hilbert problem. Mathsci (talk) 18:58, 25 June 2009 (UTC)

Group or formalism?

The initial sentence states the Butcher group, [...], is an algebraic formalism. Is a group a formalism? The whole article seems to describe a formalism, in which the group plays a part (and not, it seems, a large one). Would it be better to say that the group "is part of an algebraic formalism"? A.K.Nole (talk) 18:31, 25 June 2009 (UTC)

No. The algebraic formalism in its current state involves rooted trees, Hopf algebras and their character groups. In the same way we say that Connes-Kreimer renormalisation involves the algebraic formalism of Hopf algebras, etc. What you suggested actually doesn't really make much sense mathematically. In this paper [1] you can see Vladimir Drinfeld, the Fields medallist, using "algebraic formalism" to describe the use of quantum groups (his invention) in inverse scattering theory. Again this is the use of Hopf algebras to study a problem in physics (quantum statistical mechanics) related to Birkhoff factorization. Mathsci (talk) 18:58, 25 June 2009 (UTC)

Are we then in agreement that the group is not a formalism, but that the formalism describes, or possibly involves, the use of the group? To those of us who barely know what a group is, the initial sentence as it stands is confusing. A.K.Nole (talk) 19:05, 25 June 2009 (UTC)

No. that is not what mathematicians write. Please read the link I supplied, where you can see the phrase used in the abstract. The definition of group is here. Similarly the definitions of Lie algebra and Hopf algebra are to be found in their respective articles. Mathsci (talk) 19:14, 25 June 2009 (UTC)

The word "group" does not appear in the abstract you cite, so it didn't really help. My questions are (1) is it correct to say that this group "is" a formalism (2) what is the best way of phrasing the opening sentence of this article for the expert and non-expert alike? ("What mathematicians write" is not a complete answer to (2), methinks.) I think the non-expert would be more enlightened by something along the lines of The Butcher group is an algebraic structure used in a formalism involving rooted trees that provides formal power series solutions of the non-linear ordinary differential equations modeling the flow of a vector field. A.K.Nole (talk) 19:28, 25 June 2009 (UTC)

Supplementary. Perhaps the point I'm trying to get across is that it is not clear from the intro that the "Butcher group" is indeed a mathematical group in the sense of Group (mathematics), which is I presume the link you wanted to point to. It isn't quite obvious to the non-expert. A.K.Nole (talk) 19:39, 25 June 2009 (UTC)

Ahem, quantum groups is the title of this extremely famous article by Drinfeld. Why do you suggest there is no mention of groups? If as you suggest you barely know what a group is, it's unclear why out of the blue you are editing an article related partially to groups. "Algebraic formalism" is an established phrase as I've pointed out. The sentence you have suggested doesn't make sense mathematically. Hopf algebras are in fact the main idea underlying the article. A few senior mathematical editors who know this material fairly well have already looked over the article without objecting. Some have had experience teaching the subject. If you tried to include your sentence it would doubtless be reverted. I cannot offer very much help, except possibly to recommend an undergraduate algebra text, e.g. Serge Lang's algebra. Unfortunately you won't find much about Hopf algebras in there. Mathsci (talk) 22:17, 25 June 2009 (U

I have juggled about the sentences in the lede to make the wikilink group appear. Mathsci (talk) 22:42, 25 June 2009 (UTC)

I think your changes exactly meet the point I was making -- thank you. (It's no longer relevant since we agree that group and formalism are not the same thing, but there's no need to misrepresent me: I said the word did not appear in the abstract and hence did not help to decide that particular question.) A.K.Nole (talk) 06:26, 26 June 2009 (UTC)

Making semantic quibbles about the difference between the title and the abstract of Drinfeld's article "Quantum groups" suggests that you are not here to help improve this article. You have already twice written something wrong on this talk page, namely your "feeling" that the character group of a Hopf algebra is not a group. That seems unhelpful, since the whole article is about this group and the related Hopf algebra. You have also hinted that you have in fact no training in even pre-university mathematics. That means that much of the article, including even the lede, is something quite new and strange to you. The lede is being rewritten to reflect content changes in the main article; it will still change as the article is being written and new references are added. This talk page is not a forum for remedial mathematics. Mathsci (talk)

Where on earth did that come from? Your phrase "feeling" that the character group of a Hopf algebra is not a group does not resemble anything I have written here. Putting the word in quotes suggests you are quoting me and that simply is not true. I pointed out that the definition of the group as it then stood was not quite correct unless a certain assumption was made: you assured me that that assumption was indeed the norm. I never suggested that it is not a group. If I have been mistaken, I can accept an accurate quotation and a clear refutation, but this not a quibble but an outright distortion. I find it hard to believe that you do not fully realise all this.

Let us try to get away from your personal remarks. This article is, or should be, aimed at a range of readers, including those for whom the subject is "new and strange", because this is an encyclopaedia. Those of us who are not experts actually have a positive role to play in pointing out that what is obvious to the expert may need explanation or amplification for the rest of humanity: indeed you were good enough to acknowledge that in the section above. Why not embrace that rather than denigrate the rest of us with phrases like "remedial mathematics"? A.K.Nole (talk) 17:06, 26 June 2009 (UTC)

WP:DNFTT Mathsci (talk) 17:09, 26 June 2009 (UTC)

I'd like to second AK Nole's sentiment here and earlier at your user talk page, Mathsci; comment on content, please. Words cannot express the shock with which I view your comments in regards to this article's readership, for whom we should be making it more readable rather than obscure. I apologize for the acrimonious-sounding nature of my comments, but I fear that this may become a systemic problem with mathematics articles, and adding clannishness to the mix will never help things. I beg you to rethink your approach, and try to collaborate with the editors here to make this article comprehensible to people without an advanced degree in mathematics. —/Mendaliv/^2¢/_Δ's/ 17:00, 27 June 2009 (UTC)

Ahem, I'm continuing to edit the article at the moment. You left a message on my talk page which I read. Other editors have been editing the article helpfully. At the moment I'm trying to prepare an elementary example for the renormalization section from a secondary source (either Brouder or Kreimer). It's not that easy and will take quite a bit of time. I'm not sure Mendaliv's outlook is quite in tune with the university system: it is not expected that those who have not specialised in mathematics at high school will be in a position to understand graduate level mathematics. Wikipedia cannot change that. Printed mathematics encyclopedias, of which there are many, do not take a different point of view. Of course in all topics the aim is to try to make things as accessible as possible, but there are the limits I just mentioned. This applies just as much in real life as on wikipedia for those of us that lecture on this kind of material. Mathsci (talk) 08:33, 28 June 2009 (UTC)

It is indeed not to be expected that "those who have not specialised in mathematics at high school will be in a position to understand graduate level mathematics" but noone is arguing that. It is to be hoped that articles on Wikipedia will be of value to the widest possible range of readers, precisely because Wikipedia is not a specialist "mathematics encyclopedia". My point is that those who "lecture on this kind of material" may benefit from listening to those who do not. A.K.Nole (talk) 06:34, 29 June 2009 (UTC)

Those comments were not addressed to you. There are a huge number of WP articles on graduate level mathematics/theoretical physics. If you wish to continue discussing these issues rather than specific edits to the main part of the article, User:R.e.b. has indicated by email that you may post your comments on his talk page. I hope that is helpful for you. Good luck. Mathsci (talk) 07:10, 29 June 2009 (UTC)

Comments on an article talk page are by convention addressed to anyone interested in improving the article. A.K.Nole (talk) 18:38, 29 June 2009 (UTC)

(noindent) Also please take a look at the lede in the closely related article renormalization group:

“ In theoretical physics, renormalization group (RG) refers to a mathematical apparatus that allows one to investigate the changes of a physical system as one views it at different distance scales. In particle physics it reflects the changes in the underlying force laws as one varies the energy scale at which physical processes occur. A change in scale is called a "scale transformation" or "conformal transformation." The renormalization group is intimately related to "conformal invariance" or "scale invariance," a symmetry by which the system appears the same at all scales (so-called self-similarity).

As one varies the scale, it is as if one is changing the magnifying power of a microscope viewing the system. The system will generally make a self-similar copy of itself, with slightly different parameters describing the components of the system. The components, or fundamental variables, may be atoms, fundamental particles, atomic spins, etc. The parameters of the theory typically describe the interactions of the components. These may be "coupling constants" that measure the strength of various forces, or mass parameters themselves. The components themselves may appear to be composed of more of the self-same components as one goes to shorter distances.

For example, an electron appears to be composed of electrons, anti-electrons and photons as one views it at very short distances. The electron at very short distances has a slightly different electric charge than does the "dressed electron" seen at large distances, and this change, or "running," in the value of the electric charge is determined by the renormalization group equation.

”

Mathsci (talk) 14:04, 29 June 2009 (UTC)

Is that an example to be followed or avoided? I find it somewhat incomprehensible but either way don't see what it has to do with improving this article. A.K.Nole (talk) 18:38, 29 June 2009 (UTC)

Huh, more trolling? Mathsci (talk)

Minimal subtraction scheme

Is the minimal subtraction scheme defined under Renormalization an example of Minimal subtraction scheme? If so perhaps mutual links would be in order. A.K.Nole (talk) 17:35, 29 June 2009 (UTC)

You are likely to be blocked for these comments and those above, which seem not to be concerned with the content of the article. You have been warned about prolonged trouble-making on this page. You must now take the consequences. The minimal subtraction scheme was already described in the article. You appear to be trolling. In fact your comments seem extremely disruptive and deliberately obtuse. Here is the sentence in the article:

An important example is the minimal subtraction scheme

$\displaystyle R(\sum _{n}a_{n}z^{n})=\sum _{n<0}a_{n}z^{n}.$

If you have any problems, please ask User:R.e.b. on his talk page. Mathsci (talk) 22:28, 29 June 2009 (UTC)

I see you have accepted my suggestion [2] -- thanks for that. I thought it must be the same thing but wasn't quite sure. A.K.Nole (talk) 06:38, 30 June 2009 (UTC)

Please stop this now. [3] Any more copy-and-paste edits from one article into another will get you blocked. If you can't tell the difference between a Feynman diagram and a rooted tree, or are unfamiliar with the language of physicists (see the lede cited above), you should not try to edit articles like these. Copying-and-pasting from one article into another without sources and out of context is effectively vandalism. Mathsci (talk) 07:37, 30 June 2009 (UTC)

too specialized?

Should we have articles this abstract in wiki? Perhaps yes, sum of all knowledge. PErhaps no, OR. —Preceding unsigned comment added by 72.82.44.253 (talk) 03:46, 3 July 2009 (UTC)