3146:, paying very careful attention to exactly what it says about sourcing? Note in particular that it is much less strict than the DYK sourcing rules, which themselves require only a single footnote per paragraph. The relevant rules for GA are 2b, which does not say anything about what should be referenced, only that when things are referenced their sources should be reliable, and 2c, which says in its entirety "it contains no original research". Not that all content is actually sourced to anything, but merely that it is known material rather than something entirely made up as new material for the Knowledge article. On what basis of GA rules are you making these bizarre demands that material that merely explains the structure of the article itself must somehow have a footnote and that this footnote requirement would be satisfied by putting something inane and non-referency into the footnote? In what way do any of these statements constitute original research? Wouldn't putting an inane non-citation into a reference violate 2b, which requires that the references be reliable sources? How do you read the rules as in any way causing such a requirement to exist? What words of the rules are you following in this? —
3068:
is an argument that they should be removed (without any changes to the text). Both remaining cn tags concern the following situation: there are multiple valid conventions in the literature about how to present, display, or denote something; while each is separately valid, the conventions are incompatible with each other, and so any given expositor is forced to choose one of them. (For example, one could choose to number the rows of a matrix so that the indices increase from top to bottom, or one could choose to number them so that they increase from bottom to top.) Both sentences that are tagged explain to the reader what convention is being used in this article. This is essential information (otherwise it leaves ambiguity as to which of several incompatible choices has been made), but it is impossible in principle to cite it (because they are statements about this article itself). I suppose for the first one I could remove the figure (thereby removing the need to explain what convention is being used in the figure), but the second one seems completely unavoidable to me. Any thoughts or comments you had are welcome. --
2583:
varies between the two modes of stating things without the context of where it came from ("the affine symmetric group is ...") versus attributing particular ideas to particular people ("Shi showed that ..."). I'm not sure I can exactly justify each choice about which fact gets introduced which way, but roughly I think the division is between "folklore" and "things everyone knows" (esp. in the context where you might write a background section of a paper and begin, "for more information on these topics, see " somewhere at the beginning) on one hand, and things that are more recent, have only appeared in the original research papers (rather than being re-exposited elsewhere), etc. One might hope that mentioning particular authors also gives a sense of the dynamic of the field (something that you have noted was not otherwise conveyed well -- though now with a history section it's a bit better).
2210:. Much of your review is helpful, but I wanted to comment on this part. I do not agree with the view that the Harvard references here are deprecated since they are not references in the Knowledge sense, but are used to refer to specific works. Writing "Smith wrote in 1999" and "In (Smith 1999)" are essentially the same and don't appear (to me) to be the deprecated style, since it is merely a way of writing names, not a way of citing sources for specific information. Now, this does raise an interesting question - should the "Representation theory and an affine Robinson–Schensted correspondence" section have references, or is the textual clue that they refer to specific works enough? Dunno, but something to think about. Also, even if it were a citation style, consistency in citation styles is not required by the GA criteria. (Deprecation is a separate issue.)
1916:. The first two are summary sentences of the subsections that follow them -- they are not meant to be free-standing factual claims, but rather navigational aids that briefly give the reader an overview of what is coming. I would have thought that they did not need independent citations (for the same reason that the introductory section of articles does not). (I mean, I suppose that I could copy some lower-occuring citations up to these sentences, if you thought that would be better.) The third example is a sentence about navigating the article -- it tells the reader that we have introduced X and we have introduced Y and where in the article to find the relationship between them. It has no factual content to cite. May I remove it? Or, is there some way I can reword that you think might avoid confusion here? Thanks again,
2174:, or stand-ins for the papers being referred to, rather than at the ends of sentences as sources only visible on mouseover as we typically do, regardless of the citation style being used. This, again, recalls a common issue in GANs for me, in which it is often possible to detect the involvement of different writers at different times due to subtle and not-so-subtle changes in focus and style, often in citations ... in other words, the nominator didn't go through the whole article after expanding it to make sure the voice was consistent. But, again, that is clearly not the issue here, since the article's voice is the same throughout.
630:. Since this is a/the traditional way of writing the affine symmetric group, and it's not explained in the literature explicitly, you are right to be uncomfortable to add it to the article -- it would of course be original. I don't think mentioning S_n being the symmetric group earlier would be too helpful. I suppose I was just confused, not having much experience in this particular area, as to why a tilde was used here - but that appears to be because of my lack of exposure. :) Best of luck, I imagine you'll pass GAN with flying colors and I look forward to seeing other content you contribute to the project.
3436:, which merely stated which of multiple notational conventions the article was following, and refusing to back down from that demand after multiple editors tried to intervene. However, Case's responses to why he was refusing to back down did not make me confident in the depth of the rest of the review: as far as I can tell it amounted to checking sourcing by whether there were lots of little blue numbered footnote markers and not by whether the material needed sources and whether the sources were adequate for the material. —
2112:: a brief history section explaining how the concept came to be discovered or something like that. This article IMO could use one. As it is it just seems as if the idea of affine symmetric groups just appeared on a stone tablet lowered from a flying saucer by a beam of light. Surely this is not the case ... a reader would get a better handle on the subject understanding where it came from, who (if they are notable) developed and perhaps named the concept, when, and where. And maybe they won an award for doing this.
257:
247:
226:
21:
3263:, I think your comment has convinced me first that the description of the image should really be in the image caption, not in the text. (The reason to display the matrix this way is that if you don't, it's very hard to see the isolated 1s in a huge sea of 0s -- several references make the same choice, presumably for the same reason.) Do you think it would help if I added row and column numbers along the outer border of the matrix? --
195:
2590:, and Urve commented on it above. My view is that I'm using Harvard references in order to mention, in the text, the authors of particular works, along with pointers to those works. Some uses are unambiguously of this form (e.g., the first sentence of the section on juggling sequences). But I do agree that I'm maybe trying to have it both ways in some places, where I'm mentioning individual authors or papers and not
54:
2980:.) Yes, this really fell off my radar as end-of-semester hecticness set in. Is it possible to get a revised sense of what issues you view as unaddressed / essential to be resolved? For example, I have marked some things as "done" above, but it is not clear to me if they are done to your satisfaction. (I realize there are still a number of cn tags in the article, as well.) Thanks very much,
132:
2304:
3708:) seem to mention it -- and the article on algebraic groups does not mention the associated Weyl group. Core partitions are probably a central object in dozens of research papers, parabolic subgroups in hundreds or more, so I think notability is not a concern. (Obviously I need to convince some friends to become Knowledge editors to fix these oversights.) About
1770:
3245:
geometric flavor to the image so that you can see a pattern, but yeah. For reference, I studied math in undergrad, so it may be confusing for those even less mathematical training (though I suppose most readers of this article will have some). Perhaps an image that requires less explanation can be used, or the image might be omitted altogether as JBL suggested.
2036:
3590:, and I have not found any serious issues with the article in my reading so far. I'm a physics person rather than a pure-math person, so I'm accustomed to technical prose without being too familiar with the details of this specific topic. Speaking from that background, I think it's a pretty nice page! Kudos to all those who have put serious work into it.
2717:
2689:
2650:
2624:
2504:
2464:
2414:
2346:
5608:.) This extends the poset of regions of the affine braid arrangement (viewed as a linear hyperplane arrangement) to a lattice. It is conjectured (but not proved) that the extended weak order of any Coxeter group is a lattice. It might be worth adding something on this at some point, perhaps after more of these references are published. --
612:
discussion (that I believe may not be stated in any reliable source) into the article. However, perhaps your point is not that I should add text about why this notation is used, but instead that if I introduce the reader to the fact that the symmetric group is called S_n earlier, that by itself would be helpful? Thanks again,
3769:, it doesn't matter what type it is, but if you happen to be the kind of person who thinks about reflection groups then of course this is a crucial piece of identifying information. My first instinct is to remove both instances of "type A" from the section you mention, and to add a few words in the section
3675:, thanks for your comments! In the section on the representation theory, I could easily convert the Viennot reference to a (Viennot yyyy) reference by mild rewriting: "... introduced by Viennot in (Viennot yyyy)." That would reduce by 1 the number of distinct styles in the section. A worthwhile change?
3067:
Thanks again for all the work you've put into this. The current status is as follows: I have provided sources for factual claims as best I can; I've removed some material that ultimately I have not been able to adequately source. There are two cn tags still in the article. The rest of this comment
2582:
The mention of multiple authors amused me, because really this is my individual work (of course with several improvements by other editors since it went live). Though it's not necessary to address the issue you raised, I thought I would take a swing at explaining. Mathematical journal writing often
2374:
is a certain perspective on the study of geometry that focuses on collinearity (when points are on the same lines) and parallelism; the geometric transformations (like rotations, translations, reflections) that preserve these properties (so if several lines are parallel, they get transformed into new
2383:
gives an etymology. I would write something shorter than the preceding if I write something.) On the other hand, I don't have any source for what I just wrote: it's a cobbled together synthesis from sources that don't mention the affine symmetric group at all, plus my personal understanding of how
2018:
It seems even to me that at least one, maybe two, more grafs could be added summarizing the sections of the article past the definitions (Consider that the DYK hook fact mentioned the juggling connection ... this is something that should probably have been mentioned in the intro; some DYK reviewers,
3645:
in like that (and then including it inconsistently thereafter) might be slightly puzzling. "What makes this lattice 'type A'? Do these things also have lattices of other types? What does the type tell me?" Maybe I'd say something like, "... forms a root lattice, specifically one of type A". Perhaps
3522:
I also cannot reiterate strongly enough how personally I take your insinuation that I only cared about the existence of citations and not their quality. Assessing the quality of the sources here may have been beyond my depth, as I've said (and to put more crudely what I've said a few times already,
3412:
has stepped back from reviewing; has David
Eppstein taken over, or do we need to find a new reviewer? If the latter is the case, then the nomination should probably be set to request a second opinion, in the hopes that someone who can deal with the arcane mathematics text will be able to take over.
2444:
ever singled this group out for particular attention in his work. Unfortunately searching that out will take more time than I have available in the near future. (It is possible that in some sense the history of this object studied for its own sake rather than as one example of an important family
3244:
I will say, however, that it took me a bit to understand the image of the matrix since it uses dots for 1s and omitted zeros and lacks borders for individual rows and columns (if I understand it correctly, the lines delimit each group of three rows and columns). I know it's probably to give a more
2439:
Yes, more historical information would be good. I note that I haven't written down anywhere that the combinatorial definition of the group was first given by
Lusztig in 1986, for example. I am not sure whether it will be possible to source a full section's worth of historical content to reliable
3186:
to GA rules caused by your inventing rules out of thin air, and (3) the damage to encyclopedia content creator goodwill caused by your turning what should be a much more straightforward evaluation process into a bureaucratic nightmare. Anyway, is there a reason why you haven't answered any of the
2007:
understand from writing articles about court cases and legal topics the challenges of writing about an abstract subject, often seen as impenetrable by lay readers, where common usage within the field requires both the use of words used nowhere else in
English discourse, and some words that take a
5331:
Ref "Lusztig, George (1983), "Some examples of square integrable representations of semisimple p-adic groups", Trans. Amer. Math. Soc., 277: 623–653" is missing its DOI. Some other references are lacking DOI but have their MR numbers so it is fine by me. You might still get it pointed out at FAC
3479:
It is possible that an editor who is trying to promote an article to GA-class (good article status) might add citations to basic facts such as "...the sky is blue...". This is a good thing, and the fact that the sky is not always blue does benefit from adding a citation. We can add citations for
5335:
Two references are only arXiv preprint which might pose a problem at the FA source review. But these preprints have since been published. "Chmutov, Michael; Frieden, Gabriel; Kim, Dongkwan; Lewis, Joel
Brewster; Yudovina, Elena (2018), Monodromy in Kazhdan-Lusztig cells in affine type A" is now
3450:
Failure to assume good faith like this is why I am not interested in finishing this review. "Multiple editors" did not try to intervene on
Eppstein's behalf; in fact if there were any actions by multiple editors that might have served as effective intervention had they been more willing to take
3330:
This looks better to me too. I might quibble that the rectangular rather than square aspect ratio of each matrix cell, and pixel graphics rather than vector graphics, are both suboptimal, but those are minor quibbles that shouldn't affect the discussion here, and they may be difficult to change
563:
in some sources). When there are "linear" (or finite) and "affine" versions of some object (like a
Coxeter group), it seems very common to denote the affine one with a tilde on top (but again this is not universal; sometimes one sees a hat used, or for authors for whom the affine object is the
3468:
As for his comments about "checking sourcing by whether there were lots of little blue numbered footnote markers and not by whether the material needed sources and whether the sources were adequate for the material", well, Dave, you need to get out of the math ghetto more. This is exactly the
2078:
A lot of sentences in the definitions sections, and indeed throughout the article, use "one" in the third-person impersonal style. I do not know if this is standard in mathematics—if it is, and we use it in other math articles, and we're OK with doing that in math articles, and I can be shown
611:
is taking the intersection of these two common choices, and is widely in use in the literature. I should be clear that these are my observations from reading sources -- I do not recall seeing anyone write down anything about the choice of notation. I am reluctant to put something like this
2594:
citing those mentions to other authors or papers. I think I'd like to try to reach an agreement about which places current usage is acceptable (e.g., I feel strongly that this is the case for the section on juggling sequences) and which are not (I think Urve has a point that the section on
2114:
At the very least it would provide a brief respite from the equations and diagrams, especially if there is the potential for it to be illustrated by the inevitable image of some dead old white guy, either painted in academic regalia or photographed in black and white staring intently at the
1999:
I also was grateful for well-written prose, with few grammatical errors and no spelling errors. Nor were any facts repeated, or stated fragmentarily. I would like to think that this is coincident with having to write so precisely, and to a great extent in symbols, about such an abstract and
335:
Beautiful prose quality and attention to detail, here, with great illustration. I came across this in the GAN queue; even though I studied quite a bit of algebra, I am not in a position to determine whether the article meets the criteria. (I have no experience in combinatorics.) Good work!
4744:
I shall write here my comments rather than on your talk page as for some reason this caused your talk page to crash (it had to do with mathjax re-interpreting old stuff from your talk page for some reason). The article looks very good to me, informative and complete. I have a couple of
840:
because of the requirement that the shifts sum to zero) and the last one is the quotient from the "As a quotient" subsection? I mean, you sort of do already describe it, but without saying it's a short exact sequence and without stating that the kernel is actually a free abelian group.
3465:, which seemed to me to be rigorous enough. Again, I think anyone looking for a review of an article that had been languishing at GAN unreviewed for six months should be grateful someone went and reviewed it at all rather than raising side-eyed comments about "the depth of the review".
1958:
What about the third one: "The correspondence between the geometric and combinatorial representations for other elements is discussed below in § Connection between the geometric and combinatorial definitions.)"? This sentence has no substantive assertion beyond the structure of the
1725:, I didn't say it was deprecated here. I said use without a template (like ref, sfn, or harvtxt -- so only bare text, which this article is not) is deprecated. We're on the same page. I only mentioned it for future reference and because they mentioned there was some use of inline
3236:
and their corresponding descriptions; the first image's description there has a citation but I think that's more to support the "popular" claim. In this case (i.e. the article Affine symmetric group) the description is just kind of long and is in the article instead of under the
856:
Freeness is mentioned, but it's snuck in the very last sentence of the section. Perhaps a short additional paragraph on that section, offering the short exact sequence as a summary. Running off to lunch now, but I'll try to find a moment to think about it this afternoon.
3593:
Minor concerns: the manner in which it points to items in the literature is not perfectly uniform. I call this a "concern" rather than a "problem", because I don't think it impedes understanding, and life is too short for me to really care about the distinction between
3368:
There are several ways of getting svg from pdf, but I'm not sure which of them if any play nicely with the LaTeX fonts. The workflow I use for that (opening in Adobe
Illustrator and saving as svg) definitely doesn't work well with those fonts. Maybe someone else knows?
2737:
And to complement a history section, I wonder if it might be possible to end briefly talking about the future. What's going on in research into affine symmetric groups? Are any interesting possibilities opening up? Any new questions to be answered? Maybe something like
2741:
This is a nice suggestion but I think it would be impossible to source to secondary sources. (I could, like, go pull a bunch of conjectures and open problems from recent papers, but I think it would be impossible to justify any particular choices and avoid synth.)
2331:
Hi Daniel Case, thanks very much for this review, both the positive feedback and constructive criticisms. I plan to respond in stages over the next several days. An initial version (with some bolded quotes of your comments and my interspersed responses) is below.
1504:
I am not at home so can't easily browse for citations, but I am sure they exist; I am a bit annoyed that neither paragraph has any at the moment. (Also annoying: the
Knowledge LaTeX implementation seems to lose the correct spacing around \to if you use \overset ?
2016:. This is a common occurrence in GANs I review; usually it stems from someone's efforts to expand the article prior to the nomination and accompanying failure to commensurately expand the intro. But here this is an exception, as it came out of wikiversity that way.
1638:, no worries. Letters are usually for explanatory material that doesn't need to be in the prose proper, so they're more notes than citations. Regarding parenthetical citations: In this article, they use a template. The use of parenthetical style without templates
1866:, but ... it does not look like we can let this slide any longer. I owe about four reviews, and the type of a person I am, I prefer to take the harder reviews that are always in the top bar (not least because some of my nominations have ended up there; in fact
2956:
Merry
Christmas (two days late) and Happy New Year! Holiday greetings aside, I see that you have not been able to do any work on this article in a month. I know you're an academic and this is a busy time of year. Will you be able to resume after the holidays?
3240:
For the second one, it's just explaining convention which is common (and necessary) in mathematical writing. Since convention is important and using different conventions can be seriously confusing, it may be better to leave it in the article rather than a
3181:
The harm is not primarily to the article, but (1) to the Good
Article process caused by your dragging out this review endlessly with demands for additional footnotes where no footnote is needed and where none of the GA rules justify such a demand, (2) the
2282:
I think the juggling patterns illustration, given its long strip-like shape, would look better centered above the text, rather than off to the side sticking into the main text awkwardly, twice as wide as any other image. I did something sort of similar
4470:
1451:
5332:
though. Similarly none of the books have the publisher location listed but I think it is fine for FAC if nobody raises this in the sources review. It is actually better to have none than an inconsistent style with some that do and others that don't.
1688:. The current article uses Harvard-style references in short footnotes, and Harvard-style parenthetical text as a way of talking about authors and their publications within the article text. Neither of those things is deprecated. I would have used
3803:
can point to, then leaving it as a redlink is the best thing (as it signals there's a new topic to develop). I like your instinct about the "type A" business. And if there's no obvious way to expand the "History" section, I think it's fine as is.
3633:
There are a couple redlinks. This isn't a problem with the article itself, but if the content they should point to does exist somewhere, then they should be pipes or redirects. It's probably worth checking. No big deal if they stay as they are,
1873:
So, I will do what I usually do ... print it out, do a light copy edit (to the extent possible for me here) and come back within a week's time with my thoughts based on the article's structure and non-math aspects. I will probably ask at
5580:
3653:
Very short sections often read as unfinished to me. Here, "History" is only three sentences. It's serviceable, and I wouldn't object to a GA having a section like that, but I'd advise thinking a moment or two about whether it could be
3679:
are a wonderful combinatorial gadget that I'm surprised no one has written anything about in Knowledge, but I have looked in all the obvious places and searched in various ways and I just don't think it's here. We have content about
2032:, of "affine" so the reader has a chance to understand what it means in this context. Before I started reviewing the article I thought it might have been someone's name (some obscure French or French-speaking mathematician, perhaps?
4369:
1354:
5316:
General question: let's say a reader wants to delve more into the subject, which book shall he/she go to ? Is there no general review book on the topic or are the historical item cited the best there is to be up to date on the
3453:
I had thought my willingness to take on a GA review in this subject area, especially an article which I remind Dave had languished for months before anyone dared review it, might have been appreciated, and indeed it was, by
2023:
This wouldn't be necessary to expand the intro, as it can be done with what's there already, but it might be nice to link from the intro, as soon as possible after the bolded statement of the subject in the lede (from which
5066:(I think) where it is eminently simple to visualize. The relation to the braid group on the circle makes it easier (to me at least) to visualize the relations of the algebraic definition, in a manner similar to that given
750:
2807:
So far so good. I was wondering ... now that we've expanded the intro, is there any image in the article we could put in the intro as lede image? If not, is there one somewhere else? Can one be created relatively easily.
2398:
A lot of sentences in the definitions sections, and indeed throughout the article, use "one" in the third-person impersonal style. ... is it possible to rewrite or recast those sentences such that we don't have to use
115:
35:
5646:
5641:
3231:
For what it's worth at this point, I agree with JBL. The first CN tagged sentence is just describing the image of the matrix. It's not really making a claim that needs a citation. See for instance the images in
3732:
is introduced; in the section on Lie algebras; and finally in the section on other affine Coxeter groups) where it is alluded to, but that's it. I can't decide how I feel about this: from the point of view of
2298:
And that's it. I think these issues can reasonably be addressed within the usual week or so, and of course I can extend that time limit if it looks like you're making progress. So, I'm putting the article ...
2085:
explicitly does not rule out this use. But it also suggests that such use should be limited only to situations where that is really the only way to do it, and I'm not sure we have that here. I also note that
2595:
Representation theory and an affine Robinson–Schensted correspondence is pushing the boundary here, and maybe the section on Fully commutative elements and pattern avoidance as well) before I patch them. --
2976:, thanks for the ping and the holiday greetings -- the same to you! (I was up vaguely in your corner of the world for Thanksgiving -- took the wonderful Amtrak line along the Hudson, visiting family in
1597:
Also, what's inconsistent about them? They all appear to use short footnotes with longer references in Citation Style 2. There is one use of "Cor." rather than "Corollary" but that's easily fixed. The
4097:
5636:
1904:
for starting this review, and for your initial efforts! I will begin addressing some of the citation needed tags soon. However, there were three of them I wanted to discuss: the one at the top of
2994:
Sure, it may take a while, but I'll get back here. Generally you're more than 50% done ... if I could suggest anything right now, it would be making sure everything that isn't currently cited is.
3773:
to explicitly mention that type A is the one we're talking about. What do you think? Finally, about the short history section, if you search above on this page you'll see my comment there that
5145:
1172:
4511:
1492:
5336:
published at Selecta Math. New Ser. 28, 67 (2022) and "Monodromy in Kazhdan-Lusztig cells in affine type A; Michael Chmutov, Joel Brewster Lewis, Pavlo Pylyavskyy" is in Math. Annalen, 2022.
313:
1986:
I am glad to see that editors involved are already attending to the requested footnotes. And, since in the process of reviewing I looked at the original draft on Wikiversity and found the
1114:
1049:
4642:
4893:
for all i. Of course this is a simple observation, straightforward from the formal definition but I wonder why this isn't in the article. This quotient is compatible with the projection
1990:, by People Who Know This Stuff, with advanced degrees, actual academic chairs and all, I will defer to that and consider the math substantially correct, beyond the need for any sort of
2195:
we deprecated inline parenthetical refs a little over a year ago, there is no justification for doing things this way. Put them in tags and rewrite or recast the sentences. This is the
1583:
Can you say a bit more? I'm not too sure I understand the problem looking at that guideline. Is it that the citations themselves are inconsistent in style? If so, that's not required.
140:
5425:
1996:
review. I will say that, with the help of the various links, I was able to follow most of the article on a very minimal, conceptual level. The author(s) are to be commended for that.
2785:
Thank you for your patience. I have completed a first run through at addressing or responding to your comments. I would be interested in your assessment of this progress. Thanks,
1987:
5288:
5242:
4678:
4189:
4007:
3767:
1775:
609:
495:
409:
373:
4143:
3799:
I think reducing the number of distinct styles by 1 would be a slight but noticeable improvement, and since it's a quick fix, it's probably worthwhile. If there's nothing yet that
3432:
I have edited the article too much to take over. The mathematics has all been reviewed; the only sticking point was that Case was demanding citations for sentences like the one in
553:
5195:
809:
4947:
2620:
I have (temporarily) converted from the imperative to the third person, to be further resolved when I think about the rest of the "one can"s. So let's call this specific point
838:
3825:
Thank you! I have made those two changes (Viennot and type A). The wording of the new sentence in the section on other groups is a bit clunky, but I think it does the job. --
2109:
4593:
4383:
3966:
2019:
in fact, insist that the hook fact be in the intro. I don't think that's always necessary and don't insist on it, but here I think it could and should easily have been done).
1364:
440:
as a subgroup, as you explain later, but not when it is first introduced.) But that may be hard to do in a way that preserves prose quality and flow, so feel free to ignore.
60:
5506:
5037:
4817:
4547:
2342:
I have added two short paragraphs to the introduction; together, I believe they mention the highlights of all major body sections of the article. I'm going to call this
1621:
Sorry, that was dumb of me. I saw parenthetical refs in the text, and numbers and letters in the References section, and stupidly thought "there must be a mistake here".
5468:
4890:
3730:
438:
3102:... since we're not in a better world, I'll recommend instead what I've done for years since someone at a DYK nom accepted OI but insisted I find some way to cite it:
5064:
4974:
4844:
4774:
4709:
4233:
4034:
3530:
GA nominations I've reviewed, I have queried the reliability of the sources. And I'm sure if I looked through other GA noms I've reviewed, I could find more examples.
3451:
action (a reluctance I now fully understand) it was the many supportive emails I got from other participants in the math project about dealing with Diamond Dave here.
2294:: What's going on in research into affine symmetric groups? Are any interesting possibilities opening up? Any new questions to be answered? Maybe something like that.
522:
4994:
4911:
2917:
I did think about this a little when choosing the figure, and decided the risk of confusion was low. The figure DE linked is the one I would have chosen instead. --
5606:
3840:
1808:
3046:
That's incredibly helpful, thanks. I was able to get into my office yesterday and pick up some books; I will put in a final push this week and ping again soon. --
2044:
able to gather that it has some meaning that puts it, at least sometimes, in contrast to "infinite", but I would have liked a fuller understanding from the get-go.
5511:
5308:
3486:
774:
673:
2370:
Ok so on one hand this point is obviously correct, and I would like to write a sentence (or maybe footnote) that explains it. (The ultimate explanation is that
984:
2861:
The only thing I'm a little worried about with this image is that symmetries of the tiling itself and symmetries of the colored tiling are not the same thing. —
2535:
You are right about what MOS:MATH says, but wrong in this instance that it applies. That portion of the MOS is about definitions; for example, in the section
1983:
OK ... I had meant to get this done over the weekend, but I couldn't, and then I had to work at the polls on Tuesday so that delayed this into the later week.
497:, first let me say why this is what I chose (which I think you understand already): by far the most common notation for the (usual, finite) symmetric group is
2759:
It is a nice idea! I wish there were more people writing the kind of expository / state-of-the-field articles in mathematics that could be used for this. --
1798:
3354:
PNG to produce images (same as for all the others I created in the article); if you know how to get SVG output from LaTeX, I'd be happy to hear about it! --
4749:
There is a whole section on "Relationship to the finite symmetric group" and so I wondered why there was nothing on the relation to the group of braids on
1938:
I know how you feel, but I've been dinged myself for this sort of thing (uncited prefatory/introductory sentences) far too many times even though if I had
1867:
1780:
5656:
4238:
3167:
What harm could it possibly do to the article? If you'd like to have someone else review it, I'd be happy to let it go and have them take a crack at it.
1216:
303:
2900:. I'm not sure it's really much of a problem. Anyone who would notice that it's an issue would also know to ignore the colors for sake of the example. —
3895:
3666:
3856:
3813:
1684:
and, as an RFC that was purely about referencing, did not concern the use of the same style as a way of commenting on authors and their publications
3587:
5651:
2685:
I believe that Shi probably is notable. However, I'm not going to write an article about him in the near future, so I have removed the link.
279:
1194:, affine permutations with underlying permutation the identity, and translations); in all three settings, the natural group operation turns
5631:
3712:, I see that the classification of affine Coxeter groups is not clearly spelled out anywhere in the article -- there are a few spots (when
3918:
Here I'm going to maintain a (hopefully short) list of things that are true but that I was not able to find sources for on first attempt:
3641:
is introduced and used bothers me a bit. If a reader isn't familiar with the classification of root systems and related topics, dropping
157:
5370:
3688:
groups, and of course this concept is intimately related to the concept in Coxeter groups, but none of our articles on Coxeter groups (
5369:
All pictures must have alt text, which you can insert in the code with "| alt =" in the figure caption. This is mandatory for FA, see
2003:
That said, I do think there are some other non-math issues besides the insufficient citations (Before I begin, let me just say that I
3524:
1847:
270:
231:
2611:
To translate between the geometric and algebraic definitions, fix an alcove and consider the n hyperplanes that form its boundary
2284:
2166:
Starting at "cycle type and reflection length", the article inexplicably begins using its Harvard references not only inline and
2099:
To translate between the geometric and algebraic definitions, fix an alcove and consider the n hyperplanes that form its boundary
1826:
1803:
1601:
96:
2176:
So, honestly, I don't know why this was done here. But, and I must speak more boldly than I have otherwise in this review, it
4042:
2384:
mathematicians use words. So I'm concerned about how to go about this in practice. Thoughts or suggestions are welcome. --
2379:, and in the geometric construction of the affine symmetric group the elements are all affine transformations. The article
2897:
1676:
You are seriously mistaken about the format here being deprecated. The use of parenthetical (Harvard) style is deprecated
3352:
Hmm, I hadn't noticed that they're not perfect squares, thanks -- I've fiddled, now it's better. I am using LaTeX -: -->
2829:
Well, the natural thing is the triangular tiling of the plane; I went and found one on commons with some color in it. --
206:
3610:
is legitimate. This is a matter of taste; perhaps the most serious thing I can say about it is that having effectively
5076:
3098:. Technically, you'd say you don't need any citation, and in a better world I'd agree with you, and that would be it.
2710:
I think the juggling patterns illustration, given its long strip-like shape, would look better centered above the text
1119:
670:
In the "Relationship to the finite symmetric group" section, should there be some mention of the short exact sequence
30:
4475:
2500:, for whatever reason -- I have merged related consecutive single-sentence paragraphs there into larger paragraphs.
2250:" This to me is an implicit requirement that does not and should not need to be restated (even though, of course, it
1456:
1859:
This article has sat unreviewed for six months now, likely because it's about an arcane (to most of us) math topic.
3881:
3834:
3794:
2547:
is an affine permutation if and only if ". But the statement about permutation patterns is not a definition.
2408:
I see about 15 uses of "one has", "one may", etc. -- some are clearly avoidable, I'll see what I can do with them.
2228:
are deprecated. There's no problem with them in footnotes, like the former of the two variants you proposed above.
2977:
1054:
989:
4598:
3557:
3511:
and come in and kick everything over, over a relatively minor issue that you could have handled with much more
3441:
3374:
3336:
3192:
3151:
2905:
2866:
1713:
1639:
1626:
1612:
1573:
1524:
846:
194:
20:
2367:
it might be nice to link from the intro ... to some article, or perhaps the Wiktionary definition, of "affine"
1942:
druthers, too, it wouldn't be necessary. Your proposed solution of moving cites up has always worked for me.
5359:
5248:
3433:
3120:
endnotes. I think that would work here, and then the remaining sentence could be joined into the next graf.
3110:
As for the second one, I think, I typically put such meta references of the "In this article ..." type into
5067:
3770:
3281:
but it did not work well so I self-reverted. May try again later (1st day of classes is tomorrow) .... --
2536:
2497:
1913:
1909:
1905:
1822:
1175:
168:
5257:
5200:
4647:
4465:{\displaystyle 0\to \mathbb {Z} ^{n-1}\to {\widetilde {S}}_{n}\mathop {\overset {\pi }{\to }} S_{n}\to 0}
4158:
3976:
3736:
2093:
In other words, is it possible to rewrite or recast those sentences such that we don't have to use "one"?
1446:{\displaystyle 0\to \mathbb {Z} ^{n-1}\to {\widetilde {S}}_{n}\to S_{n}\mathop {\overset {\pi }{\to }} 0}
578:
464:
378:
342:
53:
5432:) has, in the case of the affine symmetric group, a nice combinatorial representation (Barkley--Speyer:
4102:
3891:
3877:
3852:
3845:
Do not close a review started by another reviewer without first attempting to contact the first reviewer
3809:
3662:
3571:
3566:
So you would have more respect for me if I had lied? I'll keep that in mind next time. If there is one.
3538:
3422:
3300:
3250:
3221:
3172:
3125:
3030:
2999:
2962:
2887:
2852:
2813:
2750:
2566:
2380:
2376:
2317:
2263:
2156:
2068:
1947:
1886:
1841:
527:
212:
150:
40:
5150:
2633:
Might those parentheticals at the end of a couple of grafs be better off converted into endnotes using
2290:
And to complement a history section, I wonder if it might be possible to end briefly talking about the
2279:
If J.Y. Shi's notability can be demonstrated, start at least a stub article. Otherwise delink his name.
2119:
Might those parentheticals at the end of a couple of grafs be better off converted into endnotes using
1878:
for someone not involved with the article to take a look at the math and let me know how solid it is.
779:
256:
3469:
criterion by which GANs in every other subject area are reviewed. I commend to your reading attention
3295:
Yes I think adding row/column numbers may help, and would also allow the description to be shortened.
2238:
require a consistent citation style throughout the article, but ... I would commend your attention to
4916:
4378:
4192:
3527:
3480:
things that are well-known, and the source can contain additional information to benefit our readers.
2844:
1359:
814:
4714:
Full details of the geometry for the infinite dihedral group: who are the roots, what is the alcove.
3800:
3676:
2543:
is an affine permutation if ." The admonition in MOS is that I should not have written "a function
166:
Did you know ... that juggling patterns can be encoded in terms of a mathematical object called the
5613:
5403:
5395:
5311:
4726:
3830:
3790:
3553:
3552:
rather than even the most feeble denial? (Also, please do not call me "Dave"; it is not my name.) —
3470:
3437:
3405:
3370:
3359:
3347:
3332:
3321:
3286:
3268:
3188:
3162:
3147:
3073:
3051:
2985:
2922:
2901:
2877:
2862:
2834:
2790:
2764:
2728:
2701:
2662:
2600:
2552:
2515:
2484:
2450:
2426:
2389:
2358:
2239:
1966:
1921:
1875:
1722:
1709:
1635:
1622:
1608:
1569:
1538:
1520:
1510:
862:
842:
651:
617:
131:
4560:
3925:
278:
on Knowledge. If you would like to participate, please visit the project page, where you can join
5473:
5437:
4999:
4779:
4530:
3681:
3462:
3414:
2406:(first-person plural is standard in mathematical papers, even when written by a single author).
2187:
be undone to be consistent with the citations in the first half of the article. I see nothing in
1203:
753:
262:
5310:
is" you could perhaps add the word "bivariate" in front of "generating" and you can wikilink to
3548:
Why would I apologize, when your response to my accusation of superficial reference-checking is
2136:
we should try to combine them with adjacent, longer grafs, unless there is a good reason not to.
246:
225:
4849:
2336:
First, at 60K total, it is long enough that the intro can and should be more than the one graf.
5382:
5345:
5322:
2192:
2188:
1881:
Hey, we should all try to stretch ourselves and push the limit on what we think we can do ...
1183:
44:
5575:{\displaystyle \cdots \prec -2\prec 0\prec 2\prec \cdots \prec 3\prec 1\prec -1\prec \cdots }
5447:
3715:
2079:
examples, I drop any perceived objections. However, I have seen it nowhere else on Knowledge.
5363:
3887:
3873:
3863:
3848:
3820:
3805:
3697:
3672:
3658:
3608:
to mention, in the text, the authors of particular works, along with pointers to those works
3567:
3534:
3418:
3409:
3311:
3296:
3260:
3246:
3217:
3168:
3137:
3121:
3062:
3041:
3026:
2995:
2973:
2958:
2883:
2848:
2824:
2809:
2780:
2746:
2562:
2313:
2259:
2152:
2108:
One thing that might help make for a longer intro is something many of our other conceptual
2064:
1943:
1901:
1882:
1837:
1702:
882:
875:
414:
3922:"In terms of the geometric definition, this corresponds to the reflection across the plane
3839:
Those changes look good to me. I'm inclined to say the article deserves GA status now. The
5042:
4952:
4822:
4752:
4687:
4550:
4211:
4012:
3705:
3183:
2676:
2614:
2371:
2102:
556:
500:
4979:
4896:
4776:
because as far as I can tell the affine symmetric group is a quotient of the braid group
5585:
3785:
My opinion of this hasn't really changed since then. Thanks again for your comments! --
2915:
Oops, I could have sworn I responded here yesterday, but it seems I never hit "publish".
2040:
As in so many other sciences, a lot of people leave their mark on the field that way) I
1652:
1565:
5609:
5399:
4741:
4722:
4037:
3826:
3786:
3701:
3603:
3550:"This is exactly the criterion by which GANs in every other subject area are reviewed."
3401:
3355:
3317:
3282:
3264:
3143:
3084:
3069:
3047:
3010:
2981:
2951:
2918:
2830:
2802:
2786:
2760:
2724:
2697:
2658:
2596:
2548:
2511:
2480:
2446:
2422:
2385:
2354:
2255:
2231:
2215:
1962:
1933:
1917:
1734:
1660:
1588:
1559:
1534:
1506:
858:
647:
637:
627:
613:
447:
5293:
5244:
as these forms are commonly encountered in the literature when presenting this result.
3646:
the text or the figure caption could mention that the triangular lattice is known as A
759:
339:
The only thing I can offer is to suggest that there be some motivation for calling it
5625:
5147:
I understand that this is easier for the general audience, but perhaps you could add
3689:
3508:
3019:
2471:
for now. I believe it would be possible to write a more detailed history section on
2105:. That's common in a textbook, but we generally try to avoid that in an encyclopedia.
2025:
1692:
1199:
888:
5251:
for the affine symmetric group is solvable. I think this should be stated somewhere.
4717:
The sentence "However for higher dimensions, the alcoves are not regular simplices."
4364:{\displaystyle \pi (s_{0})=s_{1}s_{2}\cdots s_{n-2}s_{n-1}s_{n-2}\cdots s_{2}s_{1}.}
1961:
Never mind, I see, it's a two-sentence parenthetical. Back to the bookshelf .... --
1349:{\displaystyle \{(1,-1,0,\ldots ,0),(0,1,-1,\ldots ,0),\ldots ,(0,\ldots ,0,1,-1)\}}
881:
is by definition the set of affine permutations whose underlying permutation is the
5378:
5341:
5318:
3114:
3095:
2637:
2123:
2090:
strongly suggests that the use of pronouns to address the reader should be avoided.
1655:, which does not require it. (Why that is the case I do not know, but so it goes.)
5617:
5407:
5386:
5349:
5326:
4730:
4373:
The relationship between the kernel, the affine symmetric group, and the image of
3847:, but in this case the first reviewer appears to have distanced from the article.
3575:
3561:
3542:
3445:
3426:
3378:
3363:
3340:
3325:
3304:
3290:
3272:
3254:
3225:
3196:
3176:
3155:
3129:
3077:
3055:
3034:
3015:
Alright. Looking everything over, I think all that's left is the six places where
3003:
2989:
2966:
2926:
2909:
2891:
2870:
2856:
2838:
2817:
2794:
2768:
2754:
2732:
2705:
2666:
2604:
2570:
2556:
2519:
2488:
2454:
2430:
2393:
2362:
2321:
2267:
2219:
2160:
2072:
2029:
1970:
1951:
1925:
1890:
1851:
1738:
1717:
1664:
1630:
1616:
1592:
1577:
1542:
1528:
1514:
866:
850:
655:
641:
621:
451:
2586:
Now let me move towards addressing the point. There was a discussion about this
275:
5394:
Note to anyone else interested in this discussion: it ended up taking place on
1607:
template uses Citation Style 1, but that appears to be more difficult to fix. —
3693:
3533:
To make such a remark so recklessly warrants an apology, but I'm not hopeful.
2200:
1651:
have inconsistent citation style, then it would be a concern. But not for the
252:
745:{\displaystyle 0\to \mathbb {Z} ^{n-1}\to {\widetilde {S}}_{n}\to S_{n}\to 0}
2211:
2013:
1730:
1671:
1656:
1584:
631:
458:
441:
144:
3775:
I believe it would be possible to write a more detailed history section on
2180:
be undone. We can't have two different cite styles within the same article.
1829:. The edit link for this section can be used to add comments to the review.
1680:. The RFC that led to deprecation of that format did not deprecate its use
3233:
2403:
2133:
2132:
There are also a fair amount of free-standing single-sentence grafs. Per
2087:
1568:, which may spell misfortune for a GA review. Might want to take a look.
1519:
Looks ok to me. I added \mathop to the formula to space the \to better. —
756:
of translations, affine permutations that don't permute the residues mod
3216:
Because I'm done with this review right here and right now, that's why.
2713:
Yes, nice suggestion, and thank you for showing me how it can be done.
2303:
1190:
is used in this article for all three of these sets (integer vectors in
3630:, where an endnote and both styles of parenthetical all crowd together.
2441:
2082:
2056:
of articles that seem like they would provide the necessary explanation
1992:
3491:
It's easier to find a citation than to argue over why it is not needed
2000:
complicated subject. If so, it speaks well of mathematics as a field.
3628:
Representation theory and an affine Robinson–Schensted correspondence
2143:. Going back to the above link to MOS:MATH, it explicitly says that "
2051:
3512:
3499:
been willing to meet me halfway on this, and I was really just that
2528:. Going back to the above link to MOSMATH, it explicitly says that "
2493:
There are also a fair amount of free-standing single-sentence grafs.
2248:
citations within any given article should follow a consistent style.
2058:
I didn't myself feel anywhere near qualified to choose the best one.
1358:
The relationship of the preceding paragraph may be expressed by the
5433:
5312:
Generating_function#Bivariate_and_multivariate_generating_functions
3771:
Affine_symmetric_group#Combinatorics_of_other_affine_Coxeter_groups
2191:
suggesting that citations can be used this way, and given that per
885:. The window notations of such affine permutations are of the form
5429:
2008:
nonstandard meaning that cannot be easily intuited from context):
4523:
The analogous combinatorial construction is to choose any subset
1910:
Affine symmetric group#Relationship to the finite symmetric group
3626:— is too choppy. This particularly sticks out in the subsection
2410:
I have significantly reduced the number of these; let's call it
5254:
In sentence "The generating function for these statistics over
5073:
In the section on the combinatorial definition, when you write
3779:, but I am skeptical that there is a lot more to be said about
2475:, but I am skeptical that there is a lot more to be said about
5247:
As a Coxeter group (or as a quotient of the braid group), the
3461:
Yes, I'm not a higher level mathematician, so I relied on the
188:
3602:. I see that this has been discussed above, and I think that
2530:
When defining a term, do not use the phrase 'if and only if'.
2145:
When defining a term, do not use the phrase 'if and only if'.
3503:
issue away from ending the review with a promotion when you
3463:
critiques of the math sourcing in the WikiJournal of Science
1708:
for the in-text ones, but that's a minor stylistic choice. —
461:, thanks very much for the kind words! About the notation
5647:
Knowledge Did you know articles that are featured articles
5642:
Featured articles that have appeared on the main page once
3025:
tags remain. Fix those and we're pretty much (ahem) good.
2050:
a link that could be made, from the third sentence; while
3914:
True statements that I was not immediately able to source
3600:
A 2019 paper by Lewis et al. proved the following formula
1729:, not that the relevant deprecation was applicable here.
3596:
In (Lewis et al. 2019), the following formula was proved
4150:
4092:{\displaystyle S_{a_{1}}\times \cdots \times S_{a_{k}}}
3969:
3278:
2682:
2525:
2498:
Affine symmetric group#Descents, length, and inversions
2140:
108:
89:
3089:
Oh ... OK ... I see what you're saying and I hear you.
1178:, that is, the isometries that shift the entire space
646:
That makes sense -- thanks again for your feedback! --
5637:
Featured articles that have appeared on the main page
5588:
5514:
5476:
5450:
5296:
5260:
5203:
5153:
5079:
5045:
5002:
4982:
4955:
4919:
4899:
4853:
4825:
4782:
4755:
4690:
4650:
4601:
4563:
4533:
4478:
4386:
4241:
4214:
4161:
4105:
4045:
4015:
3979:
3928:
3739:
3718:
1459:
1367:
1219:
1122:
1057:
992:
891:
817:
782:
762:
676:
581:
530:
503:
467:
417:
381:
345:
5416:
An object that might deserve mention in this article
3872:
been monitoring this and I'm OK with your decision.
3408:, where does this review stand? It seems clear that
3107:(Or whatever term for the image works best for you).
524:(sometimes rendered with the S in other fonts, like
274:, a collaborative effort to improve the coverage of
2183:And you have no choice of which style to adopt. It
5600:
5574:
5508:. (NB: there can be complicated order type, like
5500:
5462:
5302:
5282:
5236:
5189:
5139:
5058:
5031:
4988:
4968:
4941:
4905:
4883:
4838:
4811:
4768:
4703:
4672:
4636:
4587:
4541:
4505:
4464:
4363:
4227:
4183:
4137:
4091:
4028:
4001:
3960:
3761:
3724:
2896:Only by using a much less eye-catching image like
2460:I have made a first attempt at this, so let's say
2014:the intro can and should be more than the one graf
1486:
1445:
1348:
1166:
1108:
1043:
978:
832:
803:
768:
744:
603:
547:
516:
489:
432:
403:
367:
59:This article appeared on Knowledge's Main Page as
5366:are wikilinked to several times in the main text.
5181:
5160:
5140:{\displaystyle \sum _{i=1}^{n}u(i)=1+2+\cdots +n}
2203:after getting all the material properly sourced.
1167:{\displaystyle (a_{1},\ldots ,a_{n})\in \Lambda }
4506:{\displaystyle \mathbb {Z} ^{n-1}\cong \Lambda }
2375:lines that are still parallel, etc.) are called
1487:{\displaystyle \mathbb {Z} ^{n-1}\cong \Lambda }
3477:
3331:depending on how you're generating the image. —
2537:Affine symmetric group#Combinatorial definition
2147:" Some of the other guidance there is on point.
2110:math articles that have gained at least GA have
1914:Affine symmetric group#Combinatorial definition
5424:of Coxeter groups (general reference by Dyer:
2496:This was particularly the case in the section
559:, and I've even seen other random things like
4009:are all isomorphic to parabolic subgroups of
2679:expertise for their opinion on this question.
1174:. Geometrically, this kernel consists of the
8:
3091:These cats can be skinned in different ways.
2882:Is this something that can be easily fixed?
2524:I actually seemed to have been right before
2139:I actually seemed to have been right before
2012:First, at 60K total, it is long enough that
1343:
1220:
43:. Even so, if you can update or improve it,
39:as one of the best articles produced by the
33:; it (or a previous version of it) has been
3519:tact at all would have been an improvement.
2440:sources; for example, I don't know whether
2402:Ha, gosh, and I was so proud of myself for
2340:Yes, absolutely I agree, I will work on it.
1109:{\displaystyle a_{1}+a_{2}+\ldots +a_{n}=0}
1044:{\displaystyle (a_{1},a_{2},\ldots ,a_{n})}
4637:{\displaystyle ({\widetilde {S}}_{n})_{A}}
1758:
575:for the linear version). So the notation
220:
68:
15:
5587:
5513:
5475:
5449:
5295:
5274:
5263:
5262:
5259:
5207:
5202:
5180:
5159:
5157:
5152:
5095:
5084:
5078:
5050:
5044:
5020:
5007:
5001:
4981:
4960:
4954:
4933:
4922:
4921:
4918:
4898:
4863:
4858:
4852:
4830:
4824:
4800:
4787:
4781:
4760:
4754:
4695:
4689:
4664:
4653:
4652:
4649:
4628:
4618:
4607:
4606:
4600:
4562:
4535:
4534:
4532:
4513:is the kernel, a free abelian group with
4485:
4481:
4480:
4477:
4450:
4433:
4427:
4416:
4415:
4399:
4395:
4394:
4385:
4352:
4342:
4323:
4307:
4291:
4278:
4268:
4252:
4240:
4219:
4213:
4175:
4164:
4163:
4160:
4129:
4110:
4104:
4081:
4076:
4055:
4050:
4044:
4020:
4014:
3993:
3982:
3981:
3978:
3946:
3933:
3927:
3753:
3742:
3741:
3738:
3717:
1466:
1462:
1461:
1458:
1427:
1421:
1408:
1397:
1396:
1380:
1376:
1375:
1366:
1218:
1182:without rotating or reflecting it. In an
1149:
1130:
1121:
1094:
1075:
1062:
1056:
1032:
1013:
1000:
991:
961:
930:
905:
890:
824:
820:
819:
816:
789:
785:
784:
781:
761:
730:
717:
706:
705:
689:
685:
684:
675:
595:
584:
583:
580:
539:
533:
532:
529:
508:
502:
481:
470:
469:
466:
424:
419:
416:
395:
384:
383:
380:
359:
348:
347:
344:
3973:"The non-maximal parabolic subgroups of
3781:the affine symmetric group in particular
3413:(Or perhaps a request should be made at
2477:the affine symmetric group in particular
2026:we are strongly discouraged from linking
192:
2646:This is a chronic flaw in my writing.
2539:, the second sentence says "a function
1789:
1761:
222:
4680:of affine permutations that stabilize
4208:In terms of the Coxeter generators of
3844:
3774:
3642:
3638:
3627:
3623:
3619:
3615:
3607:
3599:
3595:
3549:
871:What about this as a starting point?
3105:See accompanying diagram</ref: -->
3094:For the first one, you're leaning on
2681:So based on David Eppstein's comment
7:
5283:{\displaystyle {\widetilde {S}}_{n}}
5237:{\displaystyle ={\frac {n(n+1)}{2}}}
4673:{\displaystyle {\widetilde {S}}_{n}}
4549:that contains one element from each
4184:{\displaystyle {\widetilde {S}}_{n}}
4002:{\displaystyle {\widetilde {S}}_{n}}
3762:{\displaystyle {\widetilde {S}}_{n}}
2033:
1678:for inline (non-footnote) references
1564:I believe the citation format isn't
604:{\displaystyle {\widetilde {S}}_{n}}
490:{\displaystyle {\widetilde {S}}_{n}}
404:{\displaystyle {\widetilde {S}}_{n}}
368:{\displaystyle {\widetilde {S}}_{n}}
268:This article is within the scope of
4138:{\displaystyle a_{1},\ldots ,a_{k}}
3187:questions in my previous comment? —
2643:? They sort of read that way to me.
2129:? They sort of read that way to me.
548:{\displaystyle {\mathfrak {S}}_{n}}
534:
211:It is of interest to the following
5190:{\displaystyle ={\binom {n+1}{2}}}
5164:
4500:
3719:
3142:can you please re-read the actual
2445:only dates back < 40 years.) --
1906:Affine symmetric group#Definitions
1481:
1161:
804:{\displaystyle \mathbb {Z} ^{n-1}}
14:
5657:Low-priority mathematics articles
3637:Something about the way the term
555:; other options are mentioned at
288:Knowledge:WikiProject Mathematics
4942:{\displaystyle {\tilde {S}}_{n}}
4884:{\displaystyle s_{i}^{2}=Id: -->
3777:affine Coxeter groups in general
3614:citation styles — a footnote to
2715:
2687:
2675:I am querying someone with more
2648:
2622:
2502:
2473:affine Coxeter groups in general
2462:
2412:
2344:
2302:
2034:
1956:Great, thanks, I will do that.
833:{\displaystyle \mathbb {Z} ^{n}}
291:Template:WikiProject Mathematics
255:
245:
224:
193:
130:
52:
19:
5358:I found a couple of duplinks :
3886:Thank you for letting me know!
3523:beggars can't be choosers). In
3458:who was working on the article.
2587:
2028:), to some article, or perhaps
1827:Talk:Affine symmetric group/GA1
1494:is the free abelian group with
1051:is an integer vector such that
564:"natural" one they might write
308:This article has been rated as
5440:) in terms of total orders on
5225:
5213:
5110:
5104:
5026:
5013:
4927:
4806:
4793:
4625:
4602:
4456:
4435:
4411:
4390:
4258:
4245:
1429:
1414:
1392:
1371:
1340:
1307:
1295:
1262:
1256:
1223:
1155:
1123:
1038:
993:
973:
892:
736:
723:
701:
680:
1:
5652:FA-Class mathematics articles
4736:Affine symmetric group review
4588:{\displaystyle 1+2+\ldots +n}
4205:(D'oh -- this is just false!)
3961:{\displaystyle x_{i}-x_{j}=k}
3896:21:00, 17 February 2022 (UTC)
3882:20:49, 17 February 2022 (UTC)
3857:20:32, 17 February 2022 (UTC)
3835:12:28, 17 February 2022 (UTC)
3814:02:26, 17 February 2022 (UTC)
3795:01:02, 17 February 2022 (UTC)
3667:20:54, 16 February 2022 (UTC)
3576:19:19, 16 February 2022 (UTC)
3562:16:07, 16 February 2022 (UTC)
3543:08:28, 16 February 2022 (UTC)
3446:06:54, 16 February 2022 (UTC)
3427:05:42, 16 February 2022 (UTC)
3004:05:39, 31 December 2021 (UTC)
2990:00:44, 29 December 2021 (UTC)
2967:01:04, 28 December 2021 (UTC)
2898:File:Uniform tiling 63-t2.svg
2871:08:21, 30 November 2021 (UTC)
2857:06:35, 30 November 2021 (UTC)
2839:00:23, 28 November 2021 (UTC)
2818:07:05, 18 November 2021 (UTC)
2795:12:25, 16 November 2021 (UTC)
2769:12:25, 16 November 2021 (UTC)
2755:22:56, 15 November 2021 (UTC)
2706:14:49, 15 November 2021 (UTC)
2605:12:25, 16 November 2021 (UTC)
2489:14:49, 15 November 2021 (UTC)
2455:11:48, 10 November 2021 (UTC)
2431:11:48, 10 November 2021 (UTC)
2394:11:48, 10 November 2021 (UTC)
1566:consistent within the article
752:where the first group is the
282:and see a list of open tasks.
5501:{\displaystyle a+n\prec b+n}
5354:Technical comments for FAC:
5032:{\displaystyle B_{n}(S^{1})}
4812:{\displaystyle B_{n}(S^{1})}
4731:11:30, 4 November 2021 (UTC)
4542:{\displaystyle \mathbb {Z} }
3379:01:53, 16 January 2022 (UTC)
3364:01:49, 16 January 2022 (UTC)
3341:02:09, 14 January 2022 (UTC)
3326:02:05, 14 January 2022 (UTC)
3305:02:35, 11 January 2022 (UTC)
3291:21:15, 10 January 2022 (UTC)
3273:21:12, 10 January 2022 (UTC)
3255:10:43, 10 January 2022 (UTC)
3226:05:59, 10 January 2022 (UTC)
3197:05:40, 10 January 2022 (UTC)
3177:05:37, 10 January 2022 (UTC)
3156:05:32, 10 January 2022 (UTC)
3130:04:59, 10 January 2022 (UTC)
2927:15:35, 3 December 2021 (UTC)
2910:07:12, 2 December 2021 (UTC)
2892:06:04, 2 December 2021 (UTC)
2733:12:17, 9 November 2021 (UTC)
2667:12:17, 9 November 2021 (UTC)
2571:18:50, 5 November 2021 (UTC)
2561:OK, no problem ... I defer.
2557:12:17, 9 November 2021 (UTC)
2520:12:17, 9 November 2021 (UTC)
2363:12:17, 9 November 2021 (UTC)
2322:06:14, 4 November 2021 (UTC)
2268:19:06, 5 November 2021 (UTC)
2230:You are correct in that the
2220:09:06, 4 November 2021 (UTC)
2161:05:47, 4 November 2021 (UTC)
2073:05:15, 4 November 2021 (UTC)
1971:23:44, 28 October 2021 (UTC)
1952:18:25, 28 October 2021 (UTC)
1926:18:01, 28 October 2021 (UTC)
1912:, and the one at the end of
1891:03:49, 26 October 2021 (UTC)
1852:03:49, 26 October 2021 (UTC)
5632:Knowledge featured articles
4099:for some positive integers
3078:00:57, 7 January 2022 (UTC)
3056:01:53, 5 January 2022 (UTC)
3035:22:54, 4 January 2022 (UTC)
2843:Looks good ... we got some
626:That makes a lot of sense,
568:for the affine version and
375:rather than another name. (
164:The text of the entry was:
5673:
5618:17:28, 5 August 2024 (UTC)
4557:and whose elements sum to
4199:that preserve the lattice
642:02:05, 30 April 2021 (UTC)
622:16:36, 29 April 2021 (UTC)
452:23:25, 28 April 2021 (UTC)
116:Featured article candidate
5408:17:09, 30 June 2023 (UTC)
4996:also the projection from
4745:questions/observations:
4235:, this can be written as
2745:OK; it was just an idea.
2256:featured article criteria
2030:the Wiktionary definition
1653:good article requirements
307:
240:
219:
179:
71:
67:
5463:{\displaystyle a\prec b}
5387:11:46, 5 June 2023 (UTC)
5377:More comments to follow.
5350:09:44, 5 June 2023 (UTC)
5327:09:44, 5 June 2023 (UTC)
4377:may be expressed by the
3725:{\displaystyle \Lambda }
2588:on the article talk-page
2224:Harvard references used
1908:, the one at the top of
1739:03:47, 8 July 2021 (UTC)
1718:21:51, 7 July 2021 (UTC)
1665:20:50, 7 July 2021 (UTC)
1631:17:41, 7 July 2021 (UTC)
1617:04:33, 7 July 2021 (UTC)
1593:02:26, 7 July 2021 (UTC)
1578:22:11, 6 July 2021 (UTC)
1543:20:33, 1 June 2021 (UTC)
1529:18:52, 1 June 2021 (UTC)
1515:17:47, 1 June 2021 (UTC)
867:15:53, 1 June 2021 (UTC)
851:19:11, 31 May 2021 (UTC)
656:15:53, 1 June 2021 (UTC)
314:project's priority scale
143:appeared on Knowledge's
61:Today's featured article
5360:infinite dihedral group
5290:simultaneously for all
3316:Indeed! It is done. --
1896:Comments from JayBeeEll
1862:I am not a math person
1642:(no longer to be used).
433:{\displaystyle {S}_{n}}
271:WikiProject Mathematics
5602:
5576:
5502:
5464:
5340:More comments to come.
5304:
5284:
5238:
5191:
5141:
5100:
5060:
5033:
4990:
4970:
4943:
4907:
4886:
4840:
4813:
4770:
4705:
4674:
4638:
4589:
4543:
4507:
4466:
4365:
4229:
4185:
4139:
4093:
4030:
4003:
3962:
3763:
3726:
3482:
2377:affine transformations
2052:we do have at least a
1602:Academic peer reviewed
1503:
1488:
1447:
1350:
1168:
1110:
1045:
980:
834:
805:
770:
746:
605:
549:
518:
491:
434:
405:
369:
201:This article is rated
169:affine symmetric group
141:fact from this article
27:Affine symmetric group
5603:
5577:
5503:
5465:
5305:
5285:
5239:
5192:
5142:
5080:
5061:
5059:{\displaystyle S_{n}}
5034:
4991:
4971:
4969:{\displaystyle S_{n}}
4944:
4908:
4887:
4841:
4839:{\displaystyle S^{1}}
4814:
4771:
4769:{\displaystyle S^{1}}
4706:
4704:{\displaystyle S_{n}}
4675:
4639:
4590:
4544:
4508:
4467:
4366:
4230:
4228:{\displaystyle S_{n}}
4193:rigid transformations
4186:
4140:
4094:
4031:
4029:{\displaystyle S_{n}}
4004:
3963:
3764:
3727:
3144:good article criteria
2381:Affine transformation
2232:good article criteria
1489:
1448:
1351:
1169:
1111:
1046:
981:
872:
835:
806:
771:
747:
606:
550:
519:
517:{\displaystyle S_{n}}
492:
435:
406:
370:
5586:
5512:
5474:
5448:
5294:
5258:
5201:
5151:
5077:
5043:
5000:
4989:{\displaystyle \pi }
4980:
4953:
4917:
4906:{\displaystyle \pi }
4897:
4851:
4846:under the relations
4823:
4780:
4753:
4688:
4648:
4599:
4561:
4531:
4476:
4384:
4379:short exact sequence
4239:
4212:
4159:
4103:
4043:
4013:
3977:
3926:
3737:
3716:
2847:action going there!
2609:And per the above, "
2097:And per the above, "
1870:is there as I type.
1457:
1365:
1360:short exact sequence
1217:
1120:
1055:
990:
889:
815:
780:
760:
674:
666:Short exact sequence
579:
528:
501:
465:
415:
379:
343:
294:mathematics articles
97:Good article nominee
5601:{\displaystyle n=2}
5422:extended weak order
4868:
4191:to be the group of
3710:type A root lattice
3682:parabolic subgroups
3639:type A root lattice
2945:Christmas greetings
2526:this partial revert
2141:this partial revert
63:on October 8, 2023.
41:Knowledge community
5598:
5572:
5498:
5460:
5300:
5280:
5234:
5187:
5137:
5056:
5029:
4986:
4966:
4939:
4903:
4881:
4854:
4836:
4809:
4766:
4701:
4670:
4634:
4585:
4539:
4503:
4462:
4361:
4225:
4181:
4135:
4089:
4026:
3999:
3958:
3759:
3722:
3509:the angry mastodon
3485:And there is also
3456:every other editor
2579:Harvard references
2170:sentences, but as
1988:peer reviews there
1484:
1443:
1346:
1164:
1106:
1041:
976:
830:
801:
766:
754:free abelian group
742:
601:
545:
514:
487:
430:
401:
365:
263:Mathematics portal
207:content assessment
72:Article milestones
5438:extended abstract
5396:my user talk-page
5303:{\displaystyle n}
5271:
5232:
5179:
4930:
4684:is isomorphic to
4661:
4615:
4441:
4424:
4172:
3990:
3750:
2916:
2151:Saving again ...
1900:Thanks very much
1817:
1816:
1533:Great, thanks! --
1435:
1405:
1184:abuse of notation
1116:, that is, where
769:{\displaystyle n}
714:
592:
478:
392:
356:
331:Beautiful article
328:
327:
324:
323:
320:
319:
187:
186:
125:
124:
90:February 17, 2022
5664:
5607:
5605:
5604:
5599:
5581:
5579:
5578:
5573:
5507:
5505:
5504:
5499:
5469:
5467:
5466:
5461:
5364:identity element
5309:
5307:
5306:
5301:
5289:
5287:
5286:
5281:
5279:
5278:
5273:
5272:
5264:
5243:
5241:
5240:
5235:
5233:
5228:
5208:
5196:
5194:
5193:
5188:
5186:
5185:
5184:
5175:
5163:
5146:
5144:
5143:
5138:
5099:
5094:
5065:
5063:
5062:
5057:
5055:
5054:
5038:
5036:
5035:
5030:
5025:
5024:
5012:
5011:
4995:
4993:
4992:
4987:
4975:
4973:
4972:
4967:
4965:
4964:
4948:
4946:
4945:
4940:
4938:
4937:
4932:
4931:
4923:
4912:
4910:
4909:
4904:
4892:
4889:
4888:
4882:
4867:
4862:
4845:
4843:
4842:
4837:
4835:
4834:
4818:
4816:
4815:
4810:
4805:
4804:
4792:
4791:
4775:
4773:
4772:
4767:
4765:
4764:
4710:
4708:
4707:
4702:
4700:
4699:
4683:
4679:
4677:
4676:
4671:
4669:
4668:
4663:
4662:
4654:
4643:
4641:
4640:
4635:
4633:
4632:
4623:
4622:
4617:
4616:
4608:
4594:
4592:
4591:
4586:
4556:
4548:
4546:
4545:
4540:
4538:
4526:
4519:
4512:
4510:
4509:
4504:
4496:
4495:
4484:
4471:
4469:
4468:
4463:
4455:
4454:
4442:
4434:
4432:
4431:
4426:
4425:
4417:
4410:
4409:
4398:
4376:
4370:
4368:
4367:
4362:
4357:
4356:
4347:
4346:
4334:
4333:
4318:
4317:
4302:
4301:
4283:
4282:
4273:
4272:
4257:
4256:
4234:
4232:
4231:
4226:
4224:
4223:
4202:
4198:
4190:
4188:
4187:
4182:
4180:
4179:
4174:
4173:
4165:
4148:
4144:
4142:
4141:
4136:
4134:
4133:
4115:
4114:
4098:
4096:
4095:
4090:
4088:
4087:
4086:
4085:
4062:
4061:
4060:
4059:
4036:, that is, to a
4035:
4033:
4032:
4027:
4025:
4024:
4008:
4006:
4005:
4000:
3998:
3997:
3992:
3991:
3983:
3967:
3965:
3964:
3959:
3951:
3950:
3938:
3937:
3867:
3824:
3768:
3766:
3765:
3760:
3758:
3757:
3752:
3751:
3743:
3731:
3729:
3728:
3723:
3698:Reflection group
3622:and also inline
3487:this section hed
3351:
3315:
3166:
3141:
3119:
3113:
3106:
3088:
3066:
3045:
3024:
3018:
3014:
2955:
2914:
2881:
2828:
2806:
2784:
2723:
2719:
2718:
2695:
2691:
2690:
2656:
2652:
2651:
2642:
2636:
2630:
2626:
2625:
2510:
2506:
2505:
2470:
2466:
2465:
2420:
2416:
2415:
2352:
2348:
2347:
2306:
2128:
2122:
2046:As it is, there
2039:
2038:
2037:
1937:
1771:Copyvio detector
1759:
1707:
1701:
1697:
1691:
1675:
1647:If this article
1606:
1600:
1563:
1500:
1493:
1491:
1490:
1485:
1477:
1476:
1465:
1452:
1450:
1449:
1444:
1436:
1428:
1426:
1425:
1413:
1412:
1407:
1406:
1398:
1391:
1390:
1379:
1355:
1353:
1352:
1347:
1212:
1197:
1193:
1189:
1181:
1173:
1171:
1170:
1165:
1154:
1153:
1135:
1134:
1115:
1113:
1112:
1107:
1099:
1098:
1080:
1079:
1067:
1066:
1050:
1048:
1047:
1042:
1037:
1036:
1018:
1017:
1005:
1004:
985:
983:
982:
979:{\displaystyle }
977:
966:
965:
935:
934:
910:
909:
880:
839:
837:
836:
831:
829:
828:
823:
810:
808:
807:
802:
800:
799:
788:
775:
773:
772:
767:
751:
749:
748:
743:
735:
734:
722:
721:
716:
715:
707:
700:
699:
688:
634:
610:
608:
607:
602:
600:
599:
594:
593:
585:
554:
552:
551:
546:
544:
543:
538:
537:
523:
521:
520:
515:
513:
512:
496:
494:
493:
488:
486:
485:
480:
479:
471:
444:
439:
437:
436:
431:
429:
428:
423:
410:
408:
407:
402:
400:
399:
394:
393:
385:
374:
372:
371:
366:
364:
363:
358:
357:
349:
296:
295:
292:
289:
286:
265:
260:
259:
249:
242:
241:
236:
228:
221:
204:
198:
197:
189:
182:Featured article
180:Current status:
134:
111:
92:
69:
56:
31:featured article
23:
16:
5672:
5671:
5667:
5666:
5665:
5663:
5662:
5661:
5622:
5621:
5584:
5583:
5510:
5509:
5472:
5471:
5470:if and only if
5446:
5445:
5436:& Barkley:
5418:
5292:
5291:
5261:
5256:
5255:
5209:
5199:
5198:
5165:
5158:
5149:
5148:
5075:
5074:
5046:
5041:
5040:
5016:
5003:
4998:
4997:
4978:
4977:
4956:
4951:
4950:
4920:
4915:
4914:
4895:
4894:
4848:
4847:
4826:
4821:
4820:
4796:
4783:
4778:
4777:
4756:
4751:
4750:
4738:
4691:
4686:
4685:
4681:
4651:
4646:
4645:
4624:
4605:
4597:
4596:
4595:; the subgroup
4559:
4558:
4554:
4551:conjugacy class
4529:
4528:
4524:
4514:
4479:
4474:
4473:
4446:
4414:
4393:
4382:
4381:
4374:
4348:
4338:
4319:
4303:
4287:
4274:
4264:
4248:
4237:
4236:
4215:
4210:
4209:
4200:
4196:
4162:
4157:
4156:
4155:One may define
4146:
4125:
4106:
4101:
4100:
4077:
4072:
4051:
4046:
4041:
4040:
4016:
4011:
4010:
3980:
3975:
3974:
3942:
3929:
3924:
3923:
3916:
3861:
3818:
3740:
3735:
3734:
3714:
3713:
3706:Coxeter diagram
3677:Core partitions
3649:
3399:
3345:
3309:
3160:
3135:
3117:
3111:
3103:
3082:
3060:
3039:
3022:
3016:
3008:
2978:Columbia County
2949:
2947:
2875:
2822:
2800:
2778:
2716:
2714:
2688:
2686:
2649:
2647:
2640:
2634:
2623:
2621:
2503:
2501:
2463:
2461:
2435:history section
2413:
2411:
2372:affine geometry
2345:
2343:
2329:
2312:Happy editing!
2126:
2120:
2035:
1981:
1931:
1898:
1821:This review is
1813:
1785:
1757:
1705:
1699:
1695:
1689:
1686:in article text
1669:
1604:
1598:
1557:
1555:
1495:
1460:
1455:
1454:
1417:
1395:
1374:
1363:
1362:
1215:
1214:
1207:
1195:
1191:
1187:
1179:
1145:
1126:
1118:
1117:
1090:
1071:
1058:
1053:
1052:
1028:
1009:
996:
988:
987:
957:
926:
901:
887:
886:
878:
818:
813:
812:
783:
778:
777:
758:
757:
726:
704:
683:
672:
671:
668:
632:
582:
577:
576:
574:
557:Symmetric group
531:
526:
525:
504:
499:
498:
468:
463:
462:
442:
418:
413:
412:
382:
377:
376:
346:
341:
340:
333:
293:
290:
287:
284:
283:
261:
254:
234:
205:on Knowledge's
202:
175:
174:
162:
109:August 22, 2023
107:
88:
12:
11:
5:
5670:
5668:
5660:
5659:
5654:
5649:
5644:
5639:
5634:
5624:
5623:
5597:
5594:
5591:
5571:
5568:
5565:
5562:
5559:
5556:
5553:
5550:
5547:
5544:
5541:
5538:
5535:
5532:
5529:
5526:
5523:
5520:
5517:
5497:
5494:
5491:
5488:
5485:
5482:
5479:
5459:
5456:
5453:
5417:
5414:
5413:
5412:
5411:
5410:
5375:
5374:
5367:
5338:
5337:
5333:
5329:
5314:
5299:
5277:
5270:
5267:
5252:
5245:
5231:
5227:
5224:
5221:
5218:
5215:
5212:
5206:
5183:
5178:
5174:
5171:
5168:
5162:
5156:
5136:
5133:
5130:
5127:
5124:
5121:
5118:
5115:
5112:
5109:
5106:
5103:
5098:
5093:
5090:
5087:
5083:
5071:
5053:
5049:
5028:
5023:
5019:
5015:
5010:
5006:
4985:
4963:
4959:
4936:
4929:
4926:
4902:
4880:
4877:
4874:
4871:
4866:
4861:
4857:
4833:
4829:
4808:
4803:
4799:
4795:
4790:
4786:
4763:
4759:
4737:
4734:
4719:
4718:
4715:
4712:
4698:
4694:
4667:
4660:
4657:
4631:
4627:
4621:
4614:
4611:
4604:
4584:
4581:
4578:
4575:
4572:
4569:
4566:
4537:
4521:
4502:
4499:
4494:
4491:
4488:
4483:
4461:
4458:
4453:
4449:
4445:
4440:
4437:
4430:
4423:
4420:
4413:
4408:
4405:
4402:
4397:
4392:
4389:
4371:
4360:
4355:
4351:
4345:
4341:
4337:
4332:
4329:
4326:
4322:
4316:
4313:
4310:
4306:
4300:
4297:
4294:
4290:
4286:
4281:
4277:
4271:
4267:
4263:
4260:
4255:
4251:
4247:
4244:
4222:
4218:
4206:
4178:
4171:
4168:
4152:
4132:
4128:
4124:
4121:
4118:
4113:
4109:
4084:
4080:
4075:
4071:
4068:
4065:
4058:
4054:
4049:
4038:Young subgroup
4023:
4019:
3996:
3989:
3986:
3971:
3957:
3954:
3949:
3945:
3941:
3936:
3932:
3915:
3912:
3911:
3910:
3909:
3908:
3907:
3906:
3905:
3904:
3903:
3902:
3901:
3900:
3899:
3898:
3801:core partition
3756:
3749:
3746:
3721:
3702:Dynkin diagram
3655:
3651:
3647:
3635:
3631:
3591:
3584:
3583:
3582:
3581:
3580:
3579:
3578:
3554:David Eppstein
3483:
3475:
3438:David Eppstein
3417:?) Thank you.
3406:David Eppstein
3398:
3395:
3394:
3393:
3392:
3391:
3390:
3389:
3388:
3387:
3386:
3385:
3384:
3383:
3382:
3381:
3371:David Eppstein
3348:David Eppstein
3333:David Eppstein
3293:
3242:
3238:
3214:
3213:
3212:
3211:
3210:
3209:
3208:
3207:
3206:
3205:
3204:
3203:
3202:
3201:
3200:
3199:
3189:David Eppstein
3163:David Eppstein
3148:David Eppstein
3058:
2946:
2943:
2942:
2941:
2940:
2939:
2938:
2937:
2936:
2935:
2934:
2933:
2932:
2931:
2930:
2929:
2902:David Eppstein
2878:David Eppstein
2863:David Eppstein
2776:
2775:
2774:
2773:
2772:
2771:
2739:
2735:
2711:
2708:
2672:
2669:
2644:
2631:
2618:
2607:
2584:
2580:
2576:
2575:
2574:
2573:
2533:
2522:
2494:
2491:
2458:
2436:
2433:
2400:
2396:
2368:
2365:
2337:
2328:
2325:
2296:
2295:
2288:
2280:
2276:
2275:
2274:
2273:
2272:
2271:
2199:thing in this
2197:most essential
2149:
2148:
2137:
2130:
2117:
2106:
2095:
2061:
2060:
2021:
1980:
1977:
1976:
1975:
1974:
1973:
1897:
1894:
1832:
1831:
1815:
1814:
1812:
1811:
1806:
1801:
1795:
1792:
1791:
1787:
1786:
1784:
1783:
1781:External links
1778:
1773:
1767:
1764:
1763:
1756:
1753:
1752:
1751:
1750:
1749:
1748:
1747:
1746:
1745:
1744:
1743:
1742:
1741:
1723:David Eppstein
1710:David Eppstein
1645:
1643:
1636:Horsesizedduck
1623:Horsesizedduck
1609:David Eppstein
1570:Horsesizedduck
1554:
1551:
1550:
1549:
1548:
1547:
1546:
1545:
1521:David Eppstein
1483:
1480:
1475:
1472:
1469:
1464:
1442:
1439:
1434:
1431:
1424:
1420:
1416:
1411:
1404:
1401:
1394:
1389:
1386:
1383:
1378:
1373:
1370:
1357:
1345:
1342:
1339:
1336:
1333:
1330:
1327:
1324:
1321:
1318:
1315:
1312:
1309:
1306:
1303:
1300:
1297:
1294:
1291:
1288:
1285:
1282:
1279:
1276:
1273:
1270:
1267:
1264:
1261:
1258:
1255:
1252:
1249:
1246:
1243:
1240:
1237:
1234:
1231:
1228:
1225:
1222:
1163:
1160:
1157:
1152:
1148:
1144:
1141:
1138:
1133:
1129:
1125:
1105:
1102:
1097:
1093:
1089:
1086:
1083:
1078:
1074:
1070:
1065:
1061:
1040:
1035:
1031:
1027:
1024:
1021:
1016:
1012:
1008:
1003:
999:
995:
975:
972:
969:
964:
960:
956:
953:
950:
947:
944:
941:
938:
933:
929:
925:
922:
919:
916:
913:
908:
904:
900:
897:
894:
869:
843:David Eppstein
827:
822:
798:
795:
792:
787:
765:
741:
738:
733:
729:
725:
720:
713:
710:
703:
698:
695:
692:
687:
682:
679:
667:
664:
663:
662:
661:
660:
659:
658:
598:
591:
588:
572:
542:
536:
511:
507:
484:
477:
474:
427:
422:
398:
391:
388:
362:
355:
352:
332:
329:
326:
325:
322:
321:
318:
317:
306:
300:
299:
297:
280:the discussion
267:
266:
250:
238:
237:
229:
217:
216:
210:
199:
185:
184:
177:
176:
163:
138:
137:
135:
127:
126:
123:
122:
119:
112:
104:
103:
100:
93:
85:
84:
81:
78:
74:
73:
65:
64:
57:
49:
48:
24:
13:
10:
9:
6:
4:
3:
2:
5669:
5658:
5655:
5653:
5650:
5648:
5645:
5643:
5640:
5638:
5635:
5633:
5630:
5629:
5627:
5620:
5619:
5615:
5611:
5595:
5592:
5589:
5569:
5566:
5563:
5560:
5557:
5554:
5551:
5548:
5545:
5542:
5539:
5536:
5533:
5530:
5527:
5524:
5521:
5518:
5515:
5495:
5492:
5489:
5486:
5483:
5480:
5477:
5457:
5454:
5451:
5443:
5439:
5435:
5431:
5427:
5423:
5415:
5409:
5405:
5401:
5397:
5393:
5392:
5391:
5390:
5389:
5388:
5384:
5380:
5372:
5368:
5365:
5361:
5357:
5356:
5355:
5352:
5351:
5347:
5343:
5334:
5330:
5328:
5324:
5320:
5315:
5313:
5297:
5275:
5268:
5265:
5253:
5250:
5246:
5229:
5222:
5219:
5216:
5210:
5204:
5176:
5172:
5169:
5166:
5154:
5134:
5131:
5128:
5125:
5122:
5119:
5116:
5113:
5107:
5101:
5096:
5091:
5088:
5085:
5081:
5072:
5069:
5051:
5047:
5021:
5017:
5008:
5004:
4983:
4961:
4957:
4934:
4924:
4900:
4878:
4875:
4872:
4869:
4864:
4859:
4855:
4831:
4827:
4819:of braids on
4801:
4797:
4788:
4784:
4761:
4757:
4748:
4747:
4746:
4743:
4735:
4733:
4732:
4728:
4724:
4716:
4713:
4696:
4692:
4665:
4658:
4655:
4629:
4619:
4612:
4609:
4582:
4579:
4576:
4573:
4570:
4567:
4564:
4552:
4522:
4517:
4497:
4492:
4489:
4486:
4459:
4451:
4447:
4443:
4438:
4428:
4421:
4418:
4406:
4403:
4400:
4387:
4380:
4372:
4358:
4353:
4349:
4343:
4339:
4335:
4330:
4327:
4324:
4320:
4314:
4311:
4308:
4304:
4298:
4295:
4292:
4288:
4284:
4279:
4275:
4269:
4265:
4261:
4253:
4249:
4242:
4220:
4216:
4207:
4204:
4194:
4176:
4169:
4166:
4153:
4151:
4130:
4126:
4122:
4119:
4116:
4111:
4107:
4082:
4078:
4073:
4069:
4066:
4063:
4056:
4052:
4047:
4039:
4021:
4017:
3994:
3987:
3984:
3972:
3970:
3955:
3952:
3947:
3943:
3939:
3934:
3930:
3921:
3920:
3919:
3913:
3897:
3893:
3889:
3885:
3884:
3883:
3879:
3875:
3871:
3865:
3860:
3859:
3858:
3854:
3850:
3846:
3842:
3838:
3837:
3836:
3832:
3828:
3822:
3817:
3816:
3815:
3811:
3807:
3802:
3798:
3797:
3796:
3792:
3788:
3784:
3782:
3778:
3772:
3754:
3747:
3744:
3711:
3707:
3703:
3699:
3695:
3691:
3690:Coxeter group
3687:
3683:
3678:
3674:
3670:
3669:
3668:
3664:
3660:
3656:
3652:
3644:
3640:
3636:
3632:
3629:
3625:
3621:
3617:
3613:
3609:
3605:
3601:
3597:
3592:
3589:
3588:only one edit
3585:
3577:
3573:
3569:
3565:
3564:
3563:
3559:
3555:
3551:
3547:
3546:
3545:
3544:
3540:
3536:
3531:
3529:
3526:
3520:
3518:
3515:... in fact,
3514:
3510:
3506:
3502:
3498:
3492:
3488:
3484:
3481:
3476:
3474:
3472:
3466:
3464:
3459:
3457:
3449:
3448:
3447:
3443:
3439:
3435:
3431:
3430:
3429:
3428:
3424:
3420:
3416:
3411:
3407:
3403:
3396:
3380:
3376:
3372:
3367:
3366:
3365:
3361:
3357:
3349:
3344:
3343:
3342:
3338:
3334:
3329:
3328:
3327:
3323:
3319:
3313:
3308:
3307:
3306:
3302:
3298:
3294:
3292:
3288:
3284:
3280:
3279:tried a thing
3276:
3275:
3274:
3270:
3266:
3262:
3258:
3257:
3256:
3252:
3248:
3243:
3239:
3235:
3230:
3229:
3228:
3227:
3223:
3219:
3198:
3194:
3190:
3185:
3180:
3179:
3178:
3174:
3170:
3164:
3159:
3158:
3157:
3153:
3149:
3145:
3139:
3134:
3133:
3132:
3131:
3127:
3123:
3116:
3108:
3101:
3097:
3092:
3086:
3081:
3080:
3079:
3075:
3071:
3064:
3059:
3057:
3053:
3049:
3043:
3038:
3037:
3036:
3032:
3028:
3021:
3012:
3007:
3006:
3005:
3001:
2997:
2993:
2992:
2991:
2987:
2983:
2979:
2975:
2971:
2970:
2969:
2968:
2964:
2960:
2953:
2944:
2928:
2924:
2920:
2913:
2912:
2911:
2907:
2903:
2899:
2895:
2894:
2893:
2889:
2885:
2879:
2874:
2873:
2872:
2868:
2864:
2860:
2859:
2858:
2854:
2850:
2846:
2842:
2841:
2840:
2836:
2832:
2826:
2821:
2820:
2819:
2815:
2811:
2804:
2799:
2798:
2797:
2796:
2792:
2788:
2782:
2770:
2766:
2762:
2758:
2757:
2756:
2752:
2748:
2744:
2743:
2740:
2736:
2734:
2730:
2726:
2722:
2712:
2709:
2707:
2703:
2699:
2694:
2684:
2680:
2678:
2673:
2670:
2668:
2664:
2660:
2655:
2645:
2639:
2632:
2629:
2619:
2616:
2612:
2608:
2606:
2602:
2598:
2593:
2589:
2585:
2581:
2578:
2577:
2572:
2568:
2564:
2560:
2559:
2558:
2554:
2550:
2546:
2542:
2538:
2534:
2531:
2527:
2523:
2521:
2517:
2513:
2509:
2499:
2495:
2492:
2490:
2486:
2482:
2478:
2474:
2469:
2459:
2457:
2456:
2452:
2448:
2443:
2437:
2434:
2432:
2428:
2424:
2419:
2409:
2405:
2404:avoiding "we"
2401:
2397:
2395:
2391:
2387:
2382:
2378:
2373:
2369:
2366:
2364:
2360:
2356:
2351:
2341:
2338:
2335:
2334:
2333:
2326:
2324:
2323:
2319:
2315:
2310:
2309:
2305:
2300:
2293:
2289:
2286:
2281:
2278:
2277:
2270:
2269:
2265:
2261:
2257:
2253:
2249:
2245:
2241:
2237:
2233:
2227:
2223:
2222:
2221:
2217:
2213:
2209:
2206:
2205:
2204:
2202:
2198:
2194:
2190:
2186:
2181:
2179:
2173:
2169:
2165:
2164:
2163:
2162:
2158:
2154:
2146:
2142:
2138:
2135:
2131:
2125:
2118:
2116:
2111:
2107:
2104:
2100:
2096:
2094:
2091:
2089:
2088:MOS:MATH#NOWE
2084:
2077:
2076:
2075:
2074:
2070:
2066:
2059:
2057:
2055:
2049:
2043:
2031:
2027:
2022:
2020:
2015:
2011:
2010:
2009:
2006:
2001:
1997:
1995:
1994:
1989:
1984:
1978:
1972:
1968:
1964:
1960:
1955:
1954:
1953:
1949:
1945:
1941:
1935:
1930:
1929:
1928:
1927:
1923:
1919:
1915:
1911:
1907:
1903:
1895:
1893:
1892:
1888:
1884:
1879:
1877:
1871:
1869:
1865:
1860:
1857:
1854:
1853:
1849:
1846:
1843:
1839:
1836:
1830:
1828:
1824:
1819:
1818:
1810:
1807:
1805:
1802:
1800:
1797:
1796:
1794:
1793:
1788:
1782:
1779:
1777:
1774:
1772:
1769:
1768:
1766:
1765:
1760:
1754:
1740:
1736:
1732:
1728:
1724:
1721:
1720:
1719:
1715:
1711:
1704:
1694:
1687:
1683:
1679:
1673:
1668:
1667:
1666:
1662:
1658:
1654:
1650:
1646:
1644:
1641:
1640:is deprecated
1637:
1634:
1633:
1632:
1628:
1624:
1620:
1619:
1618:
1614:
1610:
1603:
1596:
1595:
1594:
1590:
1586:
1582:
1581:
1580:
1579:
1575:
1571:
1567:
1561:
1552:
1544:
1540:
1536:
1532:
1531:
1530:
1526:
1522:
1518:
1517:
1516:
1512:
1508:
1502:
1498:
1478:
1473:
1470:
1467:
1440:
1437:
1432:
1422:
1418:
1409:
1402:
1399:
1387:
1384:
1381:
1368:
1361:
1337:
1334:
1331:
1328:
1325:
1322:
1319:
1316:
1313:
1310:
1304:
1301:
1298:
1292:
1289:
1286:
1283:
1280:
1277:
1274:
1271:
1268:
1265:
1259:
1253:
1250:
1247:
1244:
1241:
1238:
1235:
1232:
1229:
1226:
1210:
1205:
1201:
1200:abelian group
1186:, the symbol
1185:
1177:
1158:
1150:
1146:
1142:
1139:
1136:
1131:
1127:
1103:
1100:
1095:
1091:
1087:
1084:
1081:
1076:
1072:
1068:
1063:
1059:
1033:
1029:
1025:
1022:
1019:
1014:
1010:
1006:
1001:
997:
970:
967:
962:
958:
954:
951:
948:
945:
942:
939:
936:
931:
927:
923:
920:
917:
914:
911:
906:
902:
898:
895:
884:
877:
870:
868:
864:
860:
855:
854:
853:
852:
848:
844:
825:
796:
793:
790:
763:
755:
739:
731:
727:
718:
711:
708:
696:
693:
690:
677:
665:
657:
653:
649:
645:
644:
643:
639:
635:
629:
625:
624:
623:
619:
615:
596:
589:
586:
571:
567:
562:
558:
540:
509:
505:
482:
475:
472:
460:
456:
455:
454:
453:
449:
445:
425:
420:
396:
389:
386:
360:
353:
350:
337:
330:
315:
311:
305:
302:
301:
298:
281:
277:
273:
272:
264:
258:
253:
251:
248:
244:
243:
239:
233:
230:
227:
223:
218:
214:
208:
200:
196:
191:
190:
183:
178:
173:
171:
170:
160:
159:
154:
152:
151:Did you know?
146:
142:
136:
133:
129:
128:
120:
118:
117:
113:
110:
106:
105:
101:
99:
98:
94:
91:
87:
86:
82:
79:
76:
75:
70:
66:
62:
58:
55:
51:
50:
46:
42:
38:
37:
32:
28:
25:
22:
18:
17:
5441:
5421:
5419:
5376:
5353:
5339:
5249:word problem
4739:
4720:
4515:
4154:
3917:
3869:
3841:instructions
3780:
3776:
3709:
3685:
3624:(Smith 2005)
3620:Smith (2005)
3616:Smith (2005)
3611:
3586:I have made
3532:
3521:
3516:
3504:
3500:
3496:
3494:
3490:
3478:
3467:
3460:
3455:
3452:
3400:
3397:Status query
3215:
3109:
3104:<ref: -->
3099:
3093:
3090:
2948:
2777:
2720:
2692:
2674:
2653:
2627:
2613:" is in the
2610:
2591:
2544:
2540:
2529:
2507:
2476:
2472:
2467:
2438:
2417:
2407:
2349:
2339:
2330:
2311:
2307:
2301:
2297:
2291:
2251:
2247:
2243:
2240:WP:CITESTYLE
2235:
2229:
2225:
2207:
2196:
2184:
2182:
2177:
2175:
2171:
2167:
2150:
2144:
2113:
2101:" is in the
2098:
2092:
2080:
2062:
2053:
2047:
2045:
2041:
2017:
2004:
2002:
1998:
1991:
1985:
1982:
1957:
1939:
1899:
1880:
1872:
1863:
1861:
1858:
1855:
1844:
1834:
1833:
1820:
1809:Instructions
1726:
1698:rather than
1685:
1682:in footnotes
1681:
1677:
1648:
1556:
1501:generators.
1496:
1208:
1202:, generated
1176:translations
873:
811:rather than
669:
569:
565:
560:
338:
334:
310:Low-priority
309:
269:
235:Low‑priority
213:WikiProjects
181:
167:
165:
158:May 10, 2021
156:
148:
114:
95:
45:please do so
34:
26:
4520:generators.
3874:Daniel Case
3568:Daniel Case
3535:Daniel Case
3419:BlueMoonset
3410:Daniel Case
3312:Notsniwiast
3261:Notsniwiast
3259:Hum, well,
3218:Daniel Case
3169:Daniel Case
3138:Daniel Case
3122:Daniel Case
3063:Daniel Case
3042:Daniel Case
3027:Daniel Case
2996:Daniel Case
2974:Daniel Case
2959:Daniel Case
2884:Daniel Case
2849:Daniel Case
2825:Daniel Case
2810:Daniel Case
2781:Daniel Case
2747:Daniel Case
2657:(I think).
2563:Daniel Case
2421:for now. --
2314:Daniel Case
2260:Daniel Case
2153:Daniel Case
2065:Daniel Case
2063:Saving ...
1944:Daniel Case
1902:Daniel Case
1883:Daniel Case
1838:Daniel Case
1823:transcluded
285:Mathematics
276:mathematics
232:Mathematics
5626:Categories
5444:that obey
3888:XOR'easter
3864:XOR'easter
3849:XOR'easter
3821:XOR'easter
3806:XOR'easter
3694:Weyl group
3673:XOR'easter
3659:XOR'easter
3606:'s desire
3353:PDF -: -->
2615:imperative
2236:explicitly
2201:punch list
2193:WP:HARVARD
2189:WP:HARVREF
2103:imperative
1776:Authorship
1762:GA toolbox
1727:references
1553:References
155:column on
36:identified
4891:}" /: -->
4742:JayBeeEll
4145:with sum
3686:algebraic
3654:expanded.
3618:, inline
3434:this diff
3241:footnote.
3085:JayBeeEll
3011:JayBeeEll
2952:JayBeeEll
2803:JayBeeEll
2353:for now.
1934:JayBeeEll
1835:Reviewer:
1799:Templates
1790:Reviewing
1755:GA Review
1560:JayBeeEll
628:JayBeeEll
411:contains
145:Main Page
5317:notion ?
4976:, so is
4472:. Here
3657:Cheers,
3237:picture.
3234:addition
2677:WP:NPROF
2671:J.Y. Shi
2327:Response
2242:, which
2134:MOS:PARA
1959:article.
1848:contribs
1804:Criteria
1453:. Here
1213:vectors
1198:into an
986:, where
883:identity
203:FA-class
121:Promoted
5426:journal
5379:Iry-Hor
5371:the MOS
5342:Iry-Hor
5319:Iry-Hor
4850:}": -->
4553:modulo
3634:though.
3297:Winston
3247:Winston
2442:Coxeter
2308:On hold
2254:in the
2234:do not
2208:Comment
2115:camera.
2083:MOS:YOU
1993:de novo
1876:WT:MATH
1703:harvtxt
1505:Bah.)--
1206:by the
312:on the
147:in the
80:Process
3643:type A
3507:to be
3415:WT:GAN
2399:"one"?
2292:future
2246:say: "
2226:inline
2168:within
2054:couple
1979:Review
1864:at all
1204:freely
876:kernel
776:(with
209:scale.
102:Listed
83:Result
5582:when
5434:arXiv
5430:arXiv
5398:. --
4913:from
4879:: -->
4740:Dear
3612:three
3528:other
3184:creep
3096:WP:OI
2845:Joker
2738:that.
2172:nouns
2081:Yes,
1825:from
29:is a
5614:talk
5420:The
5404:talk
5383:talk
5362:and
5346:talk
5323:talk
5068:here
4727:talk
3892:talk
3878:talk
3870:have
3853:talk
3843:say
3831:talk
3810:talk
3791:talk
3663:talk
3598:and
3572:talk
3558:talk
3539:talk
3525:some
3513:tact
3495:JBL
3471:this
3442:talk
3423:talk
3375:talk
3360:talk
3337:talk
3322:talk
3301:talk
3287:talk
3269:talk
3251:talk
3222:talk
3193:talk
3173:talk
3152:talk
3126:talk
3074:talk
3052:talk
3031:talk
3020:fact
3000:talk
2986:talk
2963:talk
2923:talk
2906:talk
2888:talk
2867:talk
2853:talk
2835:talk
2814:talk
2791:talk
2765:talk
2751:talk
2729:talk
2721:Done
2702:talk
2693:Done
2683:here
2663:talk
2654:Done
2628:Done
2601:talk
2592:also
2567:talk
2553:talk
2516:talk
2508:Done
2485:talk
2479:. --
2468:Done
2451:talk
2427:talk
2418:Done
2390:talk
2359:talk
2350:Done
2318:talk
2285:here
2264:talk
2244:does
2216:talk
2212:Urve
2185:must
2178:must
2157:talk
2069:talk
1967:talk
1948:talk
1922:talk
1887:talk
1856:OK.
1842:talk
1735:talk
1731:Urve
1714:talk
1693:harv
1672:Urve
1661:talk
1657:Urve
1627:talk
1613:talk
1589:talk
1585:Urve
1574:talk
1539:talk
1525:talk
1511:talk
874:The
863:talk
847:talk
652:talk
638:talk
633:Urve
618:talk
459:Urve
448:talk
443:Urve
77:Date
5610:JBL
5400:JBL
5197:or
5039:to
4949:to
4723:JBL
4644:of
4527:of
4518:− 1
4195:of
4149:."
3968:."
3827:JBL
3787:JBL
3684:of
3671:Hi
3604:JBL
3517:any
3505:had
3501:one
3497:had
3489:: "
3402:JBL
3356:JBL
3318:JBL
3283:JBL
3265:JBL
3115:efn
3100:But
3070:JBL
3048:JBL
2982:JBL
2972:Hi
2919:JBL
2831:JBL
2787:JBL
2761:JBL
2725:JBL
2698:JBL
2659:JBL
2638:efn
2597:JBL
2549:JBL
2512:JBL
2481:JBL
2447:JBL
2423:JBL
2386:JBL
2355:JBL
2258:).
2124:efn
2042:was
1963:JBL
1918:JBL
1868:one
1649:did
1535:JBL
1507:JBL
1499:− 1
1211:− 1
859:JBL
648:JBL
614:JBL
457:Hi
304:Low
5628::
5616:)
5570:⋯
5567:≺
5561:−
5558:≺
5552:≺
5546:≺
5543:⋯
5540:≺
5534:≺
5528:≺
5522:−
5519:≺
5516:⋯
5487:≺
5455:≺
5428:,
5406:)
5385:)
5348:)
5325:)
5269:~
5129:⋯
5082:∑
4984:π
4928:~
4901:π
4729:)
4721:--
4659:~
4613:~
4577:…
4501:Λ
4498:≅
4490:−
4457:→
4444:
4439:π
4436:→
4422:~
4412:→
4404:−
4391:→
4336:⋯
4328:−
4312:−
4296:−
4285:⋯
4243:π
4170:~
4120:…
4070:×
4067:⋯
4064:×
3988:~
3940:−
3894:)
3880:)
3868:I
3855:)
3833:)
3812:)
3793:)
3748:~
3720:Λ
3704:,
3700:,
3696:,
3692:,
3665:)
3574:)
3560:)
3541:)
3493:".
3444:)
3425:)
3404:,
3377:)
3362:)
3339:)
3324:)
3303:)
3289:)
3277:I
3271:)
3253:)
3224:)
3195:)
3175:)
3154:)
3128:)
3118:}}
3112:{{
3076:)
3054:)
3033:)
3023:}}
3017:{{
3002:)
2988:)
2965:)
2925:)
2908:)
2890:)
2869:)
2855:)
2837:)
2816:)
2793:)
2767:)
2753:)
2731:)
2704:)
2696:--
2665:)
2641:}}
2635:{{
2603:)
2569:)
2555:)
2518:)
2487:)
2453:)
2429:)
2392:)
2361:)
2320:)
2266:)
2252:is
2218:)
2159:)
2127:}}
2121:{{
2071:)
2048:is
2005:do
1969:)
1950:)
1940:my
1924:)
1889:)
1850:)
1737:)
1716:)
1706:}}
1700:{{
1696:}}
1690:{{
1663:)
1629:)
1615:)
1605:}}
1599:{{
1591:)
1576:)
1541:)
1527:)
1513:)
1482:Λ
1479:≅
1471:−
1438:
1433:π
1430:→
1415:→
1403:~
1393:→
1385:−
1372:→
1356:.
1335:−
1317:…
1302:…
1287:…
1278:−
1248:…
1233:−
1162:Λ
1159:∈
1140:…
1085:…
1023:…
968:⋅
955:−
946:…
937:⋅
924:−
912:⋅
899:−
865:)
857:--
849:)
794:−
737:→
724:→
712:~
702:→
694:−
681:→
654:)
640:)
620:)
590:~
476:~
450:)
390:~
354:~
139:A
5612:(
5596:2
5593:=
5590:n
5564:1
5555:1
5549:3
5537:2
5531:0
5525:2
5496:n
5493:+
5490:b
5484:n
5481:+
5478:a
5458:b
5452:a
5442:Z
5402:(
5381:(
5373:.
5344:(
5321:(
5298:n
5276:n
5266:S
5230:2
5226:)
5223:1
5220:+
5217:n
5214:(
5211:n
5205:=
5182:)
5177:2
5173:1
5170:+
5167:n
5161:(
5155:=
5135:n
5132:+
5126:+
5123:2
5120:+
5117:1
5114:=
5111:)
5108:i
5105:(
5102:u
5097:n
5092:1
5089:=
5086:i
5070:.
5052:n
5048:S
5027:)
5022:1
5018:S
5014:(
5009:n
5005:B
4962:n
4958:S
4935:n
4925:S
4885:}
4876:d
4873:I
4870:=
4865:2
4860:i
4856:s
4832:1
4828:S
4807:)
4802:1
4798:S
4794:(
4789:n
4785:B
4762:1
4758:S
4725:(
4711:.
4697:n
4693:S
4682:A
4666:n
4656:S
4630:A
4626:)
4620:n
4610:S
4603:(
4583:n
4580:+
4574:+
4571:2
4568:+
4565:1
4555:n
4536:Z
4525:A
4516:n
4493:1
4487:n
4482:Z
4460:0
4452:n
4448:S
4429:n
4419:S
4407:1
4401:n
4396:Z
4388:0
4375:π
4359:.
4354:1
4350:s
4344:2
4340:s
4331:2
4325:n
4321:s
4315:1
4309:n
4305:s
4299:2
4293:n
4289:s
4280:2
4276:s
4270:1
4266:s
4262:=
4259:)
4254:0
4250:s
4246:(
4221:n
4217:S
4203:.
4201:Λ
4197:V
4177:n
4167:S
4147:n
4131:k
4127:a
4123:,
4117:,
4112:1
4108:a
4083:k
4079:a
4074:S
4057:1
4053:a
4048:S
4022:n
4018:S
3995:n
3985:S
3956:k
3953:=
3948:j
3944:x
3935:i
3931:x
3890:(
3876:(
3866::
3862:@
3851:(
3829:(
3823::
3819:@
3808:(
3789:(
3783:.
3755:n
3745:S
3661:(
3650:.
3648:2
3570:(
3556:(
3537:(
3473::
3440:(
3421:(
3373:(
3369:—
3358:(
3350::
3346:@
3335:(
3320:(
3314::
3310:@
3299:(
3285:(
3267:(
3249:(
3220:(
3191:(
3171:(
3165::
3161:@
3150:(
3140::
3136:@
3124:(
3087::
3083:@
3072:(
3065::
3061:@
3050:(
3044::
3040:@
3029:(
3013::
3009:@
2998:(
2984:(
2961:(
2954::
2950:@
2921:(
2904:(
2886:(
2880::
2876:@
2865:(
2851:(
2833:(
2827::
2823:@
2812:(
2805::
2801:@
2789:(
2783::
2779:@
2763:(
2749:(
2727:(
2700:(
2661:(
2617:.
2599:(
2565:(
2551:(
2545:u
2541:u
2532:"
2514:(
2483:(
2449:(
2425:(
2388:(
2357:(
2316:(
2287:.
2262:(
2214:(
2155:(
2067:(
1965:(
1946:(
1936::
1932:@
1920:(
1885:(
1845:·
1840:(
1733:(
1712:(
1674::
1670:@
1659:(
1625:(
1611:(
1587:(
1572:(
1562::
1558:@
1537:(
1523:(
1509:(
1497:n
1474:1
1468:n
1463:Z
1441:0
1423:n
1419:S
1410:n
1400:S
1388:1
1382:n
1377:Z
1369:0
1344:}
1341:)
1338:1
1332:,
1329:1
1326:,
1323:0
1320:,
1314:,
1311:0
1308:(
1305:,
1299:,
1296:)
1293:0
1290:,
1284:,
1281:1
1275:,
1272:1
1269:,
1266:0
1263:(
1260:,
1257:)
1254:0
1251:,
1245:,
1242:0
1239:,
1236:1
1230:,
1227:1
1224:(
1221:{
1209:n
1196:Λ
1192:V
1188:Λ
1180:V
1156:)
1151:n
1147:a
1143:,
1137:,
1132:1
1128:a
1124:(
1104:0
1101:=
1096:n
1092:a
1088:+
1082:+
1077:2
1073:a
1069:+
1064:1
1060:a
1039:)
1034:n
1030:a
1026:,
1020:,
1015:2
1011:a
1007:,
1002:1
998:a
994:(
974:]
971:n
963:n
959:a
952:n
949:,
943:,
940:n
932:2
928:a
921:2
918:,
915:n
907:1
903:a
896:1
893:[
879:π
861:(
845:(
841:—
826:n
821:Z
797:1
791:n
786:Z
764:n
740:0
732:n
728:S
719:n
709:S
697:1
691:n
686:Z
678:0
650:(
636:(
616:(
597:n
587:S
573:0
570:W
566:W
561:γ
541:n
535:S
510:n
506:S
483:n
473:S
446:(
426:n
421:S
397:n
387:S
361:n
351:S
316:.
215::
172:?
161:.
153:"
149:"
47:.
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.