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This is an old revision of this page, as edited by Radagast3 (talk | contribs) at 01:09, 3 July 2010 (→Infinitesimal calculus / Non-standard calculus: new section). The present address (URL) is a permanent link to this revision, which may differ significantly from the current revision.

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Hello, Tkuvho, and welcome to Misplaced Pages! Thank you for your contributions. I hope you like the place and decide to stay. Here are some pages that you might find helpful:

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Hi! Remember to add references to your page or the content in question may be deleted. Good luck and happy editing--Pianoplonkers (talk) 16:48, 20 October 2009 (UTC)

Talk pages

We usually permit people to remove comments they have made from talk pages, as long as nobody else has responded to the comment. Of course the edits are still in the page history either way. There's not any overriding need to prevent people from retracting a statement that they later feel was ill considered. — Carl (CBM · talk) 15:26, 2 November 2009 (UTC)

Chang's conjecture

Yes, I plan to add material and re-organize that page. Not very soon, though. Currently I am visiting a different university and am busy with other things. Cheers. Kope (talk) 14:39, 17 December 2009 (UTC)

Using Google Scholar hits or similar

Hi, if you check out Misplaced Pages:Reliable sources/Noticeboard#Can Google hit counts ever be cited as a reliable source? there is a reasonable discussion about these sorts of sources. These are excluded from articles on the basis of SYNTH and OR. You can, of course, use the information to support a discussion about notability on the article talk page. In the case of Steve Shnider it may be better to find international awards for his work or independent reviews of his books as sources to add. Cheers—Ash (talk) 12:11, 18 January 2010 (UTC)

You seem to be insufficiently familiar with notability criteria for scientific articles. The criteria explicitly state that both math reviews and google SCHOLAR provide valid indications of notability. You seem further to confuse google and google scholar, a very different engine. Tkuvho (talk) 12:31, 18 January 2010 (UTC)

Actually the Noticeboard discussion specifically discussed the case for Google Scholar. If you doubt this example discussion (just the first one I picked out) you can try searching RS/N for yourself. Notability of the article is not at issue, just the inclusion of this transient original research. As you do not seem to give much weight to my experience and have reversed my edit (again), I'll ask for an independent third opinion which may help explain the matter. I shall copy this discussion onto the article talk page for the convenience of an opinion.—Ash (talk) 13:15, 18 January 2010 (UTC)

I don't understand the logic behind your agreeing that "notability is not an issue" and at the same time insisting on placing the "notability" tag on the article. Informal discussions at the noticeboard are one thing, but guidelines to the effect that specifically in mathematics it is appropriate to use google scholar, another. Tkuvho (talk) 13:27, 18 January 2010 (UTC)

Comment

Sorry, I think we got off on the wrong foot. Let's step back from all of this and try to find a mutually satisfactory solution at Dirac delta function. Best, Sławomir Biały (talk) 12:46, 4 February 2010 (UTC)

All feet are fine. I understand that you find the new material startling (I must say it was to me as well). Nonetheless, the current version of the page does not correspond to our historical knowledge. Tkuvho (talk) 12:47, 4 February 2010 (UTC)

Let's discuss at the article's talk page

The paragraph needs expansion and improvement; we can discuss it there. Bill Wvbailey (talk) 16:18, 24 February 2010 (UTC)

Luxemburg

When you changed Luxemburg from a redirect into a disambiguation page, you may not have noticed that nearly 200 other Misplaced Pages articles contain links to "Luxemburg". When you change the page that an existing title links to, "it is strongly recommended that you modify all pages that link to the old title so they will link to the new title." --R'n'B (call me Russ) 10:44, 25 February 2010 (UTC)

History of logic

Hi - I have added to the section on post WW2 - would you have a look? Thanks From the other side (talk) 16:57, 14 March 2010 (UTC)

I still think that your comment is written from an "inside" viewpoint of a mathematical logician, rather than the way it looks to a broader mathematical observer. I am perfectly happy with "history of logic" being written from such an "inside" viewpoint, but my own opinion is that Robinson's contribution, in the eyes of a broader public, does not appear more minor than any of the other items currently mentioned. Tkuvho (talk) 17:00, 14 March 2010 (UTC)

Irrationals

Hi, I didn't see how to respond before. I have that book on request. I relied on a claim elsewhere that this is how he does it and the small amount I could see on Amazon.com I think that the 1947 "Theory of Functions" by Joseph Fell Ritt does this also (and more throughly) and am waiting to check. I finally tracked down a reference to the authoritative article Eléments d'analyse de Karl Weierstrass by Piere Dugac in the Archive for History of Exact Sciences Volume 10, Numbers 1-2 / January, 1973 Pages 41-174. Fortunately for me not all 134 pages are in French. Unfortunately those that aren't are notes in German and I don't really read either language (but the article no doubt covers much more than irrationals.) From what I have seen elsewhere he used aggregates of units and unit fractions noting that e is {1 1, 1/2,1/6,1/24,...} and {1/15, 1/15} (being 0.23333) is the same as {1/10,1/10,1/100,1/100,1/100,1/1000,...} Given one aggregate on can break a part 1/a into n parts 1/na like 1/3 into 1/12,1/12,1/12,1/12 then you have to explain operations, negatives and when x<y (when any finite subset of x can be dominated by a finite subset of y.) Now if all of his examples stick to aggregates using 1/10s, 1/00s , 1/100s etc. one has a case for decimal fractions. His students tended to describe his things as sums (additive aggregates) but he himself was careful to take them as whole infinite sets. --65.12.202.14 (talk) 05:20, 18 March 2010 (UTC)

I assume you are "gentlemath"? It would be interesting to sort this out. It is hard to believe that nobody in the English language was ever curious to find out whether Weierstrass did the reals by decimal expansions, or not. Dugac could not have been the only one to write about this. As far as the distinction between what weierstrass did and what his students did, this may be difficult to argue, since as far as I know Weierstass himself never wrote anything down in the form of either article or book. Tkuvho (talk) 10:03, 18 March 2010 (UTC)

If we continue the discussion at the talk page of construction of the real numbers other people would be able to contribute as well. Tkuvho (talk) 12:14, 18 March 2010 (UTC)

Edit summaries

Hi Tkuvho, please use edit summaries (see Help:Edit summary). Thanks, Melchoir (talk) 21:38, 22 March 2010 (UTC)

Hi Tkuvho, This looks like a good-faith edit with an edit-summary, and so deserves an edit-summary if it is to be reverted (e.g. rv: makes article inconsistent if not applied throughout). Thanks, --catslash (talk) 11:00, 12 May 2010 (UTC)

Proposed deletion of Six cross-ratios

The article Six cross-ratios has been proposed for deletion because of the following concern:

Unnecessary content fork of Cross-ratio#Symmetry. Unlikely to be a search term so redirect isn't appropriate.

While all contributions to Misplaced Pages are appreciated, content or articles may be deleted for any of several reasons.

You may prevent the proposed deletion by removing the {{dated prod}} notice, but please explain why in your edit summary or on the article's talk page.

Please consider improving the article to address the issues raised. Removing {{dated prod}} will stop the proposed deletion process, but other deletion processes exist. The speedy deletion process can result in deletion without discussion, and articles for deletion allows discussion to reach consensus for deletion. RDBury (talk) 16:13, 8 June 2010 (UTC)

I think that "six cross-ratios" can be a useful item in a list of articles in the category "projective geometry". It allows the reader to find the material he is looking for quicker. Some of the items that would be appropriate on this page would be too esoteric for inclusion at the main page cross-ratio, for instance a more detailed explanation why the sixth root of unity gives an orbit with only two elements. Giving a convincing explanation of the role of the Klein 4-group would also be too detailed for the main page, which is too long already. The topic of the present page is of independent interest, which should qualify it for a separate page. The corresponding section at cross-ratio should be shortened to include only the essentials. Tkuvho (talk) 16:25, 8 June 2010 (UTC)

In projective geometry, there is a number of definitions of the cross-ratio. However, they all differ from each other by a suitable permutation of the coordinates. In general, there are six possible different values the cross-ratio ${\displaystyle (z_{1},z_{2};z_{3},z_{4})}$ can take depending on the order in which the points z_i are given.

Action of symmetric group

Since there are 24 possible permutations of the four coordinates, some permutations must leave the cross-ratio unaltered. In fact, exchanging any two pairs of coordinates preserves the cross-ratio:

${\displaystyle (z_{1},z_{2};z_{3},z_{4})=(z_{2},z_{1};z_{4},z_{3})=(z_{3},z_{4};z_{1},z_{2})=(z_{4},z_{3};z_{2},z_{1}).\,}$

Using these symmetries, there can then be 6 possible values of the cross-ratio, depending on the order in which the points are given. These are:

${\displaystyle (z_{1},z_{2};z_{3},z_{4})=\lambda \,}$		${\displaystyle (z_{1},z_{2};z_{4},z_{3})={1 \over \lambda }}$
${\displaystyle (z_{1},z_{3};z_{4},z_{2})={1 \over {1-\lambda }}}$		${\displaystyle (z_{1},z_{3};z_{2},z_{4})=1-\lambda \,}$
${\displaystyle (z_{1},z_{4};z_{3},z_{2})={\lambda \over {\lambda -1}}}$		${\displaystyle (z_{1},z_{4};z_{2},z_{3})={{\lambda -1} \over \lambda }}$

Six cross-ratios as Möbius transformations

Viewed as Möbius transformations, the six cross-ratios listed above represent torsion elements of PGL(2,Z). Namely, ${\displaystyle {\frac {1}{\lambda }}}$, ${\displaystyle \;1-\lambda \,}$, and ${\displaystyle {\frac {\lambda }{\lambda -1}}}$ are of order 2 in PGL(2,Z), with fixed points, respectively, -1, 1/2, and 2 (namely, the orbit of the harmonic cross-ratio). Meanwhile, elements ${\displaystyle {\frac {1}{1-\lambda }}}$ and ${\displaystyle {\frac {\lambda -1}{\lambda }}}$ are of order 3 in PGL(2,Z). Each of them fixes both values ${\displaystyle e^{\pm i\pi /3}}$ of the "most symmetric" cross-ratio.

Role of Klein four-group

In the language of group theory, the symmetric group S₄ acts on the cross-ratio by permuting coordinates. The kernel of this action is isomorphic to the Klein four-group K. This group consists of 2-cycle permutations of type ${\displaystyle (ab)(cd)}$ (in addition to the identity), which preserve the cross-ratio. The effective symmetry group is then the quotient group ${\displaystyle S_{4}/K}$, which is isomorphic to S₃.

Exceptional orbits

For certain values of λ there will be an enhanced symmetry and therefore fewer than six possible values for the cross-ratio. These values of λ correspond to fixed points of the action of S₃ on the Riemann sphere (given by the above six functions); or, equivalently, those points with a non-trivial stabilizer in this permutation group.

The first set of fixed points is {0, 1, ∞}. However, the cross-ratio can never take on these values if the points {z_i} are all distinct. These values are limit values as one pair of coordinates approach each other:

${\displaystyle (z,z_{2};z,z_{4})=(z_{1},z;z_{3},z)=0\,}$

${\displaystyle (z,z;z_{3},z_{4})=(z_{1},z_{2};z,z)=1\,}$

${\displaystyle (z,z_{2};z_{3},z)=(z_{1},z;z,z_{4})=\infty .}$

The second set of fixed points is {−1, 1/2, 2}. This situation is what is classically called the harmonic cross-ratio, and arises in projective harmonic conjugates. In the real case, there are no other exceptional orbits.

The most symmetric cross-ratio occurs when ${\displaystyle \lambda =e^{\pm i\pi /3}}$. These are then the only two possible values of the cross-ratio.

Talkback: Nils von Barth, PGL fix

I have marked you as a reviewer

I have added the "reviewers" property to your user account. This property is related to the Pending changes system that is currently being tried. This system loosens page protection by allowing anonymous users to make "pending" changes which don't become "live" until they're "reviewed". However, logged-in users always see the very latest version of each page with no delay. A good explanation of the system is given in this image. The system is only being used for pages that would otherwise be protected from editing.

If there are "pending" (unreviewed) edits for a page, they will be apparent in a page's history screen; you do not have to go looking for them. There is, however, a list of all articles with changes awaiting review at Special:OldReviewedPages. Because there are so few pages in the trial so far, the latter list is almost always empty. The list of all pages in the pending review system is at Special:StablePages.

To use the system, you can simply edit the page as you normally would, but you should also mark the latest revision as "reviewed" if you have looked at it to ensure it isn't problematic. Edits should generally be accepted if you wouldn't undo them in normal editing: they don't have obvious vandalism, personal attacks, etc. If an edit is problematic, you can fix it by editing or undoing it, just like normal. You are permitted to mark your own changes as reviewed.

The "reviewers" property does not obligate you to do any additional work, and if you like you can simply ignore it. The expectation is that many users will have this property, so that they can review pending revisions in the course of normal editing. However, if you explicitly want to decline the "reviewer" property, you may ask any administrator to remove it for you at any time. — Carl (CBM · talk) 12:33, 18 June 2010 (UTC) — Carl (CBM · talk) 12:56, 18 June 2010 (UTC)

Infinitesimal calculus / Non-standard calculus

I have reverted your merge of Infinitesimal calculus with Non-standard calculus, done against the consensus at Talk:Infinitesimal calculus#Merge.

Consensus can change, but this merge should not be done unless you can get other editors to agree with you. -- Radagast3 (talk) 01:09, 3 July 2010 (UTC)

• Projective geometry