User talk:Katzmik

Welcome!

Hello, Katzmik, and welcome to Wikipedia! 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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And don't forget, the edit summary is your friend. :) – Oleg Alexandrov (talk) 15:24, 19 April 2007 (UTC)

What is the relevance of Systolic geometry to Hyperbolic geometry?

What is the relevance of Systolic geometry to Hyperbolic geometry? You have put this link into Hyperbolic geometry twice. To me it looks like link spam. What do they have in common beside the fact that they are both called "geometry"? If you do not give me a good reason, I will remove it again. JRSpriggs 07:32, 24 April 2007 (UTC)

Reply to JRSpriggs

I am sorry I am not exactly sure what format to answer you in. If you see this message, could you please indicate your email address so I can write to you? My email address can be found at the "Website for systolic geometry and topology" that you can easily find with google.

You may reply here. JRSpriggs 09:27, 24 April 2007 (UTC)

OK. First of all, I would like to apologize about duplicating the link to hyperbolic geometry. I added a number of links in the past couple of days to the new page "Systolic geometry", and did not notice that you had removed the link from hyperbolic geoemetry.

Second, if you find the link is inappropriate, I certainly have no objection to removing it.

The reason I placed the link there in the first place is because there is considerable intersection today between hyperbolic geometry and systolic geometry. The hyperbolic bibliography at the "Website for systolic geometry and topology" contains over three dozen articles, including such hyperbolic heavyweights as C. Adams and A. Reid.

I would be glad to provide further details if you are interested.

MK

The question is not whether hyperbolic geometry is relevant to systolic geometry. The question is whether systolic geometry is relevant to hyperbolic geometry. What does systolic tell us that helps us to understand spaces with constant negative curvature? JRSpriggs 10:21, 24 April 2007 (UTC)

I would be glad to answer this question. If you don't mind, I would prefer to continue this discussion via email. Otherwise I have no objection to removing the link if it bothers you. Katzmik 15:21, 25 April 2007 (UTC)

Can I just say that there has been a misunderstanding here. Your edits are welcome. Charles Matthews 16:56, 25 April 2007 (UTC)

I have restored your link at hyperbolic geometry that JRSpriggs incorrectly deleted. He means well, but has for some reason mistaken your work for some of the junk that is often added to wikipedia. Your additions are a big improvement to the article, and are most welcome. R.e.b. 17:12, 25 April 2007 (UTC)

Indeed, welcome to Wikipedia! Wikipedia:WikiProject Mathematics is always thrilled to get professional attention. I invite you to join the conversations on the talk page Wikipedia talk:WikiProject Mathematics, where your arrival was noticed and breifly discussed. linas 23:55, 26 April 2007 (UTC)

Figure in Systolic geometry

Hi, I'm trying to understand the new Systolic geometry page, and the figure has confused me. Does it show the systole of the surface? Is it importact that its a geodesic? The caption probably needs a rewrite as Fig 8 does not have much meaning here. --Salix alba (talk) 07:18, 26 April 2007 (UTC)

Thanks for your interest. The figure is an illustration of one of my favorite results in systolic geometry, namely Gromov's filling area conjecture and its proof in the hyperelliptic case. If you let me know what your background is, I will try to explain the result and the figure. I would feel more comfortable doing this via email as not all of this discussion may be relevant to the six billion readers of wiki ☺ -MK

Most readers just read the articles, not the talk pages and especially not the user talk pages. JRSpriggs 09:22, 26 April 2007 (UTC)

The best place for discussion on the article would be Talk:Systolic geometry which is the prefered place on wikipedia. My main interest is in an figure illustrating the esential point of Systolic geometry.--Salix alba (talk) 10:00, 26 April 2007 (UTC)

I agree. The "systolic geometry" page is certainly not in its final form (it only went online this week!) I am certainly pleased with the interest it generated. I am planning to expand it, both by including further details on systolic hyperbolic geometry, and a clarification of the significance of the figure-8 illustration. I will try to check the user talk page periodically. -MK

Peacock Terms

What's a peacock term you mentioned on the systolic geometry page? MK —The preceding unsigned comment was added by Katzmik (talk • contribs) 11:17, 26 April 2007 (UTC).

See Wikipedia:Avoid peacock terms --Brentt 20:41, 26 April 2007 (UTC)

Thanks for the reference. I looked over the peacock instructions. Of course, they are very appropriate and helpful. Clearly, the importance of a wiki topic has to be explained rather than merely asserted. The particular term I used was used for a different purpose. Namely, all French readers (even non-mathematicians!) know who Rene Thom is, and have a feeling for why someone would want to listen attentively to his every utterance, as Marcel Berger did. To English readers, on the other hand, he is less familiar. I therefore used the peacock term "venerable" in order to make the ensuing discussion plausible to an English-speaking layman (the details can be found by following the link to Thom's wikipage).

I don't think it is essential to put it back in, but in the future well-meaning editors should try to follow the spirit of the law and not merely the letter.

Katzmik 08:32, 27 April 2007 (UTC)

Using links instead of instructions

Several days ago, you left a message for me which included "For the time being I did not place it in wikipedia, but rather on my 'Website for systolic geometry and topology'. If you are curious to see it, go to the website, click on 'subfields', then click on 'systolic hyperbolic geometry'.". Following such instructions is rather difficult. I suggest that in the future, you instead give a direct link to the page in question. Somewhat like this:

your main page

• http://www.cs.biu.ac.il/~katzmik

website for systolic geometry and topology

• http://www.cs.biu.ac.il/~katzmik/sgt.html

subfields page

• http://www.cs.biu.ac.il/~katzmik/sgtdirectory/subfields.html

systolic hyperbolic geometry page

• http://www.cs.biu.ac.il/~katzmik/sgtdirectory/hyperbolic.html

Then it would be much less work for the readers to find it and they will be more likely to actually go there rather than giving up. JRSpriggs 06:58, 30 April 2007 (UTC)

Good idea. Now that you mention it, it seems the obvious thing to do, but I overlooked it! Thanks very much. Katzmik 11:01, 30 April 2007 (UTC)

List of inequalities

Hello. I just added a couple of pages that you created to the list of inequalities. If you know of others that should be added, could you do that? Michael Hardy 17:44, 30 June 2007 (UTC)

Thanks. I was not aware of the list of inequalities. I have a couple of other candidates, namely some inequalities of Gromov's. Katzmik 13:46, 1 July 2007 (UTC)

Too many Wirtinger's inequalities

I think something should be done about the fact that Wirtinger's inequality and Wirtinger's inequalities are on two different topics but have names so similar to each other that the similarity will cause confusion. Possibly the best way to handle it is to use the "move" button and in one case write Wirtinger's inequality (blah) and in the other, Wirtinger's inequality (blah), for two different appropriate values of "blah". Then "Wirtinger's inequalities" could redirect to "Wirtinger's inequality" which would then be made into a disambiguation page. After that, all the links to the old titles would have to get suitably directed. What do you think? Michael Hardy 22:28, 10 July 2007 (UTC)

very good idea. Could you please do that, since you are more experienced with creating disambiguation pages and the like? The current Wirtinger's inequality could be called Wirtinger's inequality for functions. The site I created is Wirtinger's inequality for 2-forms (namely, alternating, or exterior, 2-forms). It is usually stated for the standard symplectic 2-form, but in fact there is a suitable generalisation to arbitrary 2-forms, which is what Gromov used in his proof of the systolic inequality for complex projective space. Most of the literature on this Wirtinger's inequality uses the special case of the symplectic 2-form, and the literature is rather extensive, though it is not (yet) reflected in wiki. Katzmik 11:50, 11 July 2007 (UTC)

P.S. The inequality for 2-forms is usually referred to in the literature as the Wirtinger inequality, not Wirtinger's inequality. If this is compatible with wiki standards, it may be preferable to use that term. Katzmik 11:52, 11 July 2007 (UTC)

Fixing links to new disambiguation page

Wirtinger's inequality is now a disambiguation page. The next task is to go to that page, click on "what links here", and in each article that links to the disamiguation page, link to the appropriate article. Michael Hardy 02:25, 12 July 2007 (UTC)

...OK, I think that may be essentially done now. Michael Hardy 02:36, 12 July 2007 (UTC)

Thanks. Katzmik 11:09, 12 July 2007 (UTC)

Klein's quartic

This page notes that the Klein quartic is Γ(7)\H² (search for "congruence" on that page). See also the first sentence of this PDF, or Google "klein quartic congruence". You're completely right that it's a subgroup of PSL(2,Z), rather than PSL(2,R), and I've made that change to the article. Tesseran 19:50, 31 July 2007 (UTC)

There is an error here. A subgroup of PSL(2,Z) would give a complete surface, but not a compact one. To get the compact Klein quatric (as opposed to an affine slice), one needs a more elaborate arithmetic set-up. Katzmik 09:07, 1 August 2007 (UTC)

Incidentally, if you read THE SECOND CLAUSE of the first sentence of Agol's pdf at this PDF, you will notice that he is more careful about cusps. Katzmik 15:38, 1 August 2007 (UTC)

Watchlist mysteries

Jitse, could you please comment on how the personal watchlist works? For some reason items keep disappearing from my watchlist. For instance, your edit at Bolza surface appeared on my watch list, but only for a split second, and it is not there anymore. Katzmik 11:15, 6 August 2007 (UTC)

That's strange. I can think of only two explanations, neither of which seems likely. The first one is that you somehow removed the page from your watchlist. You can view a list of articles on your watchlist by first clicking my watchlist (in the upper right hand corner) and then view and edit watchlist; the URL

http://en.wikipedia.org/w/index.php?title=Special:Watchlist&action=edit

should take you there directly. The second explanation is that you have "Hide my edits from the watchlist" turned on in your preferences and looked at the watchlist after you made an edit to Bolza surface following my edit. As the watchlist only shows the last edit to an article, nothing will be shown in this case. You can check that by clicking my preferences (again in the upper right corner) and then the Watchlist tab. The only other hint I can give is to read Help:Watching pages and see whether that may help you understand what's happening. -- Jitse Niesen (talk) 12:25, 6 August 2007 (UTC)

I saw this thread by accident and wanted to share an insight: another thing that could affect a watchlist is robots modifying an article. Frequently, the watchlist will not show bot edits (this feature can be turned on and off); as a result, if a bot modifies an article right after a human, the human's edit will be struck out, but the bot's edit will not appear. Typically, this happens if someone forgets to sign a comment on a talk page, and one of the signing bots does it for him, or if a bot archives part of a page (a frequent situation at WT:WPM) Could this be happening in your case? Arcfrk 22:40, 6 August 2007 (UTC)

Liouville eqns

It is the same, I've checked with Barbashov, Nesterenko book. String4d 10:09, 13 August 2007 (UTC)

I have Polyakov's original paper and its discussion in Barbashov, Nesterenko book. Polyakov shows that string theory at D<26 contains a field described by Lagrangian $L=(\partial _{\mu }\varphi )^{2}/2+\mu ^{2}e^{\varphi }.$ Its classical equation of motion is $-\partial ^{2}\varphi +\mu ^{2}e^{\varphi }=0.$ It is equivalent to Liouville equations after variables change $\varphi =2\log f,\ \mu ^{2}/2=K.$ String4d 16:51, 14 August 2007 (UTC)

Hermite constant

Hello Michael,

I appreciate your interest in Hermite constant. A note about notation: I agree with the general principle that low-level commands at the html level are preferable to fancy math environments if the latter can be avoided. However, the square root of γ that currently appears at Hermite constant really does not look like a proper square root. There is a way of making an overline in html but I wonder if it is worth the time. In other words, one should either use proper html, or a math environment, it seems to me. Katzmik 07:51, 16 August 2007 (UTC)

I was never acting on a general principle that html is better than TeX; I was acting on the fact that TeX looks bad in some contexts where bad mismatches in alignment and size happen. Michael Hardy 03:35, 6 September 2007 (UTC)

AfD nomination of Systolic geometry for a beginner

I have nominated Systolic geometry for a beginner, an article you created, for deletion. I do not feel that this article satisfies Wikipedia's criteria for inclusion, and have explained why at Wikipedia:Articles for deletion/Systolic geometry for a beginner. Your opinions on the matter are welcome at that same discussion page; also, you are welcome to edit the article to address these concerns. Thank you for your time. Do you want to opt out of receiving this notice? tgies (talk) 12:55, 29 May 2008 (UTC)

I responded at the deletion page. Katzmik (talk) 13:34, 29 May 2008 (UTC)

Reply at User talk:Tgies

Hello, Katzmik. You have new messages at Tgies's talk page.
You can remove this notice at any time by removing the {{Talkback}} or {{Tb}} template.

Up-sizing Tex

TeX is the display language which uses "<math>...</math>" and is described at Help:Displaying a formula. You may have noticed that sometimes the formulas displayed with TeX are large (e.g. "$A_{M}:M\to J_{1}(M)\!$") and sometimes they are small (e.g. "$A_{M}:M\to J_{1}(M)$"). You can change those which are small to the large size by inserting "\!" or "\,". The "\," also adds a short blank space. It is our convention that when a TeX formula is displayed on a separate line it will be large and indented. In particular, Systolic geometry#Relation to Abel-Jacobi maps has several formulas which need to be fixed. JRSpriggs (talk) 10:37, 31 May 2008 (UTC)

Systolic geometry

I have restored the link to Systolic geometry to the article on Riemannian geometry. I do find your reasons compelling. I had removed the link because it seemed to be part of a campaign to put the link in many places where it did not belong. However, given the thrust of the Riemannian geometry article, it is clear that the link is appropriate there. Regards, siℓℓy rabbit (talk) 11:25, 24 June 2008 (UTC)

Thanks very much! Katzmik (talk) 11:29, 24 June 2008 (UTC)

Loops and spaces

Yes, that is a better solution. The link should have probably been to fundamental group instead (since there really isn't a need for an article on noncontractible loop). You are correct that as a space a loop is clearly never going to be contractible. Forgive the oversight on my part. siℓℓy rabbit (talk) 14:56, 26 June 2008 (UTC)

No problem. Let me know what you think of the vibrating introduction :-) Katzmik (talk) 14:59, 26 June 2008 (UTC)

Discussion

I hope you don't mind if I copy our discussion to Differential geometry of surfaces, its natural place, and that we continue it there, possibly at a slower pace so that I have a chance to work on mainspace edits. Perhaps you might want to write a skeleton of the examples section if you feel in the mood? Cheers, Mathsci (talk) 13:33, 5 August 2008 (UTC)

Please go ahead copy. Katzmik (talk) 13:35, 5 August 2008 (UTC)

Have done :-) Mathsci (talk) 13:38, 5 August 2008 (UTC)

BTW - quite off-topic here! - a few days ago I found a wonderful picture of Levi-Civita on the italian WP and was quite annoyed that it could not be used here for copyright reasons. One of my colleagues former graduate students in Rome has a whole story of what happened to Levi-Civita before the war, when his mother knew him. I think he was called Tullio in his honour, although I can't be sure. Mathsci (talk) 14:16, 5 August 2008 (UTC)

That's interesting. Seems like a nice tidbit to spice up the L-C page. I would read it with interest. Katzmik (talk) 14:21, 5 August 2008 (UTC)

Incidentally, since you raised the question of influential mathematicians of the 20th century, I would put Gromov right up among the top names. My additions to Orbifold were mostly about developments inspired by Gromov. I also happen to be a big fan of Arnold's classic book, which I used years ago (left invariant metrics on compact Lie groups). I hope this helps :-) Mathsci (talk) 18:04, 6 August 2008 (UTC)

Image without license

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systoles

Hi. I hope you didn't mind me expanding your references to systoles. I just used the nice chatty account in Berger - I hope it's up to date. Please correct anything if there's an error. BTW this might be a better place for your arithmetic examples, which all seem to be related to systoles. The picture of Levi-Civita seems to be resisting the image taggers so far. At the moment in RL I am busy with twisted vertex algebras, so will be less present on wikipedia. Cheers, Mathsci (talk) 08:59, 28 August 2008 (UTC)

Thanks for your systolic input. There is more at systoles of surfaces. As far as bibliography format is concerned, I just left a long message at Greathouse's talk page. Hope you change your mind. Katzmik (talk) 09:02, 28 August 2008 (UTC)

P. M. Pu

This is an automated message from CorenSearchBot. I have performed a web search with the contents of P. M. Pu, and it appears to include a substantial copy of http://www.cs.biu.ac.il/~katzmik/sgtdirectory/pu.html. For legal reasons, we cannot accept copyrighted text or images borrowed from other web sites or printed material; such additions will be deleted. You may use external websites as a source of information, but not as a source of sentences.

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Speedy deletion of P. M. Pu

A tag has been placed on P. M. Pu requesting that it be speedily deleted from Wikipedia. This has been done under section G12 of the criteria for speedy deletion, because the article appears to be a blatant copyright infringement. For legal reasons, we cannot accept copyrighted text or images borrowed from other web sites or printed material, and as a consequence, your addition will most likely be deleted. You may use external websites as a source of information, but not as a source of sentences. This part is crucial: say it in your own words.

If the external website belongs to you, and you want to allow Wikipedia to use the text — which means allowing other people to modify it — then you must include on the external site the statement "I, (name), am the author of this article, (article name), and I release its content under the terms of the GNU Free Documentation License, Version 1.2 and later." You might want to look at Wikipedia's policies and guidelines for more details, or ask a question here.

If you think that this notice was placed here in error, you may contest the deletion by adding {{hangon}} to the top of the page that has been nominated for deletion (just below the existing speedy deletion or "db" tag), coupled with adding a note on the talk page explaining your position, but be aware that once tagged for speedy deletion, if the article meets the criterion it may be deleted without delay. Please do not remove the speedy deletion tag yourself, but don't hesitate to add information to the article that would would render it more in conformance with Wikipedia's policies and guidelines. Excirial ^{(Talk,Contribs)} 10:01, 28 August 2008 (UTC)

Corensearchbot is correct but this is my own work. Katzmik (talk) 09:59, 28 August 2008 (UTC)

I included the gnu permission at my homepage as per the instructions given in the corensearchbot message. Katzmik (talk) 10:16, 28 August 2008 (UTC)

Excellent. That solves this issue then :) Excirial ^{(Talk,Contribs)} 13:25, 30 August 2008 (UTC)

Your note

Thank you for your note. I have replied. --Moonriddengirl ^(talk) 11:37, 31 August 2008 (UTC)

AfD nomination of P. M. Pu

I have nominated P. M. Pu, an article you created, for deletion. I do not think that this article satisfies Wikipedia's criteria for inclusion, and have explained why at Wikipedia:Articles for deletion/P. M. Pu. Your opinions on the matter are welcome at that same discussion page; also, you are welcome to edit the article to address these concerns. Thank you for your time. Do you want to opt out of receiving this notice? -- [[::User:Collectonian|Collectonian]] ([[::User talk:Collectonian|talk]] · [[::Special:Contributions/Collectonian|contribs]]) 16:10, 31 August 2008 (UTC)

I replied. Katzmik (talk) 08:42, 1 September 2008 (UTC)

P. M. Pu

No worries. Thanks for the note, Cirt (talk) 07:57, 5 September 2008 (UTC)

Don't take Collectonian's bitter comments (calling the Pao Ming Pu article your "school paper", saying it's only interesting to "extreme math geeks") personally. She has been very angry lately and has lashed out frequently at topics she doesn't find interesting. The writing you've done was of very good quality and your subject was obviously notable, and ultimately the community realized that. Collectonian will continue to try to delete articles like yours but she will fail every time. - McCart42 (talk) 16:47, 5 September 2008 (UTC)

On the contrary, I thanked her at the deletion page for stimulating interest in the article, and I meant it. I think that disagreement is what wiki is all about and what gives it a chance of succeeding. I recently heard that google is trying to launch a competitor encyclopedia, where every editor gets a chance to write his own version. I doubt it will succeed. Katzmik (talk) 10:33, 7 September 2008 (UTC)

Transfer principle

Hello. Any comments on the material at User:Michael Hardy/transfer principle before I try to incorporate it into the article titled transfer principle? Michael Hardy (talk) 18:41, 6 September 2008 (UTC)

(ε, δ)-definition of limit

Do you have a goal in mind for the page (ε, δ)-definition of limit? It seems to me that it just repeats information already in the article on limits. I'm not sure if anyone has pointed out the guidance on Wikipedia:Content forking. In general, it's a maintenance headache to have multiple articles covering the same subject. — Carl (CBM · talk) 13:29, 7 September 2008 (UTC)

I plan to elaborate on the quantifier use. If it bothers too many people the page can be deleted. There have to be limits to the idea that everything should be in the same article. Most mathematics articles are related to at least one other mathematics article. To follow the idea through to its logical conclusion, there will be only one mathematics article on wikipedia. My own rule of thumb is that if there is a recognizable term frequently used in the literature, it should have its own page. See for example computational formula for the variance which would be nearly impossible to find had I not created a page for it. Do let me know what you think (i.e. precisely in which ways you disagree :)) Katzmik (talk) 13:33, 7 September 2008 (UTC)

In general, I prefer to see fewer, longer articles instead of more, shorter ones. Of course there is a tradeoff. There's no need to "delete" the new article; I'm just saying that the content that I think would naturally go into it would also naturally go into the article limit (mathematics), and I think that effort is better spent improving the main article rather than splitting off a new child of it. The main article currently has the (common) problem that it presents the same definition twice but calls it the "formal definition" the second time.

The result of excessive splitting is that we end up with dozens of microscopic aritcles, which are very hard to keep straight and which overlap with each other in strange ways. It is also hard for a reader to find content when it is divided too finely. This is a continuing problem with the computability and Boolean algebra articles, for example. — Carl (CBM · talk) 13:42, 7 September 2008 (UTC)

I hear what you are saying. There is one point that could be made in favor of keeping the shorter articles (though perhaps not the microscopic ones). Namely, let's keep in mind that these are genuinely difficult concepts for a beginner. Having it all in one place can be overwhelming. Do let me know what you think of the changes I just made at epsilon-delta. Katzmik (talk) 13:47, 7 September 2008 (UTC)

Hello

Hello Mikhail, I slightly modified two of your changes to Differential geometry of surfaces, because I did not think it was good policy to evaluate your own work on wikipedia: please leave that to other editors. I hope you don't mind - the edit summaries were intended as friendly reminders (wikipedia is after all not the arxiv!).

BTW I did find out details of the story about Levi-Civita that we talked about before. Apparently Levi-Civita, although he suffered for being Jewish under Mussolini, died naturally in Rome in 1941. Just after the war his widow, a Catholic, rescued a Jewish girl from Vienna who had been placed in an Italian convent just across the Austrian-Italian border. She brought up the girl as her own daughter. Her future grandson was my colleague's Ph.D. student. He only found out much later in life about these extraordinary events and why, although unrelated, he had not only become one of the heirs of Levi-Civita, but also had inherited his first name. Cheers, Mathsci (talk) 08:00, 8 September 2008 (UTC)

Speedy deletion of Zoll surface

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John W. Dawson, Jr

Hi, here's his bio on the last page of Logical Dilemmas:

"John W. Dawson, Jr. is Professor of Mathematics at Pennsylvaia State University. Born in Wichita, Kansas, he attended M.I.T. as a National Merit Scholar before earning a doctorate in mathematical logic from the University of Michigan. An internationally recognized authority on the life and work of Kurt Goedel, Professor Dawson is the author of numerous articles on axiomatic set theory and the history of modern logic. During the years 1982 to 1984, he catalogued Goedel's papers at the Institute for Advanced Study in Princeton, and afterward he served as a co-editor of Goedel's Collected Works. An accomplished flutist, Professor Dawson resides in York, Pennsylvania with his wife Cheryl, who collaborates actively with him in his editorial and musical endeavors." (from the last page of my paperback edition of Logical Dilemmas, with a picture of him too).

I've used his Collected Works to get cc's of correspondence between Goedel and Finsler, another interesting story. I've never had any correspondence directly with Dawson (on the other hand, in the course of writing wiki articles I have corresponded directly with Martin Davis (UC Davis? can't remember), and Yuri Gurevich at Microsoft; it always (pleasantly) surprises me when these Great Men write back). I hope Dawson is still hearty and still working. Hope this helps, Bill Wvbailey (talk) 13:28, 16 September 2008 (UTC)

Yes, it does. I am surprised he does not have a wiki page. Perhaps this is due to the overabundance of john dawsons. I think we should resolve the puzzle of his absence from Penn State math department and create a page. Katzmik (talk) 13:32, 16 September 2008 (UTC)

I'm surprised, too. (Unfortunately I don't have any professional contacts to help locate him). Maybe the academic mathematicians User:CBM or User:Trovatore or User:Lambian do ... I'll sniff around Google a bit. The other thought is his publisher A. K. Peters, Wellesley MA (altho I've not had much luck with that sort of inquiry in the past.) Bill Wvbailey (talk) 19:40, 16 September 2008 (UTC)

Google indicates he and his wife have been active at a conference in 2004-2005 time frame. I wrote an e-mail to Solomon Feferman at Stanford: sf@csli.stanford.edu (Dawson appeared at a conference he organized). We'll see what happens ... Bill Wvbailey (talk) 20:08, 16 September 2008 (UTC)

Here's the response + text of Feferman's quick response!

It's John W. Dawson, Jr. <jwd7too@comcast.net> .

I imagine John would be very pleased to have something on him in Wikipedia and wouldn't mind being contacted by email.

Best,

Solomon Feferman

On Tue, Sep 16, 2008 at 1:03 PM, <pierab@aol.com> wrote:

Hi, Bill Bailey here:

I and another person want to write a biographical sketch of John W. Dawson, Jr. for Wikipedia, but we can't find his email address. The only biographical sketch I have is the one at the back of his "Logical Dilemmas" (I've used this book quite a bit for wiki-sourcing, as well as Dawson's Collected Works re Finsler and Goedel.) Do you think he would be amenable to being contacted via email? If you can help locate him, we'd appreciate it.

Thanks, Bill Bailey (I use the moniker wvbailey on wikipedia, am located in the Dartmouth College vicinity ...)

pierab@aol.com

Do you want to contact him? Or should I? I wonder what sort of questions we should ask him ... he might have a CV or more biographical information already prepared (or he might value his privacy ...). Lemme know, Bill Wvbailey (talk) 13:59, 17 September 2008 (UTC)

Completed infinity

Probably this should have its own page, or a section on the "infinity" page, or something. My impression is that the term has to do with both the issues of intuitionism versus formalism, but also recursion and induction (see below). First we need to track down where this term actually appears in the literature, and how it is used. Here's a start: In Kleene 1952/1971 it appears in the Index as

"completed infinity cf. actual."

Under the listing infinity we find:

"problem of 46; cf. actual, potential."

Under the listing

" actual infinity 48, 52, 55, 175, 317."

From page 48: "The non-intuitionistic mathematics which culminated in the theories of Weierstrass, Dedekind and Cantor, and the intuitionistic mathematics of brouwer, differ essentially in their view of the infinite. In the former, the infinite is treated as actual or completed or extended or existential. An infinite set is regarded as existing as a completed totality, prior to or independently of any human process of generation or construction, as though it could be spread out completely for our inspection. In the latter, the infinite is treated only as potential or becoming or constructive. The recognition of this distinction, in the case of infinite magnitudes, goes back to Gauss, who in 1831 wrote, "I protest . . . agianst the use of an infinite magnitude as something completed, which is never permissible in mathematics." (Werke VIII p. 216)" (Kleene 1952/1971: 48)

Under the listing

" potential: infinity 48, 62, 70, 357, 363; recursiveness 324, 331, 332."

I will have to explore each of these and other books, too. I'd suggest you (someone needs to bell the cat -- lucky you) create such a page and we can then move this "research" to the talk page. Bill Wvbailey (talk) 16:24, 17 September 2008 (UTC)

A quick read of Kleene: I see some very interesting tie-ins w.r.t. "definitions" (aka "recursion), Turing machines, Turing computability and the Church-Turing thesis as well as recursion etc etc. In particular, Kleene's conception of the Turing "tape" is as follows: "... the machine performs only acts atomic in character, but is supplied with a tape having a (potentially) infinite printing ..." (p. 363). p. 357 reflects a similar notion. Bill Wvbailey (talk) 18:28, 17 September 2008 (UTC)

My feeling it should be a separate page rather than a subpage of "infinity" as in general I think any stable idea recognizable in its own right should have an independent entry which in my mind is the whole point of an electronic database. Katzmik (talk) 16:28, 17 September 2008 (UTC)

I agree, a separate page would be better. I'll let you bell the kitty to start it off -- I guess the back- and forward-links are to the infinity page, the LoEM page, the Brouwer-Hilbert page, the Intuitionism page(s), the Foundations page. I'll fill in as I uncover stuff. Bill Wvbailey (talk) 18:18, 17 September 2008 (UTC)

It looks like someone has gotten there first. See Actual infinity; it is a well-developed page. It turns out that Completed infinity redirects to Infinity. This is not a good redirect. It should go to Actual infinity, at the least, or be a very short "disambiguation page" that offers the reader a number of choices. Hmm ... what to do? Bill Wvbailey (talk) 22:08, 17 September 2008 (UTC)

I fixed the redirect -- now Completed infinity redirects to Actual infinity

Great!

and fiddled a tiny bit with the lead paragraphs of Actual infinity. This article does not conform to wiki-standards (to say the least),

I trust such a situation will soon change :) Katzmik (talk) 14:37, 18 September 2008 (UTC)

but there's a lot of information in it. Bill Wvbailey (talk) 14:34, 18 September 2008 (UTC)

Potential

It would be good to do some justice to potential infinity, too. Katzmik (talk) 14:40, 18 September 2008 (UTC)

Massey Product

Dear Katzmik,

Sorry for the delayed response. Your edit is definitely correct but perhaps in order to avoid confusion of the reader, the paragraph should be rewritten (I don't think that it is useful to explain why a statement is wrong in an article (unless it is a common mistake)). The previous paragraph in that particular section does not fit there (for your reason) and therefore should be moved elsewhere in the article.

Topology Expert (talk) 05:28, 20 September 2008 (UTC)

Hi,

I fixed up that section on Massey product (user R.E.B made some changes to that section but it is alright now).

Topology Expert (talk) 09:35, 30 September 2008 (UTC)

Thanks

Thanks for the Wikipedia traffic page.  Declan Davis   (talk)  13:36, 22 September 2008 (UTC)

Informal Tone

Actually, let's move this to the talk page ... Bill Wvbailey (talk) 14:36, 23 September 2008 (UTC)

While we are at it, did you see the deletion page for Manifold destiny? Katzmik (talk) 14:50, 23 September 2008 (UTC)

Tempests in teapots

As I was reminded yet again when my attempt at good-natured banter came across to some as a threat of antisocial behavior, it's very, very easy to be misunderstood—and to misunderstand others. I had to chuckle, for instance, on reading your assessment that "Nsk92 is the most serious expert in the field." While that may, for all I know, be an accurate assertion, I wonder if it relegates me to the status of nonserious graph theorist. And if so, does that impose on me a professional obligation to inform the editors of the journals in which I've published of my stature as second-class authority? [This last bit has been offered in irony.]

I find it truly fascinating how easily we, all of us (or certainly me, anyway), allow our feathers to get ruffled around here. As mature and self-reliant as we are, it seems all too easy to react like offended 5-year-olds.

Ah well, onward and upward!—PaulTanenbaum (talk) 12:57, 24 October 2008 (UTC)

Hi Paul,

I did not mean to, and did not, suggest that you are not a serious graph theorist. What I did write is that, based on the wiki contributions that I personally was familiar with, Nsk92' record is impressive. I did not say that he is the most serious expert, only that he seems to me the most serious writer in this area. This does not mean other writers are not serious; I am certainly not sufficiently familiar with your contributions to make such a judgment. I just checked your record on mathscinet and noted that you certainly have a serious publication record. There is therefore no reason to write to the journals :). I did find some of Gandalf's comments somewhat impertinent. You are certainly free to disagree. Katzmik (talk) 08:44, 26 October 2008 (UTC)

Wikipedia's Expert Peer Review process (or lack of such) for Science related articles

Hi - I posted the section with the same name on my talk page. Could you take part in discussion ? Thanks ARP Apovolot (talk) 21:32, 25 October 2008 (UTC)

Reply

The discussion on uniform continuity is getting fragmented, so I wanted to point out I left some replies on my talk page. — Carl (CBM · talk) 13:40, 28 October 2008 (UTC)

Non-standard view of uniform continuity

This version has a section on the non-standard analysis point of view, which I just wrote. Michael Hardy (talk)

Merge into uniform limit.

No, I was not suggesting a merger. Thenub314 (talk) 07:57, 30 October 2008 (UTC)

Page on Kevin Houston

I didn't realize you created a page for Kevin Houston. It's explicitly not our goal to have an article about every living person, or about every professsor in every university, so we have some rules of thumb about when a person should have an article.

The informal test that people use to decide on articles about academics is: is this person more notable than the average professor? I don't know Houston, or know of him, but he appears to be a typical successful academic, not more notable than average. Turning to the written guideline, I don't think Houston meets any of the criteria at WP:ACADEMIC#Criteria. This is in no way a slight on Houston, just my attempt at an objective evaluation of the WP criteria for biographies. What do you think? — Carl (CBM · talk) 15:15, 31 October 2008 (UTC)

Thanks for your comment. Houston has at least two papers in the very top journals, as well as a book. I thought either of those would qualify him for inclusion? Katzmik (talk) 11:39, 27 November 2008 (UTC)

Wikipedia's Expert Peer Review process (or lack of such) for Science related articles

Hi - I posted the section with the same name on my talk page. Could you take part in discussion ?

User: Shotwell suggested (on my talk page) "I would endorse a WP:EXPERTADVICE page that outlined the wikipedia policies and goals for researchers in a way that enticed them to edit here in an appropriate fashion. Perhaps a well-maintained list of expert editors with institutional affiliation would facilitate this sort of highly informal review process. I don't think anyone would object to a well-maintained list of highly-qualified researchers with institutional affiliation (but then again, everyone seems to object to something)."

We could start with that if you would agree ... - could you help to push his idea through Wikipedia bureaucracy ? Apovolot (talk) 16:36, 1 November 2008 (UTC)

Non-standard calculus

You write "Believe it or not, there is a solution to the problem -- non-standard calculus."

I don't recall making such a comment on any of the wikipedia pages. Perhaps I made it on one of the talk pages? I do not recall the context. Katzmik (talk) 11:42, 27 November 2008 (UTC)

There is no problem! Cauchy put the definition of limit on a fully satisfactory foundation. Because this is based on the standard definition of the real numbers, it is widely accepted among mathematicians (like myself).

I am familiar with non-standard calculus (e.g., the book by the Henles). But it addresses a different problem (viz., Find an axiom system in which infinitesimals are well-defined), and so it is based on different axioms from ordinary analysis.Daqu (talk) 10:02, 3 November 2008 (UTC)

I am very busy right now but if you read through a few paragraphs of Non-standard analysis it might help clarify things. Katzmik (talk) 11:42, 27 November 2008 (UTC)

No, thank you. I was referring to this comment, which Wikipedia claims you placed on my User talk page:

"Hi, I just noticed your comment at the talk page of limit of a function concerning the difficulty of the definition. I think your point is well taken. Believe it or not, there is a radically simple solution to the problem; see non-standard calculus. Katzmik (talk) 11:46, 23 October 2008 (UTC)"

It is to this comment that my above comments apply.

(Also, may I suggest not interposing your comment in the middle of mine? Thanks.)Daqu (talk) 09:00, 11 January 2009 (UTC)

I responded at your talk page. Katzmik (talk) 10:27, 11 January 2009 (UTC)

You wrote: "In response to your comment at my talk page: there is a common misconception that non-standard analysis is based on a different system of axioms as compared to ordinary analysis. This is not the case. "

You're joking, right? I never said it was built on a different set theory. But it uses a number system with infinitesimals, whose existence is due to an axiom stating that they exist -- an axiom whose presence virtually defines what "non-standard analysis" means. (I didn't criticize non-standard calculus, but rather stated that it did not solve the "problem" of defining a limit, because in my view there is no problem.) Anyway, I prefer my infinitesimals to be in the context of Conway's "surreal numbers", which doesn't merely make infinitesimals axiomatic, it constructs them.Daqu (talk) 07:16, 12 January 2009 (UTC)

No I am not joking, NSA constructs them in the context of ZFC without introducing any new axioms, and in this sense it is a "conservative theory" (see Criticism of non-standard analysis). Meanwhile, the surreals lack the transfer principle, which is what makes infinitesimals relevant to calculus. Katzmik (talk) 10:24, 12 January 2009 (UTC)

AfD nomination of Kevin Houston

I've nominated this article for deletion. We have an ongoing problem at WP with biographical articles on people who aren't really notable, which is one factor I took into account. The discussion is at Wikipedia:Articles for deletion/Kevin Houston. — Carl (CBM · talk) 14:26, 10 November 2008 (UTC)

The discussion is predicated on the assumption that the page can be restored when the book is published. I find this a bit comical as the publisher already has two pages advertizing the book (one for softcover, one for hardcover). Moreover the book already has an ISBN number. We are not talking about the first rough draft :) Katzmik (talk) 13:02, 27 November 2008 (UTC)

ISBN numbers are assigned by the publisher, so it is perfectly normal for books in the publication pipeline to have them. But the discussion is about more than just the book; see my comments below. — Carl (CBM · talk) 13:56, 29 November 2008 (UTC)

Alexis Clairaut?

I have perhaps somewhat rashly added to your new article titled Clairaut's relation, the statement that it is named after Alexis Clairaut. Please attend to that if that is not correct.

I am puzzled by the part that says "the distance r(t) from a point on a great circle of the unit sphere". Usually when I read "the distance from A" I expect the next word to be "to". It's the distance from a point on the great circle to something. To what? The article doesn't seem to say. Michael Hardy (talk) 07:36, 28 November 2008 (UTC)

Houston

I agree it would be ridiculous to have a deadline-based system where Houston could not have an article until the moment his book was published. The actual system is that in order to have an article on him or her, we want a professor to be "more notable the average professor". There is a list at WP:PROF of some of the reasons this could be true - a person could have a named professorship, or win a major award, or publish an influential book, etc. In the case of Houston, it's too soon to say if the book will be influential (I'll reserve my opinion on that). Even on the day the book is released, it will still be too soon to say. The only reason people brought up the book is that they didn't see any other reason that Houston might meet the criteria at WP:PROF. The underlying issue is that Houston, although successful, just doesn't pass the threshold required to have an article. — Carl (CBM · talk) 13:36, 29 November 2008 (UTC)

Also, we probably should not have an article on the book merely because it has been published. Maybe after a year or two, if it turns out that the book is particularly notable as a math text, an article is warranted. But there are many much more notable books that do not have articles; we do not have a general policy of making an article for every published text. — Carl (CBM · talk) 13:52, 29 November 2008 (UTC)

OK, the policy is clearer to me now. Katzmik (talk) 21:35, 29 November 2008 (UTC)

License tagging for Image:Microscope4.jpg

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Knol

Are you aware of Knol? I think you are fighting against windmills here. Non-standard analysis plays a very marginal position in present day mathematics, and therefore it must also play a marginal role in Wikipedia. The general rule is: Change the real world first, then the Wikipedia articles. As you are now seeing, your attempts to push non-standard analysis into mainstream articles have led to a situation where colleagues are looking for policy violations even in articles that are specifically about the subject. It might be wise to continue your work, for a while, in an area where you have more freedom, e.g. because WP:UNDUE is not applicable. --Hans Adler (talk) 17:08, 11 December 2008 (UTC)

I believe your underlying assumption is incorrect. Google scholar lists over 900 cites for Robinson's book. Katzmik (talk) 14:05, 14 December 2008 (UTC)

Wikipedia_talk:WikiProject_Mathematics

It might have been nice if you had let me know you were discussing my edits and their reasons.

On a separate subject, I find your insistence at removing edits before they are discussed very much outside of the spirit of Wikipedia, I was simply being bold (in Wikipedia terminology). Why is it you feel my edits were hostile? Thenub314 (talk) 15:30, 12 December 2008 (UTC)

Perhaps discussing what came first the chicken or the egg will not get us anywhere, so let's see if we can improve the article in question. Katzmik (talk) 19:52, 13 December 2008 (UTC)

AfD nomination of Bishop–Keisler controversy

An article that you have been involved in editing, Bishop–Keisler controversy, has been listed for deletion. If you are interested in the deletion discussion, please participate by adding your comments at Wikipedia:Articles for deletion/Bishop–Keisler controversy. Thank you. Mathsci (talk) 05:35, 14 December 2008 (UTC)

Paul Halmos

Hi Katzmik. I probably share many of your views on Halmos' extreme conservatism, although this is not really material for wikipedia articles. He was not a fan of K-homology (Brown-Douglas-Fillmore theory) and did activate in trying to avoid using it to prove results in single operator theory. He was however ignored by those with good taste. :) Best regards, Mathsci (talk) 11:48, 15 December 2008 (UTC)

Reply

Sorry about the delay. I did receive your email, and I've offered some feedback on the article's talk page, although I don't know if it can really be of much use. Cosmic Latte (talk) 09:59, 18 December 2008 (UTC)

Talking to administrator about my edits.

I would like to point you to Wikipedia:ANI. I know you feel I am (possibly unintentionally) hostile, and tedious. I have not read the various rules associated with this, but I think it is the correct place to get administrative help when you have a problem with another editor.

I would like to resolve any personal differences of opinion by discussing, but I can see from your comments at the AfD perhaps you feel we are beyond this.

All the best, Thenub314 (talk) 18:49, 18 December 2008 (UTC)

Refraining from radical deletions without prior discussion should go a long way already. Katzmik (talk) 18:53, 18 December 2008 (UTC)

What radical deletions? I moved one section to a page where I thought it was better suited. I am sorry my style of edit first, ask questions if it discuss if the edit becomes an issue bothers you so much. Another section I moved out of the public eye until we could improve it because it was claiming something about the points of view of a person. We should make sure these are correct before they go up since it may impact someone's reputation. It is really not appropriate to demand I speak to an admin before editing, or refuse to discuss edits until they are taken off the page, etc. I have been trying to go along with it just to keep things calm, but that doesn't make it alright. Thenub314 (talk) 16:06, 19 December 2008 (UTC)

Your comment about "speaking to an administrator" is a bit misleading. If you are referring to the discussion at 0.999..., I did request that you do so, but only after you had already twice reverted an edit that, I felt, was agreed upon by 3 or 4 editors. Katzmik (talk) 18:44, 21 December 2008 (UTC)

I don't think is it misleading. We have/(are having) a similar back and forth at Criticism_of_non-standard_analysis over the section on Halmos. After reading the comments of several other editors I also felt there was some support for my decision to take out the section. But during this disagreement no party (not you, nor I, nor anyone else) demanded that anyone act in a specific way, despite several reverts in both directions. This seems like a much better way to have a disagreement. Thenub314 (talk) 08:49, 22 December 2008 (UTC)

At any rate, I never called for any "administrative action against" your edits, so with your permission I will rename this particular subsection. Katzmik (talk) 13:40, 22 December 2008 (UTC)

Are we all friends now? Take a break, folks, and enjoy the holidays. I know I will. --C S (talk) 10:27, 22 December 2008 (UTC)

Your probabilistic "Proof of Loewner's torus inequality"

Hi Mikhail, I wonder, is your probabilistic "Proof of Loewner's torus inequality" sketched in Loewner's torus inequality available anywhere outside Wikipedia? I am asking because it is mentioned in Talk:Probabilistic proofs of non-probabilistic theorems#More items?. Boris Tsirelson (talk) 21:10, 24 December 2008 (UTC)

Nice. Now it is mentioned among Probabilistic proofs of non-probabilistic theorems. Hope it is OK with you. Boris Tsirelson (talk) 21:08, 25 December 2008 (UTC)

Did you see the talk page?--刻意 11:01, 30 December 2008 (UTC)

Re: Keyi's edits at P. M. Pu

I would suggest WP:ANI if there is disruptive behavior ongoing, in order to get attention from more than just one admin. Otherwise, if it is a content issue, have you tried dispute resolution? WP:3O, or WP:RFC? Cirt (talk) 01:56, 31 December 2008 (UTC)

A. G.

Sorry, I could not help you as I know little about algebraic geomery.--Keyi 10:35, 2 January 2009 (UTC)

Emails

Hi. You had said something about starting a friendly exchange about our common interests. But you never followed up with anything specific you wanted to discuss. I am still game if you are. Thenub314 (talk) 19:40, 5 January 2009 (UTC)

I wish I could

I don't know of any applications of NSA to physics directly, at least, nothing deeper than the informal use of infinitesimals, but thats older than NSA. As far as I get it, NSA is about replacing epsilon-delta arguments with equivalent rigorous arguments using infinitesimals with fewer quantifiers. That's a proof thing, so I think that physicists wouldn't pay much attention. On the other hand, I read somewhere that there is an approach to stochastic processes which uses NSA heavily. This might have future applications to defining stochastic or quantum processes. Sorry I can't be of more help.Likebox (talk) 20:21, 11 January 2009 (UTC)

0.999...

Hi, I have recently reverted two of your edits at this page and I wanted to leave a comment to try to avoid conflict. Let me say your second insertion of the limit of a sequence was good, I liked it. As my edit summary suggested I felt the addition info was a bit too long. A few other comments as to what I thought: "In non-standard calculus," seems a bit unnecessary since we are already in a non-standard setting in this setting. The phrase "while the hyperreal number u_H=0.999... with H-infinitely many repeated 9s satisfies a strict inequality 0.999... < 1." I feel is a bit misleading, because you haven't specified all of the decimals yet. Of course I believe that you mean 0.999...;...999000... where the first zero is in the H+1st decimal place, but that is not quite (to my mind) 0.999...

I have the feeling at this point your not interested in speaking with about these subjects with me any more, but I am interested in keeping my interactions friendly so I hope this message comes as more of an olive branch then simply more criticism. Thenub314 (talk) 20:00, 13 January 2009 (UTC)

I am always open to discussion. Katzmik (talk) 16:10, 14 January 2009 (UTC) P.S. Incidentally, I personally think the underbrace under 999... does a better job of explaining which hyperreal we are dealing with, than Lightstone's notation, and hardly OR as its meaning is crystal clear. Katzmik (talk) 16:13, 14 January 2009 (UTC)

I simply put in Lightstone's notation be cause we had been using it throughout the paragraph. I am really not trying to be quarrelsome, that is why I left a message here to start a discussion. Thenub314 (talk) 16:32, 14 January 2009 (UTC)

Email about Limits

I try not to get worried about people on Wikipedia and the content of articles unless it is plainly wrong content being put in. If I got het up about people or that article are a bit messy I'd just die of a heart attack long before I should be due. So normally if things don't go my way after a couple of goes I just give up on things. So my reaction is about the calculus limits was it wasn't important enough to go further with. Dmcq (talk) 23:52, 8 February 2009 (UTC)

Infimum of the empty set

I have removed the following message from Katzmik's user page and placed it here:

Katzmik: I learned math a different way from you, and so did the writers of every textbook I've ever read that defines the inf or the minimum of a set of real numbers. The inf or minimum of the empty set is -- as I learned it -- undefined. Can you cite me an authoritative source that defines it differently?

(Certainly there are times when there is an explicitly stated convention that the inf or minimum of an empty set is expressed as infinity, but that is only an occasional convention and not the standard definition.)Daqu (talk) 08:46, 24 February 2009 (UTC)

Let S be a subset of the real numbers. By definition, the infimum of S is the largest extended real number (i.e., element of $[-\infty ,\infty ]$) x such that $x\leq y$ for all y in S. According to this definition, we do indeed get that the infimum of the empty set is $+\infty$. Confirmation can be found (for instance) on Terry Tao's webpage

http://www.math.ucla.edu/~tao/java/MultipleChoice/sequences.txt

or in the following book

http://books.google.com/books?id=mVeTuaRwWssC&pg=PA15&lpg=PA15&dq=infimum,+empty+set&source=bl&ots=6i-_Ggo4eS&sig=u2yhJF0y_aFRo1AtXVyYHfh3PWU&hl=en&ei=gWykSZrOBJ6DtwfKzNDHBA&sa=X&oi=book_result&resnum=3&ct=result

I believe the wikipedic convention is for remarks/questions like this to be placed at the user's talk page, not on the user page. Plclark (talk) 22:01, 24 February 2009 (UTC)

However, if you consider the empty set as a subset of an arbitrary ordered set, the infimum is not defined unless the ordered set has a largest element. But as a subset of the extended real number system, I agree with Plclark. --PST 06:15, 27 February 2009 (UTC)

(ε, δ)-definition of limit priority

Hi Katzmik,

Thanks for your email about priority of the (ε, δ)-definition of limit. I’d read that it was due to Cauchy, so I had been copying that.

Following your email, I looked into it, and found that in fact the (ε, δ)-definition of limit is due to Cauchy (1823), as per the references I’ve cited there.

Weierstrass certainly applied the method of εs & δs, and, significantly, saw the importance of uniform convergence (due to Gudermann), though he did not originated either of these concepts.

I’ve thus made some edits to (ε, δ)-definition of limit, Karl Weierstrass, Gudermann, and uniform convergence to reflect this (Cauchy came up with ε-δ limit, Gudermann noted uniform convergence, Weierstrass formalized and applied it), but it could surely use work.

Thanks for your comments!

—Nils von Barth (nbarth) (talk) 12:40, 1 May 2009 (UTC)

Prime geodesic

See my comments on the talk page. You probably know more about the terminology than me. 75.95.125.245 (talk) 15:23, 19 July 2009 (UTC)