User talk:Fropuff/Archive 3

Group action

Nice edits on the group action page. Keep up the good work! - grubber 02:52, 5 October 2006 (UTC)

Your question at math help desk

You asked if a top group can be connected but not path connected. I'm a tad impulsive & i just put that stuff w/ a bit more as a question at sci.math.research w/ your handle Fropuff associated. hope u don't mind.Rich 08:21, 17 October 2006 (UTC)

I don't mind at all. For anyone paying attention, Rich got an answer from Daniel Asimov on sci.math.research who pointed out that the solenoid is a connected topological group (even a compact abelian one) which is not path-connected. I believe the path-component of the identity is dense in this group but I'm not totally sure. -- Fropuff 18:40, 20 October 2006 (UTC)

Coarsest or finest?

I recently edited the disjoint union (topology) page to correct the definition, but that change was reverted so perhaps I should argue the case. The finest topology for which any given map is continuous is the discrete topology so the definition as currently stated is degenerate. We're interested in the smallest (or coarsest) topology for which the canonical injections are continuous. Tim

No, the finest topology (or final topology) is the correct one. We are concerned with maps $\phi _{i}:X_{i}\to X$ where the $X_{i}$ have a given topology. We are then trying to find a topology on the codomain $X$. If we stick the discrete topology on $X$ then these maps may not be continuous. You increase the chances of continuity by coarsening the topology. They will certainly all be continuous when $X$ has the trivial topology. So we go for the finest topology for which they are continuous. The coarsest topology (or initial topology) is used when you are trying to putting a topology on the domain of a family of functions rather than the codomain. -- Fropuff 16:55, 3 November 2006 (UTC)

Embedded submanifold

The article is indeed a special case for the Euclidean space. But when I wrote the article, there was no "submanifold" article at all - so it's better something than nothing. Also, in physics applications this Euclidean case is often good enough. Your article Submanifold is oriented primarily to the mathematicians - should we put my "Euclidean case" as the example in your article, so the physicists would also be able to find their way through? Thanks, --Sagi Harel 17:25, 12 November 2006 (UTC)

I agree that the Euclidean case is important. Certainly, mention of it should be made in the submanifold article. The present version of that article is a stub at best. The Euclidean case probably even deserves its own article — I'm just not sure what the best name would be. Any thoughts? -- Fropuff 01:08, 13 November 2006 (UTC)

criterion

Please. "This criterion is..." or "These criteria are...". Michael Hardy 04:44, 25 November 2006 (UTC)

Hi

Did you attend the colloquium at UT today? It was fantastic. linas 04:19, 28 November 2006 (UTC)

Was it? Damn, too bad I missed it. Hopefully I can catch the GADGET seminar Mark is giving tomorrow. -- Fropuff 06:51, 28 November 2006 (UTC)

Hmm, well I was going to pass as it'll be mostly over my head, but maybe I'll come anyway. You'll introduce yourself, right? Uhh, you've noticed me in these before, the irregular stranger? linas 15:17, 28 November 2006 (UTC)

Maths portal

There are some comments at Wikipedia:Featured portal candidates/Portal:Mathematics that you might be pleased to read :-) Tompw 19:17, 3 December 2006 (UTC)

Cool. I'm glad people are liking it. And kudos to you for so diligently maintaining the portal! I sort of slacked off after redesigning it last year. Let me know if you ever need help with anything. -- Fropuff 01:07, 4 December 2006 (UTC)

... and after those kind words, I duely neglect to update the pic of the month... thanks for doing it :-) I'll try and get round to setting up an automated update as per AotW. Tompw (talk) 15:31, 17 December 2006 (UTC)

Polar coordinate system

Hi Fropuff. Thanks for the comments at Talk:Polar coordinate system. I will try to address those issues. However, I admit, I really don't know about Euclidean metrics or multi-variable calculus (I only just turned 16, single-variable calculus is my limit for now). Can you perhaps add information about that to the article? Thanks. —Mets501 (talk) 12:03, 22 December 2006 (UTC)

I will see if I can find the time. No promises. It might be some fun stuff for you to learn however. None of it is overly difficult. Finding the correct integration measure (rdrdθ) comes from a simple calculation of the Jacobian of the coordinate transformation (x,y) → (r,θ). Unfortunately, our article on metric tensors isn't very good, but it at least shows you how to calculate the Euclidean metric in polar coordinates. -- Fropuff 18:17, 22 December 2006 (UTC)

Yeah, I was already starting to check it out. I basically understand it, but not enough to write about it confidently. —Mets501 (talk) 20:20, 23 December 2006 (UTC)

Flat

Hi Fropuff. I ran into your User:Fropuff/Tasklist where I saw such redlinks as flat connection and flat vector bundle. Well, let me pile on. How about adding there the following: Flat cohomology, Flat cover, Flat form, Flat manifold, Flat section, Flat space theorem, Flatness theorem, all from Wikipedia:Missing science topics/Maths10. Some of them are probably not related to geometry, but perhaps a few are. I don't mean you need to write all of them of course, but having some redlinks on a to-do list can work miracles in the long term. :)

You can reply here if you have comments. Cheers, Oleg Alexandrov (talk) 05:07, 30 January 2007 (UTC)

Cool, more articles I don't have time to write :) But yes, I should update my tasklist with some of the more important redlinks (now that you mention it, it's very surprising that flat manifold is red — I had no idea). BTW, thanks a bunch for updating the List of mathematical redlinks. I continue to find that list useful. -- Fropuff 05:29, 30 January 2007 (UTC)

Commutative diagrams howto?

Hi Fropuff. How do you convert your commutative diagrams from LaTeX to PNG? Geometry guy 20:05, 10 February 2007 (UTC)

I use John Walker's textogif Perl script. I believe the script makes use of the Netpbm library for doing the graphics conversions. I usually set the variables:

$dpi = 100;
$res = 0.25;

I do this on Windows using cygwin and MiKTeX. Of course, it should all work fine under Linux. -- Fropuff 01:56, 11 February 2007 (UTC)

Thanks. I'll try it: I made a new bundle map article and it would be nice to decorate it with commutative diagrams. Geometry guy 02:04, 11 February 2007 (UTC)

So I noticed; looks good. I'll be happy to throw together a few diagrams if you can't get it to work. Just let me know. -- Fropuff 02:08, 11 February 2007 (UTC)

Well, I won't be trying it for a day or two at least, and if the two obvious diagrams (analogous to the two in the vector bundles article) miraculously appear in the bundle map article, it will certainly save me some work :) Geometry guy 02:16, 11 February 2007 (UTC)

How to draw diagrams for category theory

I want to write a wikipedia article but I'm not sure how draw diagrams (pullbacks etc)? I was told that maybe you could help? A user suggested that I should use LaTex to draw the diagrams, but it seems a bit tedious. —The preceding unsigned comment was added by Bgst (talk • contribs) 00:54, 25 February 2007 (UTC).

I find using LaTeX to do the diagrams to be the easiest solution. I would use a vector graphics program but I've never found one that has any sort of decent mathematical typesetting ability. Once you get it all set up it's actually quite easily to do in LaTeX. You'll have to pick a diagram package to use. I use Paul Taylor's. Another one is by François Borceux. If you browse through my contributions at wikicommons you can find examples of how to write the code using Taylor's package (I've included the source with most of these files).

After you draw the diagram in LaTeX you'll need to convert it to a suitable format for Wikipedia such as SVG or PNG. I would convert to SVG but I'm not aware of a TeX to SVG converter. For instructions on how to convert to PNG see the post immediately prior to this one. I hope that helps. And welcome to Wikipedia! -- Fropuff 02:04, 25 February 2007 (UTC)

Thanks I will try it - Bgst 15:41, 26 February 2007 (UTC)

Interior product and transition functions

I'm glad I was able to blue one of your redlinks (on Interior Product) by a simple move, though the article still needs a lot of work!

Your new Clifford bundle article is great, but I have a question: why bring in transition functions? (Here and in algebra bundle.) The modern point of view is to equip the bundle with a fibrewise algebra structure. Then local trivializations are required to respect this extra structure (much like the local trivializations of a vector bundle) and are therefore transition functions are automatically algebra automorphisms (just as transition functions for vector bundles are automatically linear). Geometry guy 23:50, 28 February 2007 (UTC)

No good reason. You are quite right though. I'll reword those articles to make more sense. And thanks for moving interior product—I'd been meaning to do that. Perhaps I'll get around to working on that article sometime soon. It's in somewhat sad shape right now. -- Fropuff 01:25, 1 March 2007 (UTC)

Frame bundles

Hi Fropuff, your revision of frame bundle is looking great so far! I have some minor suggestions: for instance there is ambiguity in the notation GL(E) (as opposed to GL(M)) because it could refer to the bundle of fibrewise automorphisms, but I am happy to wait until you roll out the revised version before editing in my suggestions. This article will be useful in for the connection (principal bundle) article. I might roll out a provisional version soon, even though it needs much more editing, because I think that you (and hopefully others) will be able to do a great job of improving and rewriting it. Geometry guy 21:14, 2 March 2007 (UTC)

Good point, thanks! And I look forward to your connection on principal bundles article. -- Fropuff 21:18, 2 March 2007 (UTC)

Speed of light needs expert to straighten out conflict.

I was contacted by POM on this, but I'm the wrong person. A person who understands special relativity and understands the typical misconceptions is needed to talk lucidly and diplomatically to the editors in conflict. Can you or someone else you know is up to it take a look? Thanks!Rich 06:45, 20 March 2007 (UTC)

This seems to have settled down a bit: the undiplomatic anonymous editor has a valid point in my opinion (you can't just state formulae without saying what the terms mean) but I think it can easily be resolved. Geometry guy 22:42, 21 March 2007 (UTC)

Sorry for the delayed reply. The dispute seems to be one over an interpretation of variables. In the end it amounts to whether or not you have a minus sign in the formula. I have no problem with how the article is worded right now. Actually, the article that really could use some work is velocity-addition formula which is completely devoid of any interpretation. You might try posting at Wikipedia talk:WikiProject Physics to see if any is interested in working on it. It wouldn't take much to clean it up. -- Fropuff 06:11, 22 March 2007 (UTC)

Connections

I would appreciate your comments on the state of affine connection, since you are one of the few people here (are there just me and you still left?) who knows about Cartan connections, and I intend to use the experience gained to revise Cartan connection. Geometry guy 22:42, 21 March 2007 (UTC)

Comments at Talk:Affine connection. I'm sure there's a few other people poking around here with knowledge of such things. They probably just lack the time or desire to contribute. So for now it seems to be just us. -- Fropuff 05:51, 22 March 2007 (UTC)

I know you are working on other things right now, but I thought you might like to know (in case you've taken it off your watchlist) that I did some work on Cartan connection. I haven't done as complete a job as I did with affine connection (which itself is far from complete) because it is less clear to me what the goals of the Cartan connection article should be e.g. in terms of the applications and examples. I'd welcome your thoughts as always. Geometry guy 19:49, 12 April 2007 (UTC)

Yes, I've had my mind on other matters, but I'm still watching the connection front. (In my foolish desire to be omniscient I almost never take anything off my ever expanding watchlist. It can be difficult watching several thousand pages at once, but I do try). As soon as time permits I'll gladly offer my comments. In the meantime, keep up the good work. -- Fropuff 23:33, 12 April 2007 (UTC)

Thanks. And thanks for fixing that catlink. I feel really dumb for not spotting it myself, and instead looking for a more obscure fix, since I've already met this issue with linking categories a couple of times! Geometry guy 15:58, 13 April 2007 (UTC)

No problem. The trick works with images too. Just add a colon in front. -- Fropuff 16:24, 13 April 2007 (UTC)

It was one of those stupid situations where I had learnt how to do it right in my own catlinks, but not how to fix other people's. Anyway, thanks for the tip about images. I expect I will get that one wrong a few times too ;) Geometry guy 22:38, 13 April 2007 (UTC)

Request for diagrams

Could you whip up some proper commutative diagrams to replace the three embarassing ASCII-art ones at subobject classifier? If you have the time and inclination, the article (and I!) would appreciate it. --KSmrq^T 20:20, 2 April 2007 (UTC)

Done. I took the liberty of correcting the first diagram. -- Fropuff 02:46, 3 April 2007 (UTC)

Many thanks. And, wow, that was quick! --KSmrq^T 04:33, 3 April 2007 (UTC)

Thanks

Hey Fropuff. Thank you for fixing the math portal, I didn't know what to do about the redlinked pages. Oleg Alexandrov (talk) 03:31, 3 April 2007 (UTC)

Help needed

I'm looking for an uninvolved admin to try to get some sort of discussion going, to short-circuit a possible ArbCom case. Penultimate item on my Talk page. Interested? Charles Matthews 15:20, 25 April 2007 (UTC)

Unfortunately, I'll be a little busy for the rest of the week and weekend. I don't know how much time I'll have. If it is still an issue next week I'll be happy to lend a hand. -- Fropuff 00:47, 26 April 2007 (UTC)

Binary polyhedral groups

Hello, I've looked at the binary (dihedral, tetrahedral, octahedral, icosahedral) groups articles that you have created. It appears that they all have the same structure, and of course, for a good reason: these are all finite subgroups of SU(2) containing the center, and this dictates some common properties (one of my favorite ones is not mentioned, though: the quotient of C² by the action of any such group is a Klein-du Val singularity, with an explicit ADE description of its resolution). Here are a couple of questions: do you think it would make sense to create an over-article about finite subgroups of SU(2)? Or combine all these binary groups in one article? Or is it something that can better be done by creating a (small) category 'finite subgroups of SU(2)'? By the way, the article quaternions and spatial rotation does not seem to contain the discussion of SU(2) to SO(3) homomorphism that you are refering to. Perhaps, it disappeared after recent edits? Regards, Arcfrk 06:08, 2 May 2007 (UTC)

Comments:

1. I've long had intentions of writing an article on the binary polyhedral groups which should discuss all 3 polyhedral cases (tetrahedral, octahedral, and icosahedral). Although it will do no harm to mention the other cases (dihedral and cyclic) as well.
2. One of these years I'm going to finish my article on the unit quaternion group. This will have a section on discrete (= finite) subgroups.
3. The Klein-du Val singularities definitely deserve mention. Perhaps the best place to do this would be in the binary polyhedral groups article. Although I have no objection to mentioning these things in the individual articles as well.
4. No, the quaternions and spatial rotation article doesn't mention the spin homomorphism explicitly, but it should. It does describe the homomorphism without ever calling it such. I'll remedy this sometime.

-- Fropuff 17:52, 2 May 2007 (UTC)

All the isomorphic names for the unit quaternion group would need links. But I think "Spin(3)" is preferable to "Sp(1)" for a vital reason: symplectic groups have an identity crisis, with one author's Sp(n) being another's Sp(2n). Yes, here we have an odd n so the dual interpretation is blocked, but we'd still do better to avoid the issue. Given what you seem to want to do with the article, Spin(3) looks like the winning choice. (Although "S³" is a popular and natural choice, it forces an extended group structure discussion into an article where many readers will not appreciate it. And "SU(2)", while especially popular among physicists, brings complex numbers in prematurely.)

Various sources have influenced me to believe that Clifford algebras are the winning approach to Spin groups. One is Lecture 20 in Fulton & Harris, Representation Theory: A First Course, ISBN 978-0-387-97495-8. --KSmrq^T 19:25, 2 May 2007 (UTC)

I assume you are referring to my draft article mentioned above. I think I've come to the conclusion that the proper name for the article is unit quaternion group. At least that is the perspective from which I am writing the article.

As to the proper notation for the group, it is regrettable that such a fundamental object in mathematics doesn't have a standard notation. I think Sp(1) is the best that one can do. Yes, some people call it Sp(2) but I think these are in the minority. I also think is a bad notation. The subgroup of GL_n(H) that preserves the inner product should be Sp(n) just as the subgroup of GL_n(C) that preserves the inner product is U(n). The image of Sp(n) in GL_2n(C) should be written USp(2n) — it is the intersection of U(2n) and Sp(2n,C). In this way, Sp(1) is canonically the group of unit quaternions while USp(2) is naturally SU(2).

You are correct in saying that Clifford algebras are the best approach to the Spin groups. So Spin(3) should really be thought of as a subgroup of the group of units in Cℓ₃(R). Unfortunately, there is ambiguity here as well. Depending on your choice of sign in the fundamental Clifford identity, Cℓ₃(R) could either mean an algebra isomorphic to H⊕H or to M₂(C). In any case, the even subalgebra, in which Spin(3) lives, is isomorphic to H. So one can identity Spin(3) with the group of unit quaternions. The problem is this identification depends on a choice of isomorphism between the even subalgebra of Cℓ₃(R) and H.

Hence, my choice to stick with Sp(1) as the notation for the unit quaternion group. It is not perfect, but I think it's the best we can do. It also is as standard a notation as exists for this group.

As far as the binary polyhedral groups go, I've described them as groups of unit quaternions not as elements of Spin(3) or SU(2) so I use the notation Sp(1) there. You could, of course, provide alternate descriptions. Perhaps, the "correct" thing to do would be to regarded them as subgroups of Spin(3), but I didn't want to go through the overhead of identifying Sp(1) with Spin(3). -- Fropuff 02:09, 3 May 2007 (UTC)

One of the things that makes the "unit quaternion group" significant is the fact that it has so many different interpretations. Hence, it does not have one standard notation, it has many. Which is "best" (or "standard") depends on viewpoint, and on application. For you, Sp(1) feels most comfortable; for me, it's S³ or Spin(3); and for most physicists, it's SU(2).

Consider the following confusion. In Fulton & Harris (pp. 96–100), a standard text, we have an enlightening discussion about the Sp notation. Let Q be a bilinear form on R²ⁿ which is skew-symmetric and nondegenerate; the subgroup of GL_2nR preserving this form is denoted by Sp_2nR. Similarly, Sp_2nC preserves a skew-symmetric bilinear form, and is a subgroup of SL_2nC. In contrast, Sp(n) — or U_H(n) — has a completely different definition in terms of a form on a quaternionic vector space, implying Sp(n) = U(2n) ∩ Sp_2nC. Included is a note that SO_nR is SO(n) but Sp_nR is not Sp(n); also some write Sp_n where they write Sp_2n. The good news is, their Sp(1) actually is the unit quaternion group; the bad news is, pity the poor soul trying to make sense of all this.

I don't see how denoting unit quaternions as (some form of) a symplectic group conveys much geometric insight — something we should expect in connection with polyhedral groups. In contrast, constructing a Spin group from reflections in a Clifford algebra naturally has a very geometric flavor. Nor is Sp(1) more standard notation than Spin(3) or S³ — across all contexts. (For example, see Table of Lie groups.)

Ah well; we make our choices and muddle on. --KSmrq^T 21:38, 3 May 2007 (UTC)

I fully agree with you that the notation Sp(1) is not ideal, it can be confusing and it does require some justification. My point is that it is the only notation that I know of for the unit quaternion group. Of course SU(2) and Spin(3) are isomorphic to it, but not naturally so. And S³ lacks an algebraic structure (it is a torsor for Sp(1) to be sure, but there is no preferred identity). It would be best to give it is own name and symbol (such as T for the circle group) but we aren't at liberty to do that here. So yes, we muddle on.

And as I said before, I agree that Spin(3) is the natural setting in which to discuss the polyhedral groups. I am not at all opposed to the idea of changing the articles to reflect this. However, it is easiest to list the elements as unit quaternions (which is why I bothered with Sp(1) in the first place). -- Fropuff 23:27, 3 May 2007 (UTC)

One thing that I am truly missing is why do you not want to post your draft of the article?

By the way, the notation Sp(n) for a quaternionic group is only a good notation while you stay in a purely topological context. As soon as you want to view the group as an algebraic group (or rather, the group of points over various rings and fields ... or do base change), then the issues quoted by KSmrq make the life unbearable. In other words, if the notation we choose for a group reflects the underlying group scheme, then Sp_2n(K) (or Sp_n(K), if you prefer) has to be the isometry group for a 2n-dimensional symplectic vector space over (a field or ring) K, which of course may have nothing to do with (ordinary) quaternions! Specifically for the unit quaternion group, there are a couple of compromises: SL₁(H) and a shorter SH can both be used, as well as H₁ and U₁(H), all of which are more or less self-explanatory. Arcfrk 05:36, 5 May 2007 (UTC)

I trust it is as you say. My knowledge of algebraic groups is superficial at best. I should really remedy that someday. I admit to being a differential geometer and not an algebraic one, so my biases are such. I had forgotten that SL₁(H) also denotes that unit quaternions. However, I think that notation requires nearly as much explanation as Sp(1) since the determinant is not quite well-defined on the quaternions.

As to why I don't post my draft: it has too many gaping holes to publish right now. I realize that posting such articles is in keeping with the wiki spirit, but I usually prefer to polish articles in my own user space before posting them. This way I can make large radical edits without anyone complaining. I have renewed interest in my draft however, and will resume work on it soon. -- Fropuff 06:23, 5 May 2007 (UTC)

Mathematics CotW

Hey Fro, I am writing you to let you know that the Mathematics Collaboration of the week(soon to "of the month") is getting an overhaul of sorts and I would encourage you to participate in whatever way you can, i.e. nominate an article, contribute to an article, or sign up to be part of the project. Any help would be greatly appreciated, thanks--Cronholm144 21:26, 13 May 2007 (UTC)

Compact-Open Properties

Hey, I'm a little dubious about the second property stated on Compact-open topology. Have you a reference for the claim or a clue how to prove it?

Thanks, Tedmcpeencilcase 12:59, 11 June 2007 (UTC)

Well, it's given as an exercise in McCarty's Topology (p. 185 in the Dover edition). I think I proved it at one point in time, but I've forgotten how it goes now. -- Fropuff 18:27, 11 June 2007 (UTC)

Okay, I'll check that book out. Good old McCarty, I'm sure if I knew who he was he'd be a big hero of mine! Thanks for your time.

Tedmcpeencilcase 19:32, 11 June 2007 (UTC)

Indefinite orthogonal group

Dear Fropuff,

Thanks for your contributions to the indefinite orthogonal group article. The statement The identity component of O(p, q) is often denoted SO⁺(p, q) and can be identified with the set of elements in SO(p, q) which preserve the respective orientations of the p and q dimensional subspaces on which the form is definite. is not 100% clear to me. For example, the group O(p,q) does not leave the p-dimensional subspace on which the form is definite invariant. So what does it mean that SO(p,q) leaves the orientation of this subspace invariant?

I will try to reformulate this in the article. Best regards Pierreback 11:43, 13 July 2007 (UTC)

Spin Group

Dear Fropuff,

I've checked the spin groups for indefinite signature, and they seem to be accurate. I've therefore reverted to the old page on spin groups. Please let me know if there are errors there, but I think the groups are correct (and I am a mathematician doing research into Lie groups; but still, mistakes can happen; let me know if this is the case). All the best, Cheesfondue —Preceding unsigned comment added by Cheesefondue (talk • contribs) 09:24, 9 October 2007 (UTC)

I've replied at Talk:Spin group. -- Fropuff 16:56, 9 October 2007 (UTC)

n-sphere

Good to see you editing here again - you have been missed! Meanwhile, a number of us have been trying to turn Homotopy groups of spheres into a flagship advanced mathematics article. In doing so, I've discovered, as you did some time ago, that sphere and hypersphere are hopeless articles. I completely agree with your proposal to move hypersphere to n-sphere (and probably do the same with hypercube). I also agree with your other suggestions for improvements, and I've incorporated the introduction to your draft into the article already. Geometry guy 22:35, 29 October 2007 (UTC)

Cool, I'm glad you found my draft useful. I wish I had more time to devote to these things. There are so many wonderful and interesting things to say about spheres, but the article is in a pretty sad state. I still feel strongly that hypersphere should be moved to n-sphere but I'm sure the idea will find some opposition.

Real life has been keeping me away from Wikipedia more than I like, but I still try to edit when I can. Hopefully I'll get back to more consistent editing in the future. I need my fix. -- Fropuff

I agree with you about the move, and would just do it if I were an admin. I expect the opposition that there was has moved on by now: the articles seem pretty quiet. Geometry guy 21:40, 30 October 2007 (UTC)

My expectations have been confirmed: no response! That is a funny thing about Wikipedia... Geometry guy 20:25, 5 November 2007 (UTC)

Binary polyhedral groups

Hi Fropuff, I've linked binary polyhedral group to a section of Point groups in three dimensions and elaborated the section. I've also added a binary cyclic group stub, and I've been elaborating how the dicyclic groups are binary dihedral groups; I've a few more changes to make there. Given your interest (viz, having written the articles), I figured I'd mention them to you.

Nbarth (talk) 22:04, 25 November 2007 (UTC)

Commutative diagrams

This is connected to the above: as part of wanting to draw a diagram for the dicyclic groups, I found myself needing to draw commutative diagrams, and couldn't find documentation — your user talk page came up on a search and was useful (thanks!).

I've figured out how to do this and documented it at: meta:Help:Displaying a formula#Commutative_diagrams Enjoy!

Nbarth (talk) 22:04, 25 November 2007 (UTC)

Wiki Doctorates

Wiki Doctorate is a new scheme designed to recognise the people who "do all the work" on Wikipedia. It has been mainly developed for Wikipedia administrators however if you have done lots to keep Wikipedia on "the straight and narrow", including being members of different groups which help Wikipedia i.e "The Welcoming Committee. We have selected to email you because you can apply for the doctorate and we would be very grateful if you did and put the userbox on your user page to boost advertising. The following link will take you straight to our homepage.

Yours sincerely

--Dr.J.Wright MD (talk) 23:55, 31 December 2007 (UTC)

Preadditive for Category of rings

Howdy, I really enjoyed your category of rings article, and I think it is a nice place to record the strange details of both it and other similarly natural-sounding categories.

I noticed you recently added the comment that the category of rngs was not preadditive, and included the intuitive reason that pointwise addition of homomorphisms does not give it such a structure. It might be interesting to give a reference or work out the details of the formal proof, where one proves that there is no way to give the hom sets the structure of abelian groups such that the various axioms of preadditive category hold. At least in the category of groups, I think a longer argument is needed to show that not only is the obvious addition not sufficient to give the category of groups the structure of an preadditive category, but that no addition can do so.

If it is interesting to you also, I would be happy to share the ideas that give the Grps proof, and would be interested to see if a nice enough proof could be given to be worth including in the Category of rings article. JackSchmidt (talk) 15:46, 20 January 2008 (UTC)

Indeed. It's a natural question, but one I elected not to address in the article. If you know of a reference or nice proof I think it would be good to include it or at least mention the fact. -- Fropuff (talk) 17:56, 20 January 2008 (UTC)

Well, the idea for groups more or less works for rngs. You find a small but complex enough rng whose endomorphism monoid cannot be the multiplicative monoid of a ring with 1, so you find an object in the category which cannot have an endomorphism ring. To show that this is not silly: the endomorphism monoid of the rng Z/pZ is isomorphic to the multiplicative monoid of the ring Z/2Z, so one can define an endomorphism ring for the rng Z/pZ. Of course that does not mean one can do it for all rngs. In fact a small rng which cannot have an endomorphism ring is the rng Z/3Z x Z/3Z, but I am not sure of the clearest way to present this. The rng Z/3Z x Z/3Z has 9 rng-endomorphisms, exactly 2 of which are central. This shows there is no endomorphism ring, because the order of the center of a finite ring divides the order of the ring (it is a subgroup of the additive group), and 2 does not divide 9.

Finding the rng endomorphisms is not terribly hard (idempotents go to idempotents, the kernel is one of the 4 ideals, idempotent sums of idempotents go to idempotent sums of idempotents), and checking that only the two obvious rng endomorphisms, 0 and 1, are central is not hard, but it is a long calculation all things considered. JackSchmidt (talk) 20:28, 20 January 2008 (UTC)

I get the gist of your argument; though perhaps it's a little too involved to include in the article itself. It seems to me that one should be able to prove that pointwise addition of morphisms for both Grp and Rng is the only possible choice. I can almost prove this. In the group case observe that there is a bijection between any group G and the set Hom(Z, G). If Grp is preadditive then this identification gives every group the structure of an abelian group (let's denote the abelian group operation by #). Bilinearity shows that for any group homomorphisms f and g from G to H we must have (f + g)(x) = f(x)#g(x). Furthermore, one can show that any group homomorphism is also a homomorphism of (G, #), that is: f(x#y) = f(x)#f(y). Though this strongly suggests that # must be the group operation I don't see how to complete the proof. The same argument should work for Rng if one replaces Z with the free rng generated by a singleton. -- Fropuff (talk) 07:00, 21 January 2008 (UTC)

I would like to see this work. There were some suggestions by Michael Slone that in case one was trying to prove it was not an additive category, that the biproduct could be used to create an argument similar to this, but I never saw the complete proof. To complete your argument though, I still see it as "we have an embedding from endomorphism monoids of Grp to endomorphism rings of abelian groups," and so I would prove that one of the monoids cannot work. It seems like some groups would have no trouble. I believe you can explicitly give Ab the structure of an abelian category without defining morphism addition as pointwise, though you do take an isomorphic addition. It is this sort of non-canonical-ness that makes me think the full proof requires an explicit counterexample. JackSchmidt (talk) 08:03, 21 January 2008 (UTC)

Exponential map

Hi Fropuff. Long time no see. :) I wonder if you can look at my edit at exponential map. I hope I got it right, but my understanding of these things is rather shaky. By the way, should it be mentioned that the unit circle example is a particular case of the matrix group example below (since the unit circle is a group of rotations), or that is not necessary? Thanks. You can reply here. Oleg Alexandrov (talk) 06:22, 6 February 2008 (UTC)

Hey Oleg, thanks for stopping by. Your edits look fine to me. I wouldn't worry about mentioning this is a special case of the latter. Readers should be able to figure that out easily enough. I'll add a link to circle group to help. -- Fropuff (talk) 06:35, 6 February 2008 (UTC)

Portal:Category theory

Hi, I've created a new portal, I see that you have edited articles related to category theory, maybe it can interest you. Cenarium (talk) 17:52, 14 February 2008 (UTC)

Baez' error

Baez' piece on the Klein quartic contains an error. I have therefore removed the link. See more at the discussion page of the article. Katzmik (talk) 08:29, 15 April 2008 (UTC)

Division versus inverse

I had planned on making the same edit you just did, but had decided against it. To me either your edit or without your edit is fine, but I thought you might find the reasoning amusing.

If you check the quasigroup axiom reformulations, it says that for every x and y one has: (x*y)/y = x. In other words the function *y has inverse /y. y doesn't have an inverse, but *y does.

If x*y is not defined, just define it to be the symbol "undefined" which is not an element of the carrier set. Then * is total, but not closed.

In other words, which word you choose is just a viewpoint, and the words chosen in the table reflect the axioms of groups to which they were meant to be compared and contrasted. Hence the old way is "superior". Of course, if it takes two paragraphs to explain it, then that is a pretty good indication that the new Fropuff way is superior, so I don't personally have an opinion. JackSchmidt (talk) 18:10, 29 April 2008 (UTC)

I understand your viewpoint, but I think that changing terminology in that fashion is unnecessarily confusing. Under the usual interpretation of "inverse" the table was incorrect. Also, it seems misleading to state that the binary operation on a groupoid is not closed. It is closed wherever it is defined. Your interpretation is valid, of course, but it is not the natural one. I think if we were to stick to the old way we would need to add a footnote explaining exactly what we meant. -- Fropuff (talk) 19:05, 29 April 2008 (UTC)

"we would need to add a footnote [and this is bad]" -- exactly. Sticking to the group axiom words makes the comparison clearer, but might confuse people as to what these other structures actually are. Now that the chart has been pulled into a template, and is not just part of the groups article, the choice is clear: accuracy in the words as you have done, not artificial similarity to the group axioms as they were. JackSchmidt (talk) 19:29, 29 April 2008 (UTC)

Second countable spaces

Dear Fropuff,

When you started the article on second countable spaces, you wrote that for such spaces, compactness, sequential compactness, and countable compactness are all equivalent. Either you made a simple mistake (typo), or you thought that this statement is correct. You probably have much more experience in Wikipedia editing than I do, but here is some advice:

• Sometimes when I type a statement such as: For first countable spaces, compactness implies sequential compactness, I go through the proof carefully (in my head). This ensures that the statement is correct and if someone challenges it, I am ready with a proof.

• I also noticed that you wrote: Recall that a space is first-countable if each point has a countable local base. Given a base for a topology and a point x, the set of all basis sets containing x forms a local base at x, in the section on 'properties'. I am not very 'in tune' with Wikipedia conventions but maybe writing 'recall' is classified as 'ordering the reader about' and is something to avoid. People have told me this many times but I think you will know about Wiki conventions better than I do.

Anyway, a counterexample to what you wrote is the following space:

Let X be the product of a two-point indiscrete space with the integers (or any countable space with the discrete topology). Note that X is homeomorphic to the integers with the topology: U is open iff for every x in U, -x is also in U. It is easy to see that X is not sequentially compact. However it is limit point compact since any subset with more than one point has a limit point. Also, X is second countable. Because of this, limit point compactness implies sequential compactness of X and hence (according to you) countable compactness of X. Therefore, along with your assertion, countable compactness, limit point compactness, sequential compactness and compactness are all equivalent for second countable spaces. But X is a counterexample to this since X is second countable and limit point compact but not sequentially compact.

Actually one can 'fix' what you wrote by adding the additional requirement that the space is also T₁, i.e for second countable T₁ spaces, sequential compactness, countable compactness and compactness are all equivalent. I am not saying that you don't know how to prove this, but try to prove that sequential compactness implies compactness for second countable spaces. You will notice that the proof fails if the T₁ / Hausdorff condition is not assumed. Basically one requires that limit points of sets have the property that every neighbourhood of the point in question intersects the set at infinitely many points.

Anyway, I will change your mistake now. I hope you understand that I just want to point out this mistake. I have made several mistakes in the past and I have now learnt that bactracking through the proof of one's assertion is the best way to prevent mistakes.

Thanks

Topology Expert (talk) 13:47, 2 August 2008 (UTC)

You haven't convinced me that what I wrote is wrong—in a second-countable space, compactness, sequential compactness, and countable compactness are all equivalent. Each of these then implies limit point compactness (this last implication reverses if the space is also T₁). Your space X (the product of the integers with a two-point indiscrete space) is certainly limit point compact, but it fails to satisfy the other three compactness axioms. How does this invalidate my statement?

Your explanation above is a little confusing. At one point you say that X is not sequentially compact, and at another you claim that it is. You also seem to claim that limit point compactness implies sequential compactness when, in fact, the implication goes the other way.

My statement is equivalent to one made in Steen and Seebach (p. 22). See also Figure 5 on page 23. This was my reference for the statement. I haven't attempted the proofs. Perhaps I should.

-- Fropuff (talk) 18:34, 4 August 2008 (UTC)

Sorry, if my explanation was a bit confusing. Also, please ignore my earlier example; I do not think that it is satisfactory. Also, I wrote that for first countable spaces limit point compactness implies sequential compactness. Therefore, the same holds for second countable spaces.

Here is the proof that limit point compactness implies countable compactness for second countable Hausdorff spaces.

Proof:

Suppose that X is not countably compact and let {G_n} be a countable open cover of X that has no finite subcover. Also let H_n = G_1 U G_2...... U G_n ; let K_n be the complement of H_n relative to X for each n. Note that K_(n+1) is a subset of K_n for each n and that each K_n is closed and nonempty. Choose, for each n, x_n that belongs to K_n. We assert that no point of X can be a limit point of A = {x_n | n is a natural number}. If x is in X, note that x belongs to H_p for some p. Then note that for all i > p, x is not in K_i. Therefore, there is a neighbourhood of x that doesn't intersect any K_i for i > p (since K_i is closed). But this means that this neighbourhood of x can not contain x_i for any i > p. This means that this neighbourhood intersects A at only finitely many points. Therefore, x cannot be a limit point of A for in Hausdorff spaces, any neighbourhood of a limit point of a set, intersects that set at infinitely many points. Therefore, X is not limit point compact.

Q.E.D

Notice that the proof collapses if the Hausdorff property is ommitted (in fact one can replace Hausdorff by T_1 in the theorem and it will still hold). Perhaps I will put the following explanation in bold since I believe it is the most convincing:

Sequential compactness implies countable compactness. Therefore, for second countable spaces, limit point compactness implies seqeuntial compactness and for any space sequential compactness implies countable compactness. You assertion implies that limit point compactness implies countable compactness for second countable spaces. This implies (according to you) that the Hausdorff condition can be ommitted. I am not in the position to give an example showing that the Hausdorff condition cannot be ommitted. However, if you can give me a proof that limit point compactness implies countable compactness for second countable spaces (i.e show me that the Hausdorff condition can be ommitted), I will retract my claims that your assertion is incorrect.

Thanks

Topology Expert (talk) 11:21, 5 August 2008 (UTC)

I never said anything about limit point compactness for second-countable spaces, and I certainly never stated that limit point compactness implies countable compactness. It does for T₁ spaces, but this is certainly not true in general. It is easy to find counterexamples which are second-countable (non-Hausdorff, of course).

I believe your confusion stems from the assumption that, for first-countable spaces, limit point compactness implies sequential compactness. This is not true — counterexamples are easy to find (e.g. your space X above). What is true is that countable compactness implies sequential compactness for first-countable spaces. But, as you point out, limit point compactness implies countable compactness only if the space is T_1.

Let me summarize the implications and see if you agree. For all spaces the following hold:

1. Compactness implies countable compactness.
2. Sequential compactness implies countable compactness
3. Countable compactness implies limit point compactness

Arrow 2 reverses for first-countable spaces. Both arrows 1 and 2 reverse for second-countable spaces. Arrow 3 reverses for T₁ spaces. The proofs of all these are reasonably straightforward.

--Fropuff (talk) 05:54, 6 August 2008 (UTC)

First of all, I am just listing down the properties I believe to be correct:

For first countable T_1 spaces:

• Limit point compactness implies sequential compactness
• Limit point compactness implies countable compactness

Also:

• Compact first countable, T_1 spaces are always sequentially compact (trivial)
• Limit point compactness implies compactness for first countable T_1 spaces.
• Sequential compactness implies countable compactness and therefore limit point compactness (as you said)

At this point, I believe that you are correct and I was wrong.

Topology Expert (talk) 09:21, 6 August 2008 (UTC)

I agree with all of your points except the fourth one, which should read:

• Limit point compactness implies compactness for second-countable T₁ spaces.

There are spaces which are first-countable, Hausdorff, and countably compact (and therefore limit point compact), but not compact. An example is the ordinal space [0, ω₁) where ω₁ is the first uncountable ordinal. -- Fropuff (talk) 18:11, 6 August 2008 (UTC)

Concentric circle topology

Dear Fropuff,

You said that you have the book, 'Counterexamples in topology' (first edition)'; am I correct? In any case, I thought you might find it interesting to note that according to this book, the concentric circle topology is compact. However, I found that this is not the case and here is my (somewhat 'visual' but still formal) proof:

Proof

The topology given by the book is generated by a subbase. Therefore, finite intersections of such subbase elements constitute a base. In particular, an open interval on C₁, together with one point of C₂ that is not the midpoint of its radial projection (but belongs to its radial projection) onto C₂ is a basis element. If A is an open interval in C₁, refer to such a basis element as A U {x} (where x is restricted to certain points as described earlier). Choose two open intervals in C₁ that cover C₁ and call these open intervals A₁ and A₂. Then consider the open sets B₁ = A₁ U {x₁} and B₂ = A₂ U {x₂} where x₁ and x₂ belong to the radial projection of A₁ and A₂ respectively but do not equal to the midpoint of these respective projections. Then the sets:

B₁, B₂ and all the remaining one-point sets of C₂ form an open cover of X that has no finite subcover.

Therefore, X is not compact.

Q.E.D

Either this book has made a mistake or the authors refer to a subbase differently to how I do (the topology generated by a subbase (according to me) consists of all unions of finite intersections of these subbase elements). Note also their proof that X is compact assumes that what they call 'subbase elements' are actually 'basis' elements. Could you have a look at this? I know this is not related to the Wiki issues but it is an interesting thing to note.

Topology Expert (talk) 10:25, 10 August 2008 (UTC)

This space you mention is, in fact, compact. I see no errors Steen and Seebach's proof. You appear to misidentify the basis elements in your proof above. As you say, the basis elements are formed by taking finite intersections of subbasis elements (Steen and Seebach assume this as well, although they don't mention it explicitly). Any such basis element which includes an open interval on C₁ will include infinitely many points of C₂. If you think not, explain to me how to obtain your basis element A ∪ {x} using only finite intersections.

--Fropuff (talk) 15:44, 14 August 2008 (UTC)

Fibre Bundle

Dear Fropuff,

With the concentric circle topology, it was not the finite intersection part where I made my mistake; I just was not thinking correctly. Sorry for the late response.

Could you please have a look at fibre bundle? I have included some comments as to why I believe that this article is unsatisfactory on the talk page of User:OdedSchramm who is sadly dead. If you have time, could you please have a look at these comments. Additional comments are (which is more evidence that the article is unsatisfactory):

• In the 'examples' section, an example of a fibre bundle is given and then it is written that the corresponding trivial bundle would be the cylinder, the torus etc... This is ridiculous since a fibre bundle (at least for locally trivial fibrations without structure groups) are ordered quadruples and not just a space. The reader may be able to infer the meaning of these statements but the mathematical language should be more precise

• If I am not mistaken, a G-atlas should be maximal with respect to certain properties (such as the composition of a chart over an open set with the inverse of another chart over the same open set induces a homeomorphism of the fibre when restricted to {x} X F for a fixed element x of the open set. This homeomorphism should be given by the action of a certain element of the structure group on F for every such pair of charts); the maximal bit is missing in the article

Thanks

Topology Expert (talk) 09:53, 10 September 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) 01:05, 27 October 2008 (UTC)

Second countable space

Dear Fropuff,

You wrote in second countable space that the finer the topology, the less likely it is to be second countable. But one can impose a topology on a countable space (that is not discrete) but is still not second countable (even not first countable!). The discrete topology is finer than this topology but still happens to be second countable. So second countability is somewhat like regularity and does not really depend on the number of open sets a space has. I removed the statement you wrote, but if you think otherwise, could you please give me your opinions?

Thanks

Topology Expert (talk) 12:51, 2 November 2008 (UTC)

The statement I made was meant to be vague and serve as a guide to intuition. It was not meant as a precise statement. Certainly, when you altered the statement to make it precise, it became incorrect—as you note above. Perhaps it is best to leave the statement out although since intuition does not always serve one well here. -- Fropuff (talk) 20:04, 2 November 2008 (UTC)

Thanks for the quick response and I did leave out the statement.

I wrote earlier that there was a mistake in counterexamples in topology which was my misunderstanding, but I have found two mistakes since then (which I am quite certain are mistakes!):

1. See Wikipedia:Reference desk/Mathematics for the first one

2. In the first edition, and in the example 'Arens-Fort space'(simply the cartesian product of the non-negative integers with themselves), the example quotes: 'open sets containing (0,0) are precisely those that contain all but a finite number of points in all but a finite number of columns'. Prior to this, it is written that a set, U, containing (0,0) is open if and only if S_m = {(m,n)|n is in U} is finite for all but finitely many m. But both of these definitions of an open set containing 0 contradict each other because in the second definition, U is open if it contains finitely many points in all but finitely many columns whereas in the first definition, U is open if it contains all but finitely many points in all but a finite number of columns.

If you have time, could you please have look at these two (both of these are the first edition so if they are mistakes, they might have been changed by the second edition)?

Topology Expert (talk) 02:57, 3 November 2008 (UTC)

Both of these are corrected in the second edition (only $10 on Amazon—go grab a copy). In the case of the Arens-Fort space the open neighborhoods of (0,0) are the sets U for which the sets

$S_{m}=\{n:(m,n)\not \in U\}$

are finite for all but finitely many m. Sorry, for the delay response; sadly, I haven't been on Wikipedia much lately. -- Fropuff (talk) 02:27, 12 November 2008 (UTC)

Comma Category

The image Image:CommaCategory-02.png you had on the page Comma category was lost in September and has not been recovered. I was wondering if you still had that image and were willing to re-upload it. Thanks. Gandalf (talk) 18:12, 5 November 2008 (UTC)

I restored the original from my hard drive. Thanks. Fropuff (talk) 03:15, 7 November 2008 (UTC)

image loss

Hi,

due to a software problem the following image which you uploaded was lost on Wikimedia Commons (maybe except for a preview image):

• Image:Commacategory-01.png
• Image:Commacategory-02.png
• Image:CommaCategory-02.png

If you still have this file on your hard disk or if you can recover it from other sources please upload it again. If this image was transferred from a local Wikipedia to Wikimedia Commons it can be even recovered by an admin on the local Wikipedia.

Best regards, -- Ukko (talk) 20:14, 6 November 2008 (UTC)

I've restored the last of these. The first two appear to be duplicates of my originals (the ones named with CamelCase). They should be deleted if not used anywhere. -- Fropuff (talk) 03:15, 7 November 2008 (UTC)

Great, thank you! --Ukko.de (talk) 12:08, 13 November 2008 (UTC)

File source problem with File:CategoricalPullback-01.png

File Copyright problem

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n-connected

Hi Fropuff,

I’ve move the article on n-connectedness to n-connected, as there is both an absolute notion (of a space) and a relative notion (of a map), so I wanted to include them both; I’ve also expanded the article to discuss n-connected maps; hope you enjoy!

—Nils von Barth (nbarth) (talk) 02:00, 9 March 2009 (UTC)

Killing vector field

Hi Fropuff,

September 28, 2004 you wrote in the article Killing vector field that for compact manifolds, if the Ricci curvature is positive, then a Killing field must have a zero. Unfortunately, I cannot find this statement in the books available for me now. Could you, please, give a precise reference either here or in the text of the article?

Yours sincerely, Victor Alexandrov (talk) 16:13, 4 June 2009 (UTC)

Two points:

1. I didn't write that statement, I merely copied it from Killing field where it had existed since that article's creation.
2. The statement is wrong as written and has since been removed. I pointed out an explicit counterexample on the talk page.

Hope that helps. -- Fropuff (talk) 02:46, 5 June 2009 (UTC)

Userspace draft moved into mainspace

Hi,

I recently moved one of your userspace drafts into article space in order to repair a Cut and Paste move than an editor had performed earlier. You can read the notice on our administrators' noticeboard here. They copied the editable text from your draft, pasted it into a new window and made an article. This is bad for us because it means that you, the creator of that content are not properly attributed. In repairing the move I had the option of either simply deleting the content from article space or moving your draft page over their article. Since the draft seemed good enough to be an article, I chose to move your draft page over their article.

This is reversible! You can view your draft as it was when you last edited it by clicking on the page history and viewing the revision which bears your name. If you feel that the draft does not belong in mainspace, contact me or any other admin and we will simply move the article into your userspace and delete the redirect. You can reverse the move yourself if you choose, but please remember to place a speedy deletion template (such as {{db-house}} or {{Db-r2}}) on the resulting redirect from the article to your subpage.

I'm sorry that your work got mixed up like this. Please let me know if there are any problems or if you need help with anything. Protonk (talk) 06:14, 18 August 2009 (UTC)

Okay, thanks. I'm fine with leaving it in the main namespace. It's still somewhat stubbish, but I think it's a decent starting point for an article (of course, I would think that, having wrote it). -- Fropuff (talk) 01:57, 20 August 2009 (UTC)

Suspension of admin privileges due to inactivity

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You might be interested in this...

Talk:Eisenstein_prime#An_issue_with_the_image as it relates to an image you created.--SPhilbrick(Talk) 01:38, 2 February 2013 (UTC)

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