User talk:Stca74

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Mathematics CotW[edit]

Hey Stca, 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 00:13, 14 May 2007 (UTC)

Hello[edit]

I thought say hello, and thank you for your interesting comments at WT:WPM, even if we are not entirely in agreement. Actually, I was prompted to stop by because I saw your comment at User talk:Edgerck on the cross product. I was about to make precisely the same point, but wouldn't have done it nearly as well as you did: your comment about the product of real numbers being a pseudoreal was particularly nice! I hope you continue to enjoy it here. Geometry guy 14:40, 22 May 2007 (UTC)

Hello, and thanks for the compliments. I do quite enjoy it here, yes. Let's see, however, for how much longer I manage to have this luxury of some spare time to devote to Wikipedia. In any case, plans for what to do here keep accumulating like unread novels at my bedside... As you've seen in my comments elsewhere, I've become convinced that Wikipedia's coverage of quite a few (if not all) areas of maths require more top-down planning and structure: we have now too many gaps and overlaps as well as seriously uneven coverage due to (often very good) individual articles springing up on random topics based on authors' impulses to write them up. I've started to work on a plan on how the needed reorganisation could look like for algebraic topology (I thought about algebraic geometry first as a topic closer to my own turf but came to the conclusion that editorial complications there are harder and that topology should work as a test case). I've been planning to post a note to the Wikipedia talk :WikiProject Mathematics page with link to the outline once I get it done. If you wish, feel free to have a look at a work in progress and comment — the page is here. Stca74 17:50, 22 May 2007 (UTC)

FLT[edit]

Do you have any comments on Fermat's last theorem. I put it up for A-Class review. It seems that it needs expert help! Geometry guy 21:34, 12 June 2007 (UTC)

Not exactly my expertise (the modular form side of things is not one of my strengths...), but I'll see what I can do. Maybe start adding the structure to the description of the proof. However, this will not likely happen over the next few days. Stca74 18:05, 14 June 2007 (UTC)
Anything you can do will be much appreciated! Geometry guy 19:22, 14 June 2007 (UTC)

Heian Palace[edit]

First of all, congratulations on your GA on this article. This is a very nice piece of work, that should easily go through FAC. However, as it was pointed at GA/R, you will definitely need the page numbers for that. If you do have the book but don't have time to find the exact place, you might try through Google Books, as in this example: by using the research thingy, you can find that buraken is talked about in page 713. I hope this can help you in building other great articles on very interesting subjects.

Sincerely, --SidiLemine 18:33, 16 July 2007 (UTC)

Thanks for kind words and the encouragement to push the article towards FA. I agree with the page number point for FA articles, and will put those in order. As for Google Books, I think that's an excellent advice. Indeed, that's the source I used to locate a few of the references in books read but not at hand.
Regards, Stca74 08:30, 18 July 2007 (UTC)

WikiProject Japan taskforces[edit]

In order to encourage more participation, and to help people find a specific area in which they are more able to help out, we have organized taskforces at WikiProject Japan. Please visit the Participants page and update the list with the taskforces in which you wish to participate. Links to all the taskforces are found at the top of the list of participants.

Please let me know if you have any questions, and thank you for helping out! ···日本穣? · Talk to Nihonjoe 02:06, 8 August 2007 (UTC)

Heian Palace[edit]

Hi Stca74, and congratulations on bringing Heian Palace to Featured Article status. It's remarkable that the article is so purely the work of one person. Again, congratulations! Fg2 06:07, 25 September 2007 (UTC)

Hi, Fg2, and thanks for the kind remarks! This one started in May with the modest intention of creating a stub for the old palace, but grew quite quickly... The motivation was first just to correct a few statements made in the article on Kyoto Gosho, which made it appear that the current palace dates from the Heian period. It seems that some topics are sufficiently esoteric that it is at the same time possible and necessary to work on your own on them — the same appears to be happening with some more technical maths articles like Fibred category that I'm editing at the moment. But while edits may be mostly mine in Heian Palace, it certainly would be much worse without the comments received in Peer Review, the GA process and now in FA process. Despite some moderately excessive requests at times, the Wikipedia collaboration model appears to work nicely. Stca74 10:53, 25 September 2007 (UTC)

Limit superior and limit inferior[edit]

The general definitions in Limit superior and limit inferior for the limits superior and inferior of sets and filter bases look too general to me. In particular, they impose no conditions whatsoever on the relationship between the topology of the space involved and its partial ordering. Without some sort of order compatibility, the limits defined don't seem particularly meaningful. Also, for the set definition, the article suggests ambiguously that the ordering should be a complete lattice, which may be too restrictive as it still makes sense to talk about limits superior and inferior even in contexts where they are not guaranteed to exist. Do you know the most common definitions for these terms? If so, could you make sure the article matches them? Dfeuer (talk) 20:46, 30 December 2007 (UTC)

In fact the whole article seems to be a bit of a mess. The section on sequences of real numbers contains way too much secondary trivia, lim inf and lim sup of real-valued functions is not even defined, the metric space (why metric?) definition does not impose any order structure on the codomain and the definitions do not even make sense,...
As for the definition for filters, first of all the most important definitions, those of lim inf and lim sup of a real-valued function with respect to a filter (base) are missing. And you're right, to have a meaningful theory one should link the topology and order structure together - a natural way would be to require that the topology be the order topology (generated by open intervals) specified by the order structure, which one would assume to be a linearly ordered complete lattice (the latter being equivalent to the topological space being compact). Then lim inf and lim sup of a filter as well as of a function with values in such ordered topological space with respect to a filter would behave as expected. Whether it makes sense for some purposes to consider more general order structures I do not know.
Given that there has apparently been quite some heated argumentation on the talk page, and given the amount of work the page would need, I'm somewhat hesitant to jump into editing it. I could do some expansion and fixing around filters, though (seems that by staying way from the usual undergrad curriculum one avoids most useless edit wars...). Stca74 (talk) 11:05, 31 December 2007 (UTC)
I completely forgot to watch for a response here. I'm sorry about that. I'm not yet convinced that the space should be required to be a complete lattice, although requiring it to be bounded-complete is likely sensible. Of course, the big question is how this actually is defined by working mathematicians, which is something I don't know. The same goes for what sort of order is required. Requiring the order topology certainly works, but it's conceivable that a weaker condition would suffice to give interesting results. One possibility might be requiring that for each subset X with an upper bound, \sup X \in \overline X, or something vaguely like that.Dfeuer (talk) 03:18, 10 January 2008 (UTC)

unrelated to wikipedia[edit]

Hello, you seem like someone who's interested in sharing your knowledge. I see you have a PhD in alg. geom. and work in the financial sector. I may soon be completing mine, studying 4-manifolds. I had some questions about your vocational experiences. If you're curious or willing to talk to me could you send me an email - jwilliam at math . utexas . edu? Orthografer (talk) 01:38, 4 April 2008 (UTC)

groups FAC[edit]

Hi,

thanks again for your FAC comments. I've replied to all of your points. Most of them are covered, I think, but I would like to have your updated opinion, especially 5) and 8), once you have a free moment. Jakob.scholbach (talk) 22:43, 5 September 2008 (UTC)

Thanks for the message; it's a pleasure to contribute, if only through comments this time - Groups will apparently be the first "real" mathematics article to get to FA (General relativity is more physics, and the rest either biographies or rather trivial maths). Comments just left on the FAC page. Stca74 (talk) 14:55, 7 September 2008 (UTC)

Easy as pi?: Making mathematics articles more accessible to a general readership[edit]

The discussion, to which you contributed, has been archived, with very much additional commentary,
at Wikipedia:Village pump (proposals)/Archive 35#Easy as pi? (subsectioned and sub-subsectioned).
A related discussion is at
(Temporary link) Talk:Mathematics#Making mathematics articles more accessible to a general readership and
(Permanent link) Talk:Mathematics (Section "Making mathematics articles more accessible to a general readership"). Another related discussion is at
(Temporary link) Wikipedia talk:WikiProject Mathematics#Making mathematics articles more accessible to a general readership and
(Permanent link) Wikipedia talk:WikiProject Mathematics (Section "Making mathematics articles more accessible to a general readership").
-- Wavelength (talk) 01:42, 29 September 2008 (UTC)

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

Hi - I posted the section with the same name on my talk page. Could you take part in discussion ? Thanks ARP Apovolot (talk) 01:12, 27 October 2008 (UTC)

Vector spaces[edit]

Hi Stca,

I have asked for a GA review at the round table, but people are busy/dizzy with LateX formatting and icon questions ;) But I remember your thorough review of the group article, so if you have a moment, could you review vector spaces? This is the page. Thanks a lot. Jakob.scholbach (talk) 15:28, 28 November 2008 (UTC)

Hi Jakob,
Thanks for confidence. I've unfortunately been too busy at work to respond earlier. The article appears to be very comprehensive - I just left a few minor comments on tensor products on the talk page. Great work! Stca74 (talk) 10:50, 7 December 2008 (UTC)
I was reading your recent post at the talk page with interest. I had not realized that such a staple of dualities is lurking behind the harmless bidual of vector space... Just out of curiosity, one related question: the key point in Poincaré or Verdier duality is, as far as I understand, not only the existence of the 6 functors, but also f!f! = (R)ff(d)[2d] (twist and shift) when f is smooth of relative codimension d, right? What would be the analogous "normalization" in the coherent/Serre duality situation? (Coherent duality is a mess, unfortunately) Cheers, Jakob.scholbach (talk) 21:09, 13 December 2008 (UTC)

(←) Yes, in the topological (and étale, and D-module) theory the key is that f! is right adjoint of R f! (roles of "!" and "*" mixed for D-modules, however). For the coherent duality, one usually deals with the case where f is proper, hence R f! = R f*. Now in more precise terms, one has the following local version of the adjunction:

 \mathbf{R} f_* \mathbf{R} \mathcal{H}om(\mathcal{F},f^!\mathcal{G}) \cong  \mathbf{R}\mathcal{H}om(\mathbf{R}f_*\mathcal{F},\mathcal{G}).

If f is smooth of relative dimension n, then

 f^!\mathcal{G} = \omega_{X/Y}[n] \overset{\mathbf{L}}{\otimes} f^*\mathcal{G},

where ωX/Y[n] is the top exterior power of the sheaf ΩX/Y of relative differentials, shifted n spaces to the left. More generally f! can be more complicated. To recover the ordinary Serre duality for smooth projective X over a field k apply the above to the unique morphism to S = Spek(k) and on S to the structure sheaf, which is just k sitting on the only point of S. Then applying RΓ to both sides of the adjunction one gets

 \mathbf{R}Hom(\mathcal{F},\omega_{X/Y}[n]) \cong \mathbf{R}Hom(\mathbf{R}\Gamma(X, \mathcal{F}),k) ,

whence by taking cohomology and minding the shift

 Ext^p(\mathcal{F},\omega_X) \cong Hom_k(H^{n-p}(X,\mathcal{F}),k).

To link this to biduality, define first dualising complexes: if X is a (locally Noetherian) scheme, then an object \mathcal{K} of the bounded coherent derived category \mathbf{D}_c^b(X) is a dualising complex if (i) it is quasi-isomorphic to a bounded complex of injective sheaves, and (ii) the duality functor

 D_{\mathcal{K}} = \mathbf{R}\mathcal{H}om(\cdot,\mathcal{K})

satisfies the following biduality: the natural map of functors

\mathrm{id}\rightarrow D_{\mathcal{K}} \circ  D_{\mathcal{K}}

is isomorphism. Notice that the simple algebraic biduality of vector spaces becomes the statement that on Spec(k) the strukture sheaf k is dualising.

The dualising functors and complexes and the various duality theorems are related in a number of ways, both in this coherent sheaf set-up and in the other contexts mentioned above. First, the functor f! takes dualising complexes to dualising complexes. Next, the dualising functors switch between the functors f! and f* and between R f! and R f*. More precisely, let \mathcal{K} be a dualising complex on Y and \mathcal{L} = f^!\mathcal{K} a corresponding dualising complex on X. Denote by DY and DX the corresponding dualising functors. Then:

 D_Y\circ \mathbf{R}f_! \cong \mathbf{R}f_*\circ D_X

and

 f^! \circ D_Y \cong D_X \circ \mathbf{L}f^*.

For the coherent duality the first statement says that the derived direct image commutes with the duality functors, and is simply an application of the adjunction property expressing the general duality result (first displayed formula above). Similarly, taking into account the biduality property, one has the following representation of the functor f!:

 f^! = D_X\circ \mathbf{L}f^* \circ D_Y.

Hope this clarifies the picture. Hartshorne's Residues and Duality and Grothendieck's Exposé I in SGA 5 are good sources for further information in the coherent and étale settings, while Iversen's book treats the topological (locally compact spaces) case and several texts (e.g., Mebkhout or Björk) on D-modules cover the (somewhat more complicated) picture in that context.

And you're right, the article on coherent duality needs work. As do the ones on Verdier duality, and Poincaré duality and most other duality theorems. Stca74 (talk) 14:57, 14 December 2008 (UTC)

Thanks muchly. I think it might be good to more or less copy this to one of the said articles. Also, Arcfrk and PaulTanenbaum seem to be interested in duality, so perhaps we can write something together. Currently I'm on matrices, and I also have to resolve whether I'll try and push v.sp. to FA status, but the duality topic is indeed intriguing, and would (should we get there) be the first maths GA/FA article appealing to experts.
Is there a similar sheafish/category-minded characterisation of reflexive top.v.sp.? Jakob.scholbach (talk) 09:39, 15 December 2008 (UTC)

Measure (mathematics)[edit]

Hi Stca4!

I noticed your recent improvements to measure (mathematics) and am glad that someone has taken the time to improve the article! But I just wanted to confirm some dubious points in the lead:

a) It is written, "there are in general infinitely many different measures on a given set, each assigning different "sizes" for subsets". This seems incorrect because the empty set has only one measure on its power set (the measure of the empty set must be either 0 or infinity, depending on the conventions used). Maybe, there should be a discussion on the distinction between a "measure" and a "set function". I know you have noted this later on, but maybe it should be emphasized.

b) There is also the problem (which was there before you edited it) of the definition of a measure. In the first sentence of the lead, it is written that every subset has a measure. Without confusing the reader, there should be emphasis that the domain of the measure must be a sigma-algebra and not neccesarily be equal to the power set in question (or a sigma-ring depending on convention).

On the other hand, your lead is far better than the initial one because it gives the applications of measure theory to probability. P.S Could you please respond on the talk page of the article? --PST 10:22, 28 February 2009 (UTC)

Hi,
And thanks for the message. I'll reply briefly here to avoid having to copy your comments to the article talk page (to simplify, perhaps you could leave the comments on article talk page, and just leave a notification to user talk page to alert an editor?).
For the number of measures on the empty set, you are of course right. However, I feel the lead is not the best place to elaborate on (essentially trivial) technical exceptions such as this; I would not object to qualifying the "given set" by adding "non-empty", even though that comes in my mind close to being a bit pedantic.
As for the issue with "each" in the first sentence, you're obviously right again. I left it there after considering the options — given that the true state of affairs is revealed in the following paragraph, I preferred slight sloppiness to a convoluted sentence structure or introducing too many concepts in the first sentence of the lead. However, I did now add "suitable" to qualify the subsets on which a measure is defined. While I'm afraid this is not going to help the reader too much, at least the next paragraph clarifies the situation and the meaning of "suitable". This way someone reading only the first paragraph is not left with a technically incorrect claim.
The article is still quite seriously incomplete (I would classify it as Start class rather than B). In particular, relation between integral and measure should be developed. In addition a host of topics is still missing: discussion of Lebesgue-measurable sets (and non-measurable ones!), measurable functions, products of measures, outer measure, signed measures and Hahn decomposition, complex and more general vectorial measures, absolute value of a complex measure, key properties of bounded measures (including the norm), vague and other topologies on spaces of measures, support of a measure.
Br, Stca74 (talk) 16:59, 28 February 2009 (UTC)

Duality (mathematics)[edit]

Hi, I'm trying to gather some people working on duality article. Are you up to it? I'd like to develop the article to Good Article standard, but I think this is a broad topic so more hands/eyes would be good. Jakob.scholbach (talk) 16:47, 8 March 2009 (UTC)

Hi Jakob,
Thanks for the invitation to work on Duality, this is a worthwhile and interesting initiative. However, I'm afraid the time I can give this work short term is extremely limited, perhaps some random hours in the weekends. I'll see what I can do. Initial thought about the article is that while developing a comprehensive framework to discuss the various duals and dualities would be great, one could easily move into original research. This may make it necessary to make the article largely a summary, with wikilinks to individual dualities. I cannot think of many sources that would provide an overall framework covering all relevant dualities. Best, Stca74 (talk) 12:08, 9 March 2009 (UTC)