User talk:Silly rabbit/Archive 1

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I will keep in mind the lager offer. Oleg Alexandrov (talk) 03:31, 7 November 2005 (UTC)


 * I intend to honor that offer! Silly rabbit  kicks are for trigs.

Torsion, specifically Torsion (differential geometry)
I don't know if you do requests, but I've always wanted to gain a better understanding of torsion in differential geometry, and I suspect you might be able to write something interesting about that. Books on diff geom typically state "Let the torsion be zero. Then..." and that is the last one hears of it. I personally have never seen a good writeup. Once, long long ago, I even had a professor express confusion over the issue (although it was in the context of supersymmetry): he stated something like "... and in this gauge, the curvature can be made of vanish, but then the torsion is not zero; and so it is not clear if this is a physical theory..." Ever since then, I've wondered if or when one can trade torsion for curvature (was the profs statement only true for supersymmetry?), and what that might mean. I'm hoping you can help.


 * Your former professor's confusion over whether the theory was physical most likely resulted from the fact that torsion can only be specified in the presence of some additional data besides a (pseudo-)Riemannian metric (or a discrete symmetry, as with a spin structure).  I'm not much of a super-symmetrist myself.  But many (ten?) years ago I attended a lecture in which the speaker axiomatically characterized the Einstein equation from GR, and then hedged by saying (roughly):  "Of course, this only holds if the connection is torsion-free.  And there may or may not be a good physical reason for this, but the issue deserves to be investigated."  The reason for relativity theorists to reject the idea of connections with torsion is clear (see below): torsion usually depends on some preferred class of frames (or higher-order frames, jets).  I think the speaker was referring to some structure above and beyond the usual relativity principle.  In my opinion he was trying to say, in an obscure and carefully couched fashion, that the current theory was incomplete.  I don't know if this helps to vindicate your former professor. Silly rabbit 18:50, 8 November 2005 (UTC)


 * We do have an article Parallelizable manifold but it never mentions torsion. The article on jet bundles mutters something about holonomic coordinates, which I think are supposed to be torsion-free coordinates on a G-bundle, but I'm not sure ... Are there smooth homotopies that take one from coordinate systems on manifolds with torsion but no curvature to manifolds with curvature but no torsion? Are there obstructions (homologies) to this? As I said, I'm quite confused on this. linas 14:22, 8 November 2005 (UTC)


 * Assuming I understand you correctly: There are obstructions, but they probably all arise in the existence of a connection with torsion but no curvature.  As a first-guess approximation, in the Riemannian case I would say that the manifold must be parallelizable.  Silly rabbit 15:42, 8 November 2005 (UTC)

And, as Oleg says: Welcome! Glad your here! linas 14:03, 8 November 2005 (UTC)


 * How deep should I go, then? Torsion is a tricky thing.  Silly rabbit 15:23, 8 November 2005 (UTC)


 * Ok, let me try to explain at least some of the difficulties with torsion. Let's begin with the case of torsion on a (pseudo-) Riemannian manifold.  We know that it is possible to eliminate all torsion using the Levi-Civita connection.  But suppose instead that we wish to eliminate all curvature.  Locally, this is possible.  In a coordinate patch, pick a particular trivialization of the bundle of orthonormal frames, and simply declare the connection form to be the Maurer-Cartan form of O(n).  There will be lots of torsion, but no curvature.    Of course, this isn't of much use since the torsion has no nice properties (just like the "connection" we've chosen, it too depends on our preference of frame).


 * Well, sure, but I feel comfortable with the connection because I know how it transforms, and I know how to build various things from it. So the fact that the torsion is frame dependent is not a problem, per se. Perhaps I can build invariants from the torsion; I just don't know what these invariants are.


 * In (pseudo-)Riemannian geometry, the torsion is basically just frame-dependent garbage. Unless you have something else to pin it down with (for instance, a class of frames or sections of the frame bundle which is already preferred).  That's why the torsion is usually eliminated from the connection.


 * For other geometrical structures, there are various integrability conditions which can lead to the presence of intrinsic torsion. For example, consider a complex analytic manifold with a hermitian metric.  Here there is a natural tension between the data of the metric and the integrability condition of the manifold (Cauchy-Riemann equation).  The U(n) connection can still be put in a normal form, but this form will have some associated torsion.  In fact, a complex manifold admits a torsion-free U(n) connection if and only if it is Kähler.  (I believe this is due in large part to Nijenhuis.)


 * Ah! Well, there!


 * In general, torsion is the stuff you try to get rid of in Cartan's equivalence method. Of course, as in the case of complex hermitian manifolds, it isn't always possible to eliminate all of the torsion.  So, just as curvature is an invariant of Riemannian manifolds, there are geometric structures with torsion invariants as well. Silly rabbit 16:07, 8 November 2005 (UTC)


 * So none of this really answers the question: What is torsion?  Even in this more general context, answers like "stuff you can't get rid of", or even "just another invariant of the structure, like curvature" are unsatisfactory.  It is an unfortunate deficiency of mathematicians that we tend to overlook the forest for the trees.  I need some time to think about this.  Silly rabbit 16:15, 8 November 2005 (UTC)


 * Yes, thank you, that was a good reply. And there's no hurry or urgency nor is a personal reply even needed; its just that I noticed you were working on nearby articles, so I thought I'd ask. I'm not sure I know what my question is. Perhaps it is this: "when is torsion important, and when isn't it?", and you've partly answered it: its important for complex manifolds, and not important for Riemann surfaces. Seems like its important when Lie groups enter the picture (e.g. in fiber bundles) because these are in a sense naturally "parallelizable"; but I have no particular intuition.  I have a few questions which I am too embarrassed to ask since I suspect they're shallow and can be answered with only a minor bit of effort.


 * As to supersymmetry, I don't exactly understand it myself. It seems to be the result of a tension between the adjoint rep (for example, that thing that acts naturally on a tangent bundle) and the fact that adjoint reps are decomposable into sums involving the fundamental rep (which then tie to spinors in a strange related way). And so there's this soup of ingredients that promises a geometrical outcome, but I haven't quite made sense of it.  linas 02:57, 10 November 2005 (UTC)


 * For a truly complete answer of the question of torsion, one needs to incorporate Spencer cohomology. This is the cohomological data for a Lie group G that can yield a unique prolongation of G-structures on a manifold.  Spencer cohomology captures the ambiguity in selecting the torsion for prolongation equivalence problems.  Anyway, it's very hard stuff for intrinsic geometry, although it can always be described in local coordinates using jets.  The precise higher-order behavior is controlled by the curvature of the connection and its torsion.


 * Consider, for example, the first order differential equation for geodesics on a (pseudo-)Riemannian manifold. This is well-understood.  But what if we only have a conformal structure to work with?  In this case, then, there will be torsion in the first order because of the fact that the first-order connection involves some derivatives of an arbitrary function.  But this torsion is not intrinsic, since the connection itself changes if we change our representative of the conformal metric.  Nevertheless, it is possible  to absorb all of this torsion in higher-order.  This yields a connection on a higher-order G-structure: a Cartan connection.  Furthermore, the associated geodesic equations require a second-order initial condition, as opposed to a first-order condition (as is the case in a Riemannian manifold.)


 * The only conclusion I can make is that torsion is such a tricky thing that it can't possibly be explained to someone who isn't an expert.    I mean no offence, but ultimately it has no interpretation apart from that of D.C. Spencer.  Cheers, and sorry for dumping all of this on my talk page, Silly rabbit 20:15, 15 November 2005 (UTC)


 * No offense taken, and no apologies needed; by contrast, thanks. These questions have been a stone in my shoe for far too long; I'm slowly getting around to actually doing something about it.  Please don't delete the above, I'll have to read it a few more times before I am through.  linas 22:55, 15 November 2005 (UTC)

The trick to editing wikipedia sites is...
Indtrouce as mnany splelling erors as posible. Thenn you ar shoor too get attencion!

But seriously, I'm sorry about the simplectic versus symplectic. I was wrong, it was late at night (for me anyway), etc. I hope some good content comes out of my error by bringing more people to the page cotangent bundle.
 * --Silly rabbit, kicks are for trigs!

Char poly to trace
Not exactly the right place for such a question (although it does pertain to how the total Chern class and the Chern character can be expressed in terms of one another). Is there an article which might be contain a sort of determinant to trace formula? The formula is simple enough:
 * $$ \log \det(I-tA)=-\sum_{i=1}^\infty t^i\frac{\hbox{tr}(A^i)}{i}$$

(where A is a matrix with entries in a commutative rational algebra and blah, blah, blah...) I've had occassion to use this formula in the past (with Pontrjagin classes).

'Proof.' This follows from the identity for an n&times;n matrix A
 * $$\det(A-tI)=\sum_m (-t)^{n-m}\sum_\pi (-1)^{\epsilon(\pi)}\frac{tr(A)^{\pi_1}tr(A^2)^{\pi_2}\dots tr(A^m)^{\pi_m}}{\pi_1!\pi_2!\dots\pi_m!1^{\pi_1}\,2^{\pi_2}\dots m^{\pi_m}}$$

The inner sum extends over all partitions &pi;=(&pi;1,...,&pi;m) of m of the form &pi;1+2&pi;2+...+m&pi;m=m with each &pi;i a non-negative integer. The sign is determined by $$\epsilon(\pi)=\sum_{i=0}^{[m/2]}\pi_{2i}.$$ This can be proven combinatorially. ;-)

Alternatively, we can prove it using Cauchy's theorem. Let z and t be complex variables, and let f(z)=det (A-zI). Let w1,...,wn be the zeros of f(z). These zeros are just the eigenvalues of the matrix A. Let &gamma; be a contour enclosing all of the wi. We have
 * $$\frac{1}{2\pi i}\int_\gamma t^k z^k\frac{f'(z)}{f(z)}dz=t^k\sum_iw_i^k=t^k\hbox{tr}(A^k)$$

Summing over all k and employing the geometric series
 * $$\frac{1}{2\pi i}\int_\gamma \frac{1}{1-tz}\frac{f'(z)}{f(z)}dz=\sum_kt^k\hbox{tr}(A^k).$$

By Cauchy's theorem, the LHS is equal to
 * $$-\left.\frac{1}{t}\frac{d}{dz}\right|_{z=1/t}\log f(z)$$

from which the result follows after some manipulation.


 * Techniques like this are also used to study functional determinants. I should learn more about such things.

Silly rabbit 21:13, 13 November 2005 (UTC)

I haven't seen an article on WP that resembles this. Seems that th article on determinant doesn't link to such a thing. WP does have the marginally-related Jacobi's formula. The intersting but troublsome case is where A is infinite-dimensional, possibly trace-class or a nuclear operator when its not hermitian. WP also does not currently have an article for the resolvent formalism, which deals with 1/det (A-\lambda I) and related constructions (the resolvent formalism is often used in quantum mechanics and dynamical systems, and is often accompanied by physics-type hand-waving, because its often not clear if the proofs are valid for the operators in question.)

One WP search technique is to click on "what links here", which shows all the other articles that reference this one. It can be an interesting read sometimes. This works even if the target article hasn't yet been created! (i.e. it will reveal the articles with the "red links").

Rather than posing questions on your talk page, I can recommend posting questions on Wikipedia talk:WikiProject Mathematics, which is followed by many, and will elicit a broader reply. linas 23:18, 15 November 2005 (UTC)

Differentiation in Fréchet spaces
So what actually is the difference between Differentiation in Fréchet spaces and the Gâteaux derivative? According to the article you've written, they look identical. Are you sure that we don't have two names for the same thing (in which case, would it not be better to have only one article?). -lethe talk [ +] 01:25, 9 June 2006 (UTC)


 * There is no difference between the the definitions, except that the Gâteaux article defines them on locally convex topological vector spaces so it won't have very many nice properties. Over Fr\`{e}chet spaces, the chain rule holds.  Over a certain subclass, you get an inverse function theorem.  More on this later. Silly rabbit 03:39, 9 June 2006 (UTC)


 * Ah, I see. I'm still not convinced that it shouldn't all be in the same article, but I'll wait and see what you have up your sleeves. -lethe talk [ +] 04:03, 9 June 2006 (UTC)


 * Perhaps I should rewrite a bit of the intro to say something like: "The Gâteaux derivative has many happy properties on the category of Fréchet spaces..." I gather that most folks around here know about the Gâteaux derivative on comparatively nice spaces such as Hilbert spaces (via quantum field theory no doubt).  But the uglier spaces are important for variational calculus, embedding problems, and (in my own case) diffeomorphism groups.  So I'll give the Fréchet stuff my top priority, and I hope not to disappoint.  Thanks for the feedback, Silly rabbit 04:26, 9 June 2006 (UTC)

Hello?
Is this a "silly rabbit" with any connection to Boston Herald internet forums in the last 5 years?. Many thanks, Nesher 21:45, 14 June 2006 (UTC)


 * Sorry, no. It's good to know that I'm not the only sylvilagus sillius around, though. Must be a close cousin.  Silly rabbit 03:09, 15 June 2006 (UTC)

invitation
I'm greatly surprised you aren't yet listed on WikiProject Mathematics/Participants. Dmharvey 03:21, 18 June 2006 (UTC)

Cartan-Karlhede algorithm
You should write a review paper on this topic. Right now WP lacks needed background articles which might make it difficult to incorporate some of your suggestions, but a readable review paper would make an excellent source for the next revision. ---CH 20:06, 21 June 2006 (UTC)


 * Thanks. I've already added it to my todo list.  Unfortunately, this list seems to grow rather than shrink with time ;-). Silly rabbit

Infinite-dimensional holomorphy
Hi Silly rabit. Thank you for your edits to Infinite-dimensional holomorphy. By the way, do you have the time to add a section about trully infinite dimensional holomorphic function, that is, not only with values in a Banach space, but also defined on a Banach space (instead of the complex numbers)? In that case by the way weak holomorphy is not equivalent to strong holomrphy (Gateaux &ne; Frechet). I know too little about this topic to attempt to write myself. Oleg Alexandrov (talk) 22:21, 21 June 2006 (UTC)
 * Actually I added there the little I knew. You are welcome to take a look and expand on my blurb. Oleg Alexandrov (talk) 04:29, 22 June 2006 (UTC)


 * Thanks. I'll see what I can do with this article.  There are many different notions of holomorphy out there.  For one thing, a substantial part of the theory carries over to Frechet spaces (with the Gateaux derivative).  Although in the f:X->Y case, it may be best to stick with Banach spaces because the two theories do have substantial differences, and most people only seem to be interested in the Banach case anyway.  Silly rabbit 11:50, 22 June 2006 (UTC)

The state of connections
Hi all. I'm a bit concerned about the state of affairs in connection theory on Wikipedia. I've collected some of the pages here:
 * connection (mathematics) -- this is a good starting point, but perhaps could be turned into Category:Connection (mathematics)
 * connection form -- strangely enough, the article actually fails to define the connection form (in either the sense of Ehresmann or Cartan). Perhaps that's the source of some of the confusion in this discussion.
 * covariant derivative -- This article ought to be about the covariant derivative in any old vector bundle. Instead it focuses exclusively on affine connections.
 * gauge covariant derivative -- This article seems to be about what covariant derivative ought to be about (but isn't), perhaps with a few physical idiosyncracies which may justify its existence as a separate article.
 * Cartan connection -- I abstain from comment, since I have been trying to bring this up to par lately. See the talk page.
 * Levi-Civita connection -- A fine stub, but this definitely needs some expansion. At least a brief discussion of parallel transport and geodesics are warranted.
 * affine connection -- This stub fails to distinguish between the infinitesimal notion of a connection, and the local notion of parallel transport. I like the fact that it at least makes clear what is affine about the connection, but it fails to show how the affine-ness also admits an infinitesimal interpretation which distinguishes the connection from a garden-variety linear connection.
 * parallel transport -- Why does this only discuss the parallel transport of tangent vectors? A notion of parallel transport can be defined for any connection in any vector bundle along any C1 curve.  (More generally, it could talk about development in principal bundles too and thus tie in with holonomy.)
 * Riemannian connection -- The same thing as a Levi-Civita connection, but it's allowed to have torsion. Ok.
 * Christoffel symbols -- This is probably the best article in the entire group. (No mention of generalized Christoffel symbols or jet transformations, but you can't have everything.)
 * Weitzenbock connection

And here's a wish list: Silly rabbit 00:06, 27 June 2006 (UTC)
 * linear connection -- everything presently in covariant derivative actually belongs here. The Koszul connection associated to a linear connection is then precisely what "covariant derivative" means in the context of (pseudo)Riemannian geometry and relativity theory.
 * projective connection -- a nice example of a Cartan connection which is not an affine connection.
 * Koszul connection -- This is the general notion of a connection as a linear differential operator which has been so frequently reiterated in many of the articles here. Perhaps it deserves its own place?  (Or a redirect to covariant derivative once that has been totally rewritten.)
 * Ehresmann connection -- This is the notion of a connection form which connection form tries to capture.
 * Grothendieck connection -- Not something I know about, but connection (mathematics) suggests that it might be worth exploring.
 * Anything I can do to help? Maybe you should make a "to-do" list.  -lethe talk [ +] 03:27, 27 June 2006 (UTC)
 * Hi Lethe. I've started [Category:connection (mathematics)].  I've collected everything I could find which is of direct relevance to connections there.  I'm also trying to write an intro which is (1) agnostic to whatever the connection du jour happens to be, (2) reasonably accessible to someone who is seeing connections for the first time, and (3) gives a broad overview of the history of connections and the various relationships between the different kinds of connections out there.  I know that there is already a connection (mathematics) article.  I hope to merge some of that into the category page, and assimilate the rest of it elsewhere. Silly rabbit 05:11, 27 June 2006 (UTC)

Todo List
Here's a (non-exhaustive) to-do list
 * Categorize everything which is closely related to connection theory in [Category:Connection (mathematics)]. (I think I have mostly finished with this task.)
 * Write a stub for Koszul connection. This is just the "usual" idea of a connection in a vector bundle (i.e., $$\nabla_X$$.)
 * Make the other de facto references to a Koszul connection point back to Koszul when appropriate.
 * Fix connection form. Define the connection form as a vertical vector valued form at the outset.  Do it on a fibre bundle first, and then discuss the G-equivariance for a principal bundle separately.  (Otherwise there is simply too much going on in the definition to digest at once.)  The Vector bundles section should be more clearly delineated as different from the usual definition of the connection form.  A couple of things need to be fleshed out here.  First, the Koszul connection is indeed "invariant".  So I don't like all the hedging at the beginning of the section.  Second, the differential form which ultimately crops up is the "connection form" under consideration.  I might call this a gauge connection (which is, I believe, what some physicists call them).  Indicate that it comes from a connection form (in the sense defined earlier in that article) upon choosing a gauge.
 * Fix parallel transport. This article needs a complete, and highly detailed rewrite.  A topic as important as parallel transport shouldn't be marginalized like this.  Parallel transport is, afterall, one of (if not the) centerpiece of connection theory.  Specifically, (1) talk about the parallel transport along a Koszul connection, (2) parallel transport for a linear or affine connection and how this leads to geodesics, (3) parallel transport as a lifting of curves to a principal bundle, (4) some (albeit brief) mention of holonomy -- and maybe the relationship with curvature, (5) the more general notion of development along curves or families of curves (at this point, we enter various topics which are probably best treated elsewhere: monodromy, Cartan-Darboux theorem, and who-knows-what-else).
 * Someone figure out what the heck a Grothendieck connection is, please ;-)
 * Cartan connection already has an extensive to-do list.
 * Projective connection needs to be started. This is a good application of Cartan connection.

More to come later. Silly rabbit 05:11, 27 June 2006 (UTC)

Some discussions with Lethe

 * I am unfamiliar with the name Koszul, but your description makes it sound like the definition we have at covariant derivative is the Koszul connection. Perhaps a redirect instead of a stub?  Clean-up and point changing still needed, I suppose.  Similarly, Ehresmann connection should probably redirect to connection form. -lethe talk [ +] 05:28, 27 June 2006 (UTC)
 * A covariant derivative (defined as it is in covariant derivative) is a linear connection. A Koszul connection, named after Jean-Louis Koszul, is a connection on any vector bundle (not just one associated to the bundle of linear frames for the manifold).  So a redirect certainly won't do the trick.  Furthermore, even the Koszul version of the covariant derivative (with respect to a linear connection) is different from the classical one.  Classically, one defines the connection in terms of some generalized Christoffel symbols &Gamma; (which may be defined for a general linear connection by postulating their transformation law).  This is unwieldy, as anyone who has ever tried to work with Christoffel symbols knows only too well.  Koszul provided the required algebraic framework for describing the &nabla; algebraically.  Most modern approaches to covariant differentiation define the covariant derivative as a type of Koszul connection.  Silly rabbit 06:17, 27 June 2006 (UTC)
 * Then again, I may be just arguing for its own sake here. The two ideas are compatible enough that, with a little effort, they can easily be merged.  I do worry a little that the more general Koszul connection is a lot to hit first-time readers with.  I could say something like: "More generally, covariant derivatives also make sense on general vector bundles" in the intro.  But I really hate articles that do that sort of thing.  Silly rabbit 07:23, 27 June 2006 (UTC)
 * I agree with the Ehresmann connection redirect to connection form. There is the possibility of confusion with a Cartan connection, but that (remote) possibility can probably be handled easily. Silly rabbit 06:30, 27 June 2006 (UTC)
 * Gauss-Manin connection, I've added it to Category:Connection (mathematics). -lethe talk [ +] 15:24, 27 June 2006 (UTC)
 * Thanks. I think I saw that yesterday, but managed to take away the mistaken impression that it wasn't really a connection.  (It was late, I was tired.  I can't be bothered to read the first sentence of the article ;-D.)  Anyway, yes, thanks. Silly rabbit 15:28, 27 June 2006 (UTC)

Torsion redux
I started reading the various articles on connections. I note you added Koszul connection yesterday, including the statement:


 * There exists a local trivialization of the bundle E with a basis of parallel sections if, and only if, the curvature vanishes identically.

Although you linked "parallel" to parallel transport, I mentally linked it to parallelizable manifold, which is perhaps closer to the intended meaning?

I may be able to explain what I found confusing about torsion. It's well known that $$S^3=SU(2)$$ is parallelizable. That means that there exists a connection on $$S^3$$ for which the curvature vanishes, right?. However, this connection has a non-zero torsion everywhere, right? But, of course, $$S^3$$ also has a connection whose curvature is that of a sphere; this curvature is non-zero everywhere, but the torsion vanishes identically everywhere (its the Levi-Civita connection).

So ... are these two connections, one flat, and one not, "isolated points", or do there exist a continuum of connections interpolating between the two, having a little bit of curvature and a little bit of torsion? If there's a continuum, is there a canonical path through it (i.e. the path where each connection on the path has curvature distributed uniformly, and torsion distributed uniformly)? If there's a continuum of connections between these two, is this space of connections simply connected? Or is the structure more complex? What is the "shape" of the space of all possible connections? If there are homotopically inequivalent paths in the space of connections that takes one from $$S^3$$ to parallelized $$S^3$$, then one might argue that, in this sense, the parallelization isn't unique.

Since all bundles are locally parallelizable (as far as I know; its part of the definition of charts and atlases for a bundle, right?) does this mean that the geometry of the space of connections is trivial if and only if the bundle is trivial? In particular, one should be able to read the homotopy groups of one straight off from the other, right? Or, perhaps, this is more complex: did I just make the Poincaré conjecture generalized to parallelizable manifolds?

I am an amateur at geometry, and infrequently, at that. I suspect I could answer my own questions if I retreated and ground out some calculations, read a few books. However, being both lazy and social, I thought I'd pose them here. linas 15:02, 28 June 2006 (UTC)


 * Well, I re-read our earlier conversation on this topic. Perhaps I should shut up and just do the calculations. I probably just can't get farther without investing some hard work. linas 15:14, 28 June 2006 (UTC)


 * The confusion has to do with the two uses of the term parallel (both of which are, rather unfortunately, used in differential geometry). I meant parallel in the sense of parallel transport: that is, sections which are utterly annihilated by the connection.  To be sure, a vector bundle will always be (locally) parallelizable in the topological sense by just giving a basis of local sections.  But requiring these sections to be parallel with respect to the connection is a further constraint which will not be satisfied in general.  Stated another way, a local parallelization is just a local section of the frame bundle.  If we have a connection (on the frame bundle), then a basis of parallel sections of the tangent bundle defines a horizontal section of the frame bundle.  So parallel for connections might roughly be equated with horizontal.  (In fact, I might just do that in the article Koszul connection...)


 * To address your question about the shape of the space of all connections, it may get a little hairy depending on the particular "brand du jour" of the connection, the gauge group under considerations, and other sundry details.


 * For principal connections, i.e. Ehresmann connections, the space of connections admits parititions of unity over the base manifold and the space of connections over a fibre is convex. See the Chern-Weil homomorphism for at least one important topological consequence of these facts.  Furthermore, a principal bundle admits a flat connection if and only if it is equivalent to a principal bundle whose transition functions are constant.  This certainly implies some topological constraints on the bundle, but it is unclear (to me) what they are.


 * For Cartan connections, things get a little clearer. Basically, a connected manifold admits a complete flat Cartan connection if and only if it is (diffeomorphic to) the quotient of a homogeneous space by a discrete group.


 * Regards, Silly rabbit 15:59, 28 June 2006 (UTC)


 * Oh, I guess I didn't really answer your questions. I need to think about it. Sorry,  Silly rabbit 16:05, 28 June 2006 (UTC)


 * Well at least you can glean one thing from my response: the space of connections is convex.  So indeed, SU(2) will admit a continuous, even a linear, family of connections interpolating between the two.  (Just take t&nabla;1 + (1-t)&nabla;2.) Silly rabbit 16:11, 28 June 2006 (UTC)


 * Oh, another thing. The Maurer-Cartan connection on SU(2) is flat and torsion-free.  Silly rabbit 16:44, 28 June 2006 (UTC)
 * That didn't make a whole lot of sense, did it? Ok, I clearly need a break. Silly rabbit 19:02, 28 June 2006 (UTC)


 * I added something equally untrue to an article on my first days at WP, and boy, did I ever get walloped for it!. Thanks, though, this was helpful; I think I finally understand my question, whih is a big step. linas 23:34, 29 June 2006 (UTC)

More on holomorphy
Hi Silly Rabit. Thank you for working on infinite-dimensional holomorphy. I have a question. I think sections 3 and 4

3 Holomorphic functions defined on a Banach space 4 Weak holomorphy

need to be deleted or integrated I think as that stuff was already discussed in the sections above. Wonder what you think. Thanks. Oleg Alexandrov (talk) 19:35, 30 June 2006 (UTC)


 * They definitely need to go at some point. I'm not quite done fleshing some of the stuff out, but I've been engrossed with a major overhaul over at Category:connection (mathematics).  I'll get back to it now and then.  In the mean time, I'll see what I can do about trying to merge in the older material.  Best regards, Silly rabbit 19:46, 30 June 2006 (UTC)

better off
We'd probably be better of with you here. At any rate, I intend to sit down and figure out a few things about Fréchet and Gâteaux derivatives, and I need you around for that, so you can't leave! :-) But I can certainly understand and appreciate how edit-warring and arguing can get you frustrated. -lethe talk [ +] 01:57, 4 July 2006 (UTC)

a good edit
I'm usually loathe to remove content from articles. It's someone else's good faith work that we're supposed to respect. But sometimes, it just has to go. This was a good edit. The Lie derivative is not a generalization of the exterior derivative (nor the other way around), and there's pretty much no way I can see to save that text to make it right. So good work, sir, and I salute you. -lethe talk [ +] 20:19, 5 July 2006 (UTC)

Infinite-dimensional holomorphy
Hi. It is me again. I wonder if you could work more on that article. There is good stuff in there now, but it does not yet feel coherent. Would be nice to integrate the bottom two sections too. Thanks. Oleg Alexandrov (talk) 17:16, 16 July 2006 (UTC)
 * Anybody home? :) Oleg Alexandrov (talk) 04:14, 9 August 2006 (UTC)

Today's featured article
Tobacman 00:37, 25 October 2006 (UTC)

Spinor
Glad to see you're having a wrestle with the spinor article.

Can I request something else in the first line rather than "spinors are ... similar to spatial vectors".

Spinors are *not* like vectors; presenting them as being like vectors is putting people on the road to horrible confusion, right at square 1. I can't see that there is any usefulness in the comparison. All the things that are meaningful about spinors are things that vectors can't do.

I'd be much happier with "even graded members of a Clifford algebra built over a vector space", even if that leaves quite a lot for the article to then explain, because at least (i) it is a definition, and (ii) it isn't actually misleading.

Jheald 18:48, 17 April 2007 (UTC)


 * I see where you're coming from, but I think we both agree that (with the difficulties in saying what a spinor is) the first paragraph needs to draw some kind of analogy in order to be at all meaningful to the entire pool of readers. I think the most (if not the only) "easily" accessible thing about spinors is their peculiar behavior under rotations.  In fact, historically this is how they first came about, as projective reps of SO(n).


 * Yes, that's a lot better. Thanks for fixing it!  Silly rabbit 19:15, 17 April 2007 (UTC)

I don't know if there's anything from the opening paragraphs of this version that might be useful. It got rather annihilated by a sequence of 3 anon editors last week, but I think it is useful to feature the iconic spinor "sandwich" as a way of doing rotations earlier rather than later.

I hope I've fixed the link in Spinor to where you were trying to get it to go.

But I'm a little puzzled by the statement "The action of the Clifford algebra on S is complicated and unnatural, and so the rotational properties of S are obscure"; possibly because I am more familiar with spinors in real Clifford algebras rather than complex ones. For the real Clifford algebra it seems rather natural that the spinors are the even half of the algebra (corresponding to rotations rather than reflections), and to follow how these elements of the algebra themselves transform under rotations. I'm not sure I follow, as to which of the rotational properties are meant, that are supposed to be obscured ?

But is that because I'm thinking of a different explicit construction for the spinors, and not thinking of them as living on S ?

Jheald 23:10, 17 April 2007 (UTC)


 * The spin group lies in the even part of the Clifford algebra. I'm not aware of a way to find the spinors themselves lying in it though.  Perhaps what you're thinking of is the fact that the even Clifford algebra can be split into the single (in odd dimensions, complex) or a pair (in even dim, complex) of complete matrix algebras.  In the real case, more reductions are possible and I'm no expert on real spinors, but I've never seen a result of the sort you're thinking of.  Anyway, I think it's important, though, to draw a distinction between the spinors themselves and the representations (endomorphism rings).  Silly rabbit 01:27, 18 April 2007 (UTC)


 * Oh, the link. Thanks.  No, that wasn't what I was trying to link to.  Wikipedia doesn't allow for nested namespaces apparently, so I can't link to #Explicit constructions below.  Silly rabbit 01:31, 18 April 2007 (UTC)

Aha. No, but it does let you link to Spinor, which is what you wanted ?

Now, "the distinction between the spinors themselves and the representations (endomorphism rings)". Maybe this is where I'm not yet on the right page. Surely the most fundamental objects are the even-graded elements of the Clifford algebra of the space, considered in purely algebraic terms, regardless of specifying any specific method of possible numerical representation. These are surely the spinors in the most general sense? The column spinors are an (in general unfaithful) representation of these elements, which may or may not distinguish the effects of particular symmetry operations; but in general, surely, the column representations are secondary objects compared to the representation-free elements of the algebra themselves. One can also construct matrix-valued representations, which are more flexible than column-valued ones; but similarly, these objects in specific representations are surely secondary entities compared to the more general more fundamental representation-free objects, which they may or may not faithfully represent.

In a nutshell: I think spinors aren't just column spinors. I think column spinors are just one type of representation of the more general object.

Would you agree? Jheald 08:21, 18 April 2007 (UTC)

Action on isotropic subspaces
I started a new section: partly to sandbox some of these ideas about the action of the Clifford algebra on S=$$\bigwedge^\cdot W$$ where W is a maximal isotropic subspace of V.

In even dimensions, there is a splitting V=W1&oplus;W2 is a direct sum of two copies of W. Let iv denote the interior product operation, and &epsilon;v the exterior product. For v &isin; V, we write v = w1 + w2 be the splitting corresponding to the direct sum decomposition of V. Then
 * $$\rho:v \mapsto \epsilon_{w_1} + i_{w_2}$$

defines a linear map from V into the endomorphism ring of S. Finally, note that &rho;(v)2 = g(v,v). So, since &rho; preserves the Clifford relations, it descends to the full Clifford algebra.

In the odd case, there is a supplementary subspace U which is not isotropic and is orthogonal to W. Then there is a direct sum decomposition V=U&oplus;W1&oplus;W2. Using &rho; as in the even case, but with &rho;(u) = +1 on $$\bigwedge^{even}W$$ and =-1 on $$\bigwedge^{odd}W$$ for a unit vector u &isin; U.

Spin matrix
Are you sure "spin matrix" is right? What if you're not using a matrix to represent these objects? Or considering the objects algebraically, without any representation. Typically the matrix representations aren't sparse, so it's more efficient, eg in a computer, to represent these objects as sums of the base elements of the Clifford algebra, and to do multiplication by hard-coding the way those base elements combine - the way one would do for complex numbers - without any matrix being invoked.

So I'm not sure calling these spin matrices is helpful. Surely (the point I made a couple of sections above), these are the spinors, as abstract entities. Matrices are just one way to represent them. Columns are just another.

No? Jheald 00:04, 19 April 2007 (UTC)


 * The spinors are the things acted upon by the even Clifford elements. In physics, traditionally these elements are called spin matrices, although I admit that the terminology is less than ideal in situations where you don't want to fix a basis (or represent them abstractly in terms of the Clifford relations, as done in the article).  But something must be done to distinguish them from the proper spinors that the article addresses.  Silly rabbit 00:09, 19 April 2007 (UTC)


 * No. Spinors can be acted on by any Clifford elements.  Spinors are represented by even Clifford elements.  It's a matter of choice as to whether you want to think of the elements acting on the spinors, or the spinors acting on the elements.  Jheald 01:23, 19 April 2007 (UTC)


 * Yes, you're quite right. Thanks for pointing it out.  But still, spinors cannot themselves be identified with Clifford elements.  Silly rabbit 01:28, 19 April 2007 (UTC)


 * Ok, I fixed it (more or less). The locution "even-graded element" is a bit awkward.  But I have never seen a standard name for these.  Silly rabbit 01:55, 19 April 2007 (UTC)

User talk:Silly Rabbit\Sandbox\Affine connection
Please don't create subpages in the mainspace. If you want to test, edit over a longer period, or whatever you want a sandbox for, do so on either projectspace or in your user space. I have created the above page as a copy and deleted the article sandbox. This has by the way nothing to do with the contents of the page, which I haven't judged but which looked to be serious contributions. So there is no objection to add this to the main article page once you are finished polishing or expanding them. Fram 09:42, 23 April 2007 (UTC)


 * Thanks for letting me know, and for moving the page. Silly rabbit 10:44, 23 April 2007 (UTC)

Spinors
Hello, I apologize if my comments in the Spinor talk page came across as antagonistic. I can assure you that I assumed good faith in all your edits, and I definitely respect your putting time and effort in improving the article. Moreover, I didn't really mean that you have to revert to the old version of the article (as it appears on the cursory look you have done), merely that the structure of that version seemed superior. You don't have to agree with me on that, and the reason I wrote my long critique was (strange as it may sound to you) to avoid confrontations over editing the article, as you clearly have a much better grasp of it as nearly anyone else, and are best suited to modify it appropriately, or state the reasons why certain things should be as they are! In hindsight, I should have just written to you directly on your talk page, but I was a bit frustrated with a seeming loss of structure, and maybe wanted to state some things for the record. Again, I am sorry if I offended you.

I wasn't aware that Charles moved out some of your earlier contributions, but I was and am of the opinion that his edits raised the article from an obscure state ca March 2007 to a decent, even good one. Yes, there were many things stated incorrectly even after his edits, yes, especially concerning the difference between spinors and the elements of Clifford algebra. It is far from obvious (to me, at least) how to introduce spinors in a meaningful way as to serve both mathematicians and physists, who may both need them. For mathematicians, some statement along the lines that the even Clifford algebra is almost simple and contains orthogonal group, hence has a spinor (or half-spinor) representation may be more preferable. Physists may want to see more concrete realization via matrices, especially for the SO(3) and SO(1,3) cases. Both approaches have their pitfalls and are not quite compatible. Ideally, I would like to see the article that is simple enough (in the sense of the absense of technical jargon) that both can use, and that more advanced facts can be explained. For example, the parallel between the Clifford algebra (CAR)/spinors and spinor representation of the orthogonal group and the Heisenberg algebra (CCR)/oscillator representation of the symplectic group; or the Dirac operator (mathematically). I am sure that you have good ideas on how to do it, and I encourage you to try them! Cheers, Arcfrk 06:03, 28 April 2007 (UTC)

Coffee cup
Congratulations, you made the CuisineLog (see Orientation entanglement, links to). Charles Matthews 17:02, 29 April 2007 (UTC)


 * A dubious distinction, no doubt. But entertaining, at least. Silly rabbit 17:08, 29 April 2007 (UTC)


 * Does wikipedia discriminate against coffee cups? Arcfrk 06:52, 30 April 2007 (UTC)

Spinors and Clifford algebras, idempotents, and left ideals
SR, if it's all the same to you, I think I'd prefer the Spinors and Clifford algebra section not to be reverted completely to how it was before. I think it's quite useful, having introduced the Clifford algebras, to explain where spinors fit into them -- namely as a subset of the whole set of elements of the algebra, a minimal left ideal that the algebra is projected onto by an idempotent. I think this is quite a standard notion -- for instance it's the way Lounesto goes about identifying the spinor spaces of real valued Clifford algebras ("Clifford Algebras and Spinors", 2nd ed, C.U.P., 2001, page 226); on a cursory glance it also appears to mesh in with Wikipedia's page on Modular representation theory.

I think it serves also as a useful bridge between the Clifford algebra approach and the traditional column spinors. By identifying the matrices corresponding to the idempotents in the Mat(k,C) representation, it confirms how right multiplication by them projects the whole representation onto a single column, linking to column spinors. But it also (correctly) suggests that the spinors do not have to be representated as complex column vectors - other representations may also be possible; or, more abstractly, simply to think of spinors as particular types of elements of the Clifford algebra, without necessarily specifying any particular array-like representation.

It's also nice I think to see how the various orthogonal primitive idempotents f1, f2 ... fn sum to the scalar 1, so Cln(C) {f1 + f2 ... fn} = Cln 1 = Cln -- ie why the direct sum of n spinors is isomorphic to the whole Clifford algebra.

The relevant idempotents can have quite a geometrical significance too. For example in the space-time algebra Cl1,3(R), the idempotent (1/sqrt 2)(1+e0) is, as a rotation, associated with Lorentz boosts onto the light cone.

I need to think some more about the significance of isotropic subspaces, and about whether or not you are correct to assert that "The space of spinors Δ can be identified with a maximal anticommutative graded subalgebra of the Clifford algebra." ((I'm not sure, on the face of it, that this looks right -- perhaps you can explain why it should be so?))

But I utterly disagree that the Clifford algebra approach to spinors is a "kludge". On the contrary, by giving a more abstract, general, representation-independent geometrical view of things, I think it is actually rather more fundamental, and if anything the column representation of spinors is the "kludge", or at least the happy accident. To the extent that the "explicit construction" works, its effect should be representation-independent, so it should be possible to understand how it is achieving what it is achieving purely in terms of elements and operations in the Clifford algebra -- and that gives a useful, more general viewpoint.

So, summary: I think there is more in what I added to those paragraphs that is useful and should be preserved, rather than just reverting the whole thing back. Jheald 22:29, 5 May 2007 (UTC)


 * A correction. I got the relevant idempotent wrong for Cl1,3 above.  A primitive idempotent for C &otimes; Cl1,3 is f = ½(1+γ0)½(1+iγ1γ2).


 * This leads to a spinor space of elements of the subalgebra spanned by f, -γ13f, -γ03f,  -γ01f.  (Lounesto p.138), where γ12 is a shorthand notation for γ1γ2. The idempotent f can be recognised as the product of idempotents corresponding to the energy operator, and the spin projection operator, respectively (according to Lounesto).


 * A primitive idempotent for C &otimes; Cl3,0 is f = ½(1+e3), leading to a space spanned by f, e2f, -e1f, e1e2f which corresponds to the space of Pauli spinors (ie is isomorphic to C2 (Lounesto p.60).


 * I'm not sure I understand what that all signifies interpretatively yet; but I may have been over-quick to identify it with your "explicit construction". However according to Lounesto at least, these are the parts of the Clifford algebras that the Pauli and 1+3D Dirac spinor spaces correspond to. Jheald 13:02, 7 May 2007 (UTC)

Redirect of HFGW
I was bold and was combatting the POV pushing of a certain notorious pseudoscience editor. The discussion is clear from the pages that there is nothing worth salvaging from the article. --ScienceApologist 13:50, 11 May 2007 (UTC)


 * I have the page on my watchlist, and if we can get a few more people to watch it, we should be able to combat any further problems. --ScienceApologist 13:54, 11 May 2007 (UTC)

Expand? abstract index notation
Hey Rabbit, that was a very quick response, Since this is not merge worthy(I am on a ratings tear and didn't even bother to double check to see if they were compatible and didn't sign my comment). Could you or someone else qualified give the article some meat, thanks--Cronholm144 01:37, 13 May 2007 (UTC)

Quick work, I have upgraded the article status, feel free to delete my merge suggestion--Cronholm144 04:34, 13 May 2007 (UTC)

Mathematics CotW
Hey S Rabbit, I am the one who miscategorized The Inverse Galois Group, It was late and I didn't look at it closely enough, Mea culpa. Anyway that is not why I am here. 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 00:12, 14 May 2007 (UTC)

Editing My Comments? What?
Nah, just kidding. Thanks for the fix and the link on the Null (SQL) talk page. SqlPac 19:16, 18 May 2007 (UTC)

Reverts of edits by 81.84.238.135
Hello, Silly Rabbit ... BTW, may I call you, "Thilly Wabbit"? :-)

I was making some edits to and noticed that you had done a revert of the link added by  to the site www.ChessBook.net as "link spam to online blitz games" ... I don't see how you could call it spam, as it provides exactly the same kind of "Chess Openings" information as the link to  that appears immediately above it on that page and on Queen's Gambit, to name just one other.

Just take a look at any article that links to ChessGames.com, follow the link in that article to the site, then press either the "SEARCH" or "KIBITZ" buttons at the top of the ChessGames.com page, and BANG! ... "Ads by Google.com" for travel insurance, chess supply retailers, and even their own online forum ... there is no such advertising or self-promotion on the ChessBook.net site, nor any links for "online blitz games" ... just a "Contact" link for email inquiries.

Anywho, I don't want to get into a revert-war with you, but I felt compelled to explain why I went back and restored all 13 of 81.84.238.135's edits that you had summarily reverted ... after changing "External links" to "External link" (singular) in those cases where they had created a new section, and omitting a rather POV "Summary" section they had added to one of them. :-)

It must have been the nugget's ommision of edit summaries and the "Typical moves in online chess games" as commentary in the external links that made you suspect that they were spam ... apparently, the information is culled from a data base of games played online at Some Other Site(s), but that is not what I inferred from my initial reading of their additions, either.

Upon closer examination, I found the ChessBook.net pages to be very useful (and more informative) as a statistical reference for reply moves to the basic positions, and a welcome companion to the ChessGames.com pages ... for example, compare the respective pages for the Queen's Gambit:


 * ChessBook.net
 * ChessGames.com

Please note also that the ChessGames.com page has a link to a Javascript chess program, which might very well divert a reader's attention from the task that brought them to Wikipedia in the first place, while the ChessBook.net site has no such distractions ... and FYI, I had never heard of either site before today's encounter!

Happy Editing! &mdash; 21:20, 23 May 2007 (UTC)


 * I know you don't care all that much about this, but I just wanted to let you know that I was the "other editor" making the reverts, and I've explained my reasons here. youngvalter 04:34, 24 May 2007 (UTC)

Spinor discussion
I saw that on the article "spinor in three dimensions" you wrote: "given a unit vector in 3 dimensions, for example (a,b,c), one takes a dot product with the Pauli spin matrices to obtain a spin matrix for spin in the direction of the unit vector."

Could you explain to a would-be physicist like me what "a spin matrix in the direction of the unit vector" is? How can one imagine it? (Physical examples from the real world are welcome...)


 * The article has become a bit fractured of late. The article quaternions and spatial rotation should provide some hints, and it may be worth linking that in the spinors in three dimensions article. Silly rabbit 12:55, 24 May 2007 (UTC)

Re: Calculus
I agree with your suggestion, so I went ahead and added Liu Hui and Zu Chongzhi in the History of calculus article. I might also mention them in the main Calculus article. Jagged 85 11:18, 24 May 2007 (UTC)

Vector
Not very comfortable with your most recent edit to vector. Article as it stood had a lot of faults, but was pitched at a fairly consistent level, introducing things gently idea by idea, with a progression, and at a relatively simple level of language and sophistication, well gauged for schoolchildren meeting the concept for the first time.

I worry that what you've added breaks this gentle progression, being rather more sophisticated and advanced much earlier in the article. But for the moment let me hold off editing till I see where you're going with it. Jheald 07:51, 26 May 2007 (UTC)


 * Re your latest edit, I think I would do "length" before "addition and multiplication", as the article previously was; and separately from "dot product", which I would hold over to products of vectors, with cross product and (perhaps) wedge product. Yes I know that length requires a metric, or at least a norm, that addition doesn't, and that vector spaces don't necessarily have.   But against that, consider that we're introducing vectors in simplest terms as objects with a magnitude and a direction.   It's useful to do the magnitude first, right up near the top of the article; and the multiplication (scalar product) much later.   AFAIR, it was probably sever years after I knew about the magnitude of a vector, and of matrix transformations of vectors, before I first heard of the scalar product.  (Though some time has passed!)  Jheald 15:16, 26 May 2007 (UTC)

Connection form: Torsion
Hi Silly rabbit. Thanks for the catch on torsion; I corrected the discussion at Connection_form to specify that it's only defined on the tangent bundle, or in the presence of a solder form.

From reading the above, torsion seems very subtle. I just thought that parallel transport gave you an affine map between fibers, and that curvature and torsion were the infinitesimal linear and displacement parts of this (and that this makes sense for arbitrary affine bundles).

And agreed, the connection form article certainly needs rewriting; I can see that you (y'all?) have been working on this section for some time. I doubt I can help -- as you can tell, this isn't my expertise. Nbarth 23:52, 27 May 2007 (UTC)

Codazzi-mainardi article
Hi rabbit, No need to be concerned about stepping on toes. I am a student so I yield to the real mathematicians when it comes to decisions like these. Jhausauer 02:01, 28 May 2007 (UTC)

Warnings after a "final" warning?
Just curious: why you warned a vandal after he or she had already been issued a "final" warning? Why not just go straight to AIV? It seems to undermine the entire idea of warning editors if we continue to just warn and warn, particularly after "final" warnings. He or she hasn't edited again so it's not a big deal. --ElKevbo 01:24, 30 May 2007 (UTC)


 * Was it a final warning? That wasn't clear to me. Silly rabbit 01:31, 30 May 2007 (UTC)
 * Yeah, it was. In general, the user warning templates with the stop sign are "final" warnings.  It's not a big deal.  Happy editing!  --ElKevbo 01:53, 30 May 2007 (UTC)


 * Thanks for the clarification. Silly rabbit 01:56, 30 May 2007 (UTC)

Silly rabbit
Trix are for kids! theanphibian 07:52, 30 May 2007 (UTC)

Image (mathematics)
Hi SR,

I agree that the notion is extremely important, but does the article inherit that importance? It seems to me that there's just not that much to say in the article. There are a lot of these articles whose main function is to be able to link to them so that we don't have to explain terminology inline. I tend to think that their maximum importance should be "mid", just because they'll never be very interesting as standalone articles, irrespective of how many places the concept is used. --Trovatore 22:00, 30 May 2007 (UTC)


 * I see your viewpoint, but at the time I was thinking of the isomorphism theorems which, as some of the most vital and pervasive theorems in mathematics, are certainly deserving of "High" stature. Bearing in mind your objections, I will downrate image to mid.  But I don't agree that it's impossible to write an interesting article about images.  Has anyone tried?   Silly rabbit 00:11, 31 May 2007 (UTC)
 * Well, I haven't. Go for it. --Trovatore 02:33, 31 May 2007 (UTC)

Frenet-Serret
Kudos on your work on the Frenet-Serret formulas. It has become quite a respectable article. Jhausauer 22:18, 30 May 2007 (UTC)

Affine connection...
...desperately needed an image like that! Nice one! And thanks also for the other improvements you have been making to this article, and other articles in the connections category. I have been watching with interest! Geometry guy 01:15, 2 June 2007 (UTC)

PS. I also appreciated your supportive remarks at WT:WPM. Good luck with the FAQ ;)

Your reply at Wikipedia talk:WikiProject Mathematics
Hello, one importance upgrade that you have made which illustrates my statement is. I can explain it in more detail for this particular instance, but if you at all followed my argument, I believe that it's rather pointless to discuss individual cases of ratings since we have not agreed on the overall criteria, and everyone comes with their own set of them. (And moreover, the ratings are a means to an end, not an end in themselves.)

Please, take trouble not to twist around the meaning of my words or ascribe your opinions to me, so that you can make an offended posture (as you have also done in the past). Reread the actual text of my comment.
 * I did not say that anyone discussed your rating changes. We discussed rating scheme, and did not arrive at a definite conclusion, which I lament.
 * I did say that we had a broad agreement that only few articles be rated top importance, and a limited number be rated high importance. Here is the discussion.
 * I did not object to people adjusting the ratings. On the other hand, it would have been preferable, in my opinion, to start by formulating the criteria, reaching a broad agreement, and then going through with rating the articles.
 * Moreover, in earlier comments on the rating project I pointed out that the ratings are likely to be both subjective and contentious. Taking the accusatory/offended tone, as you did, is not helpful.

Best, Arcfrk 02:29, 2 June 2007 (UTC)

Don't undo everything!! Your edits are helpful. This is a collaborative process. Undoing them all doesn't help anything. --Cronholm144 02:53, 2 June 2007 (UTC)

Of course, you should act as you see fit. If the rating project provides such an ample source of frustration, then by all means, do what is best for your health. I did not single you out and 'ridicule' you, however, it is clear that you do not want to listen, so I have erased a detailed, rational reply to your childish message on my talk page. Knock yourself out. Arcfrk 03:23, 2 June 2007 (UTC)

The R word again
Hi there - I took the liberty of going over your contributions to maths ratings. I actually agreed with nearly all of your upratings, but, in the spirit of simulated annealing, I didn't uprate all the ones I agreed with! Since then, as I think you noticed, I've been working on WikiProject Mathematics/Wikipedia 1.0/Importance (comments welcome). This can now be transcluded onto another page, and only the main importance table will be included. I've done this at WikiProject Mathematics/Wikipedia 1.0 for example. Geometry guy 17:41, 2 June 2007 (UTC)

Descriptive or prescriptive?
I am examining with great interest and optimism your progress on WikiProject Mathematics/Wikipedia 1.0/Importance. Another way to describe importance, which you seem to have overlooked, is as a family resemblance concept (as Ludwig Wittgenstein would put it). As mathematicians, we like to pin things down precisely; perhaps too precisely for something as contentious as a nebulous idea of importance. That's why I linked Top, High, Mid, and Low to their respective categories in the table on the page. It seems that one very effective way to determine the importance of an article is to examine it in relation to other articles which are of potentially equal importance, and such an approach is likely to appeal to naive editors (such as myself).

However, this method is not without its pitfalls, as I just discovered in my disastrous recent debut into the math ratings community. For instance, if group (mathematics) and commutative ring are both Top, then clearly ring (mathematics) must also be Top. It's a slippery slope from then on, though. (Kernel and image to High, for instance, etc.) Reflecting on my reasons for upgrading some of these "fundamental ideas" in the light of your descriptive classification of importance, I still see no reason that group/ring/commutative ring should be categorized as any less than Top. So, given the opportunity, I would likely make the same errors in judgement all over again.

I propose that instead of attempting to describe the categories of the importance scale in detail, that we should be attempting to create a prescription for how importance is assigned. The importance ratings can be roughly described as follows:

1. Top


 * Description. The major fields of mathematics, and ideas of mathematics without which mathematics would fail to exist as we know it today.


 * Prescription. If you could save just ten books from the burning library of Alexandria, what would they be? Be sure to check that no other editor has saved a book treating the same or similar subject.


 * Examples. Calculus, mathematical analysis, abstract algebra, Fundamental theorem of calculus, function (mathematics), addition, multiplication, etc.


 * Bad examples. probability density function, probability space &mdash; since someone else already saved the book on probability and probability theory. fraction (mathematics) &mdash; superceded by arithmetic.

2. High


 * Description. The most significant ideas and most sweeping theorems of mathematics.


 * Prescription. If you could only save one hundred mathematical concepts, which ones would they be?  Again, be sure to check that someone else hasn't saved a concept which supercedes one of your choices!


 * Examples. isomorphism theorem, Lie group, homomorphism, integral

3. Mid


 * Description. The more significant components of a particular mathematical field.


 * Do you mind if I copy this over to Wikipedia talk:WikiProject Mathematics/Wikipedia 1.0/Importance? It is pretty lonely over there, and this is great stuff. I have plenty of comments to add, but maybe the guideline talk page is a better forum. Geometry guy 19:37, 2 June 2007 (UTC)


 * Copy away. Be advised that I'm still sorting it out myself, so its very much a work in progress (hence the template).  Cheers, Silly rabbit 19:43, 2 June 2007 (UTC)

Image:Vector AB from A to B
Hi rabbit. Hope you're well and the grass is green. :) I moved Image:Vector AB from A to B.svg to commons, at commons:Image:Vector AB from A to B.svg. Usually free images are better off there, as then they can be used by other projects too. Cheers, Oleg Alexandrov (talk) 23:15, 2 June 2007 (UTC)


 * Thanks for letting me know. Perhaps I should set up an account over there so I can do the same for more of my images? Silly rabbit 03:04, 3 June 2007 (UTC)
 * Perhaps. Setting up an account on commons is just as easy to making one here. Oleg Alexandrov (talk) 05:55, 3 June 2007 (UTC)


 * too true, I created my account not 3 days ago, It is very handy! On a related note, have you looked at the talk page of image (mathematics) lately? I doodled a picture for the article but I don't know if it is applicable with the terminology the article uses. Take a look Image:Group_homomorphism.svg and Image:Codomain.svg. thanks!--Cronholm144 07:55, 3 June 2007 (UTC)


 * Looks good to me. Except the two small circles in the domain should have the same size ;-).  Anyway, I'm thinking of starting an (admittedly small) category Category:Isomorphism theorems to include the kernel, image, cokernel, coimage, isomorphism theorem, rank-nullity theorem, and other sundry applications.  Incidentally, are you familiar with the "least-squares picture" that Gilbert Strang uses in his Introduction to Linear Algebra book?  Presumably this is a standard diagram, and I'm curious if it has a name: it's like rank-nullity on steroids.  Silly rabbit 13:00, 3 June 2007 (UTC)

I'll take a look and get to doodling another SVG. Ironically enough I "took" one of his classes on the MIT open courseware when I was first learning linear algebra...interesting guy. BTW I also made a picture for coset multiplication, let me know if I need to tweak it.

I just realized that the only copy of Dr. Strang's work that I have is from 1988 and only has a picture of a vector's projection onto its column space. Can you get my the picture by E-mail or point me in the right direction to find it? Thanks. --Cronholm144 15:44, 3 June 2007 (UTC)


 * Here's my weak attempt (for a handout I made on least squares awhile back... in LaTeX):

To answer my own question, it's called the four subspaces. Silly rabbit 16:04, 3 June 2007 (UTC)
 * Ah. Here we go: fundamental theorem of linear algebra.  No pic, though.  I'll add my poor attempt in the sincerest hope that someone comes along and replaces it.  ;-) Silly rabbit 16:06, 3 June 2007 (UTC)

I am so very close. How do I get unicode to render the $$\mathbb{R}$$? If I can do that I am done.--Cronholm144 18:18, 3 June 2007 (UTC)


 * Try unicode . This produces &#8477; (in which I increased the font size).  --Lambiam Talk  19:03, 3 June 2007 (UTC)

It seems that WP knows what I want more than I even know. When I used the unicode and uploaded the picture it rendered in WP like I had originally wanted. Thanks Lambiam!--Cronholm 144 19:31, 3 June 2007 (UTC)


 * Hey, cool. And thanks for fixing my awful picture.  ;) Silly rabbit 19:41, 3 June 2007 (UTC)

Re: To Count Iblis
At first the situation was not clear to me. But having engaged him in discussions (together wit some other editors), I can only explain Ed Gercks behavior as part of the test he is performing. I simply don't believe anymore that he is serious when he insists that "the invariant mass of an isolated closed system can change".

I wrote on the special relativity talk pages that the article passed his test, refering to what I wrote on his talk page, so that we can ignore him and proceed with improving the aticle. But he reverted that, perhaps he wants to keep it a secret...

Now, I can't be sure that the recent edits on the GR page were made by him. If Ed Gerck wanted to test the GR pages then he would need to do that from a new wiki account as editing from his Edgerck account would be too transparant. Also, Ed Gerck seems to have stopped editing from his regular account after I exposed him.

But even if it isn't Ed Gerck, we could still call it an (perhaps unintended) "Ed Gerck" test :)   Count Iblis 17:09, 3 June 2007 (UTC)

Relativity
I keep getting a blank response to my simple question. This seems to be a way to quieten our disruptive friend: the problem with long answers is that he/she can pick something out of them to argue with. Geometry guy 00:24, 4 June 2007 (UTC) PS. I've notice that user contribs have not been recording post 22:00. Do you have the same thing, or is it just my set-up?


 * Yes, I just noticed that. It's quite inexplicable: He posted to the page after that time, but its not showing up in his contribs.  What's going on, Sherlock? Silly rabbit 00:28, 4 June 2007 (UTC)  Actually, it seems to be true for everyone.  Oh no!  Wikipedia is broken! Silly rabbit 00:33, 4 June 2007 (UTC)


 * Seems to be a Wikipedia wide problem: check out WP:AN. Whatever remains, however improbable... Geometry guy 00:39, 4 June 2007 (UTC)


 * You're probably right. I'm considering removing the page from my watchlist for a little while so I can rest easier.  It's not good for my health.  :-) Silly rabbit 00:30, 4 June 2007 (UTC)


 * Yeah, I think several of us noticed that this has been upsetting you: you definitely have friends here! But, as the WP advice says, don't feed the trolls. Meanwhile I will continue with my tactic. Geometry guy 00:39, 4 June 2007 (UTC)

For your information
Administrators' noticeboard/Incidents. Exploding Boy 19:28, 18 June 2007 (UTC)

Warning: WP:BLP violation
Hello, and welcome to Wikipedia! We welcome and appreciate your contributions, such as Jerry Lawler, but we regretfully cannot accept original research. Please find and add a reliable citation to your recent edit so we can verify your work. Uncited information may be removed at any time. Thanks for your efforts, and happy editing! Burntsauce 18:12, 20 June 2007 (UTC)


 * Since you are obviously not in the habit of replying to messages left on your own talk page, I will reply to you here. It would be nice if you could give a precise stetement as to how what I did was a WP:BLP violation.  There seems to be no consensus as to what makes a given section contentious, and I think that many editors on Wikipedia do not share your interpretation.  You say that you are sick of explaining yourself on User_talk:Burntsauce.  But I think you will find that it is quite often the case that one needs to explain one's self on Wikipedia.  For starters, we usually use an edit summary.  Furthermore, if a particular edit is likely to be controversial, it is conventional to explain it in more detail on the talk page of the article rather than to take unilateral action.  Wikipedia is a collaborative project, and the hard work of editors &mdash; even misguided editors such as those you abhor for not providing references &mdash; is not to be trivialized.  But your unilateral actions in the name of nebulous BLP policies do precisely that.  Finally, if you find some material objectionable, then by all means delete it.  But if you find nothing wrong with the material, then a more constructive approach would be to seek out some references of your own.  Silly rabbit 00:27, 21 June 2007 (UTC)

I wouldn't bother SR, BS never replies and never explains, and mainly never enters talk pages to explain, and the one thing BS never does is source. He has used BLP to justify all kinds of behaviour. Darrenhusted 12:42, 21 June 2007 (UTC)


 * Also, I would appreciate it if User:Burntsauce would start using edit summaries. This has already been brought up several times at WP:ANI here and again here.  Check out the red in Kate's tool for him: Moreover, his applications of WP:BLP on the removal of uncontroversial information has been discussed here. Silly rabbit 18:57, 21 June 2007 (UTC)

If you look on the front of WP:PW there is a section for pages he has deleted and on the talk page for the project on of the things to do on the top section is check his contribs, the problem is that he has at least three admins who back him up. Darrenhusted 23:53, 21 June 2007 (UTC)

Calculus
I am wondering why you reverted my edits, on the basis of WP:CIVIL. Not only did I not make such edits as an act of incivility, I justified my version while the other version went (rightly) unjustified. Please explain yourself, and please have the courtesy to accuse me of violations of WP:CIVIL to my face. I made a justified, explained, and civil change, and restated my justification when it was reverted without good cause (while being personally berated by the original reverter). I hardly consider my actions out of line, and would like to know why you reverted my edits and why you accused me of policy violations. Thank you. --Cheeser1 05:35, 21 June 2007 (UTC)


 * I hardly think Arcfrk was berating you. He merely said that university curriculum is a standard idiom, or so he thought.  Your edit summary replied that he should check a dictionary!  That, to me, is uncivil.  Anyway, I have replied in more detail on your talk page. Silly rabbit 05:46, 21 June 2007 (UTC)


 * An excerpt from his berating me: "I dare say, you've expressed your personal opinion (based on misunderstanding of the subject) and wrapped it into an insult. If you cannot see that, you've got some serious personality issues...Being defensive about your opinion is usually a sign of insecurity." If that's not berating, I don't know what is. Just if you know, you thought that I was being the mean one or whatever. --Cheeser1 06:02, 21 June 2007 (UTC)

Hi Rabbit, thank you for your intervention. I have no inclination to engage in this silliness, and have reported him to ANI, Best, Arcfrk 07:07, 21 June 2007 (UTC)

Testing template
Silly rabbit 17:06, 21 June 2007 (UTC)

Dipole gravity
An anonymous IP editor has been inserting material on dipole gravity into various high-profile relativity and astrophysics articles. The following debate already addresses why these issues are not suitable for wikipedia:. 18:04, 24 June 2007 (UTC)


 * At this time, the offending IP is . Previously, possibly a sock of  and . Silly rabbit 18:12, 24 June 2007 (UTC)


 * Two of these IPs are from Austin, Texas, the "last known siting" of Eue Jin Jeong, the author of the papers referenced by the recent additions. Note the POV-pushing tone adopted by the author (Eue Jin Jeong) in this post responding to a NASA press release.  See this dialog with the author, and how these ideas are being used to argue the existence of aliens. Silly rabbit 12:01, 25 June 2007 (UTC)

Natural topology
Thanks for your comments. If "natural topology" is a meaningless term, why is it used in so many contexts in mathematical articles? Perhaps you could explain what is meant by "natural" in these contexts?

For example, when a mathematical paper says: "I will now show that, under certain conditions, the natural topology is a locally compact topology." what exactly is meant? Presumably, some specific meaning is indended in the use of the words "the natural topology", and presumably something that is definable in some way. Is "natural" a qualifier, like "locally compact"? Should we have an article natural (topology), or natural (mathematics)? -- Karada 19:50, 24 June 2007 (UTC)

Thank you for your reply. "Natural topology" is clearly, at the very least, a mathematical jargon term, like "up to", even if it does not have a precise formal meaning. If we can't explain what we mean by this, we are doing our readers a disservice. If it hasn't got a meaning which can be articulated, we shouldn't be using it at all.

The problem is that the phrase "natural topology" is used, but nowhere defined, throughout many mathematical articles, and is indeed used in ways specific enough to be able to write papers with titles such as "There is no natural topology on duals of locally convex spaces". Clearly, "natural topology" must mean something in this context, or the paper wouldn't get published in a peer-reviewed journal: but what? Saying "it's informal" is like saying "it's a secret, we can't tell you", or "we're handwaving here, please ignore this bit". It's clear that "natural" means something a bit more precise than just "obvious" or "simple": my best guess is that "natural" in this context means something like "induced by the partial order(s) used in the construction of this structure". Is this correct? If not, can we find someone who does know what "natural" means in this context? -- Karada 20:07, 24 June 2007 (UTC)

Hi SR, I've looked at a few of these links and they seem to be clearly inappropriate, no matter what is written in the page titled Natural topology. If you have time, I'd ask you to revert all of them as they would do more damage than good (see also my comment | here). Arcfrk 00:43, 25 June 2007 (UTC)


 * Thanks for your support. I'll wait until the situation simmers down a bit, and have another look at things.  Silly rabbit 00:47, 25 June 2007 (UTC)


 * That would have probably been wise. But I went ahead myself and unlinked them all, as they seemed so blatantly inappropriately linked in every single case. Arcfrk 01:24, 25 June 2007 (UTC)

Rodney Anoa'i
Burntsauce's edit to this page has been reverted despite consensus otherwise and BLP violations. I recommend you taking a look at it and see if it needs to be reverted back.61.238.5.79 23:22, 25 June 2007 (UTC)

Silly Rabbit?
You like to eat Trix cereal for breakfast?--  PNiddy  Go!  0 15:02, 26 June 2007 (UTC)


 * No, I actually don't. As a matter of fact, I don't usually eat breakfast at all.  The nickname is not, in any way, a reference to the cereal or its plucky rabbit mascot.  Silly rabbit 15:07, 26 June 2007 (UTC)


 * You don't get it do you? Check your user page edits :) 阿修羅96 18:16, 26 June 2007 (UTC)

Whoa. This is kinda scary. Where did you two come from anyway? Silly rabbit 19:17, 26 June 2007 (UTC)

I think I fixed the problem. Your page linked to the Trix rabbit.--Cronholm144 09:12, 27 June 2007 (UTC)

Spin the bottle
Thank you for removing the remainder of that unsourced garbage from the article. I have located a worthy source and hope to expand it further in the near future, and have provided a stepping stone for others who are interested in proper research as well. Burntsauce 00:05, 27 June 2007 (UTC)


 * Good. I was beginning to be worried that you were only interested in removing unreferenced or poorly-referenced material, and not in actually providing any references.  Apparently I was wrong.  (Still: Edit summaries, please?)  Happy editing. Silly rabbit 00:12, 27 June 2007 (UTC)

consistent contradiction
you may think it does not make sense but just think about it for a moment. I appreciate your concern so please hear me out.

A contradiction is always a contradiction. Therefore a contradiction is always consistent. Therefore it's proven true by it's own disproof. I know it's crazy but once you understand it it will blow your mind. It is the same problem presented by the incompleteness theorem, the axiom of choice, Russell's paradox, and the number 1/0. It is the solution to the theory of everything. Come on bro, I know what I'm saying or I wouldn't be saying it. I went to school too but this is nothing they can teach in school.

Dear Silly Rabbit
Dear Silly Rabbit,

Thank you for being so helpful and understanding. I wish to drop you a line because I'm trying to make friends on wikipedia, so that I will no longer be misunderstood. Please let me tell you a little bit about myself and why I am here so that you will understand me better.

I, like you, am a mathematician, searching for the truth. Not that it really matters but in case you want to know, I scored %99 in the Math section of the ACT and I am currently an undergraduate studying physics and environmental engineering at New Mexico Tech. That having been said, I am now on a quest for ultimate understanding of everything.

I read the link you provided, and I do not believe in discordian philosophy, but rather, I believe in accord and rationality. I believe there is a very set order to the universe which is a harmonious order and which can be understood to the greatest iteration. Notwithstanding, I believe that the universe at the most fundamental level is composed out of north and south magnetic poles. I believe that all that it takes to create the universe is these two things.

Think about this math for a moment, since you are a fellow mathematician. Division is the opposite of multiplication, right? Therefore division by zero should be the opposite of multiplication by zero. If multiplication by zero results in nothing, then division by zero should result in the opposite, everything! This makes perfect sense if think about it because the unified field, aka everything, means the same thing as an undivided field. And what is an undivided field if not "one thing divided by nothing." So you can see, I do not believe in discordant ideas but only in pure logic spelled out in plain english.

Continuing on, if 1/0 means the same thing as an undivided, or unified field, then perhaps this long misunderstood number should receive a little more credit. Perhaps defining 1/0 is akin to defining the theory of everything. Now let us examine why it is that 1/0 has heretofore been considered undefined.

1/0 is considered undefined because if you evaluate it using limits you can show that it is equal to both positive and negative infinity. And something that is equal to both positive and negative infinity would appear to be a consistent contradiction, right? Because no number should be both positive and negative. Or should it?

Remember what I said, about the possibility of the entire universe being formed out of nothing more than north and south magnetic poles. If this is true, then it makes sense for the number 1/0 to be both positive and negative, if indeed 1/0 is the definition of the unified, or undivided field. Imagine that if everything is made out of north and south pole magnetism, then north pole represents positive infinity and south pole represents negative infinity. Therefore it makes sense that there should be a number that is both positive and negative, if indeed that number is the definition of the unified field.

Let us delve into this a little bit further shall we? Let's assume that this definition is true, that 1/0=everything. This would be a key factor in understanding the "theory" of everything, would it not? Would this provide a long awaited paradigm shift? Imagine that 1/0 is the total amount, or value, of all the energy in the universe in all it's iterations. If 1/0 is the total amount of energy in the universe, it would mean that energy in all it's iterations is truely limitless. This would seem to explain why the universe is expanding faster and faster - because there is a limitless amount of energy that lies at the core of everything. If mankind could learn how to harness this energy, we would be able to travel faster than the speed of light and also have a non-polluting energy source that uses no fuel. This of course is already known to many people, and they call it zero-point or free energy, but I do not want to confuse you with that.

Now I know all this sounds heretical, but please just bear with me for a moment. It may be true that much of what we know about physical science and all of mathematics is "destined" to be rewritten in theory or else we will never know the theory of everything. It may in fact be true that the only way mathematics will ever be complete is if we define the number 1/0, in theory. By defining the number 1/0 we may possibly complete the number line and turn it into a number circle - where positive and negative infinity are joined together at one point. Aha! This would give us the solution to what happens after the big rip. Allow me to explain.

Some scientists theorize that if the universe keeps expanding faster and faster then eventually all matter will be torn apart. The question which the originator of this theory has been wondering, however, is what will happen after that. If 1/0 truely does have a definition, and if the number line is actually a number circle, it would imply that the arrow of time will completely reverse after the big rip. For that is when infinite expansion instantly flips and becomes infinite negative expansion. In this case, the point 1/0 would be considered the "end" of time (actually just when time starts moving backwards because it has gone so far forwards). By combining the implications of quantum mechanics and general relativity, I can then theorize that this great reversal will happen when two points in space seperated by one planck distance begin expanding away from eachother at faster than the speed of light.

So there it is; that's me in a nuthsell. Hopefully you can see now that I am not a troll, and that I believe there is a very strict order to the universe which can be fully understood. It is true that 1/0 is a consistent contradiction but maybe it is treu that that makes perfect sense. If 1/0=everything, than the theory of everything can in fact be fully understood. Please tell me what you think about all this, and also I would like to know a little bit about yourself if you don't mind me asking.

cheers! Field