User talk:Linas/Archive15

Riemann zeta universality
Hi, I just found your intriguing Zeta function universality article. I was wondering if we have to assume that the interior of U is non-empty? Also, does "non-vanishing" mean "not identically zero" or "nowhere zero"? Thanks and cheers, AxelBoldt (talk) 20:53, 19 December 2007 (UTC)


 * I don't know, I'd have to go to the source. I may have read a review I had found on arxiv, not sure. I can certainly use intuition to guess: emptiness) I don't think it matters: if the statement holds for interior of U non-empty, then one has, as a lemma, the special case of a collection of points (i.e. where int U is empty), right?


 * Zeros) Intuition says that a finite number of zeros shouldn't matter: for a holomorphic function that has a finite number of zeros, I can find another holomorphic function that approximates it arbitrarily well on U, but has no zeros in U (right? I think so ..) A similar statement holds true, for a countable number of zeros, although that is clearly "harder". Right? Finally, there are no holomorphic functions that have an uncountable number of zeros, right? So, by this lemma, the question of zeros doesn't much matter.


 * But I find what I wrote strange, since, if that is the case, then is Voronov saying that the Riemann zeta does everything except get arbitrarily "flat", on any U? So I'm not sure what the "non-vanishing" means.


 * Of course, its late at night, I might be spouting nonsense. Caveat Emptor. linas (talk) 03:50, 20 December 2007 (UTC)


 * I thought a bit about it, and I think I have the answers.


 * First of all, a function that's identically zero on U can be approximated by the zeta function: just approximate the non-zero holomorphic function f(s)=&epsilon;/2 to accuracy &epsilon;/2. So the statement of the theorem applies to those holomorphic functions that aren't zero anywhere. And this makes sense: if you have a holomorphic function with an isolated zero, then every "nearby" function will also have an isolated zero; but the Riemann hypothesis says that the zeta function doesn't have any zeros in that strip, so it could never approximate such a function.


 * And we don't have to assume that U has non-empty interior. Suppose for example that U is a line segment, and we're given a continuous function f:U&rarr;C. Then by Stone-Weierstrass there exists a holomorphic function g that's defined on an open neighborhood of U and that is close to f (and we can also assume that g doesn't have any zeros). Then our theorem yields a spot where &zeta; approximates g and therefore also f. AxelBoldt (talk) 21:33, 21 December 2007 (UTC)


 * Right; I was wrong, I think. So then: Thm: if f(s) has a zero inside of U, then *every* sufficiently nearby function g(s) will also have a zero inside U. This sounds like a perfectly plausible theorem that would seem to date to the classical period, but for the life of me I can't think of a name for it. Some variant on the "three circles theorem", or something like that?  Again, its late at night, so I'm babbling. linas (talk) 05:25, 24 December 2007 (UTC)


 * This is a consequence of the open mapping theorem. If f is defined on the open set V, holomorphic, non-constant, and f(z0)=0, then f(V) is an open neighborhood of 0. Pick &epsilon;>0 such that {|w|<&epsilon;} is in f(V). Then every g:V&rarr;C that approximates f to accuracy &epsilon; will also have a zero in V. AxelBoldt (talk) 06:48, 24 December 2007 (UTC)


 * Actually, the last sentence is not correct; it's more complicated than that. It's better to use Rouché's theorem, applied to a small circle C around z0 on which f is non-zero, and pick &epsilon;=min {|f(z)| : z in C}. AxelBoldt (talk) 17:35, 25 December 2007 (UTC)

Newton Series for Riemann Zeta
Hi Linas, It was real pleasure for me to read your paper (with P.Flajolet) "On differences of Riemalnn Zeta". So I would like to give you a reference to another 2 papers (1985-88) on (nearly) the same subject  Also wish you Happy Xmas & New Year. —Preceding unsigned comment added by 193.233.210.174 (talk) 16:22, 28 December 2007 (UTC) I.Kaporin (Erdos number 4)


 * Thank you for this Christmas present! I am writing a letter to Flajolet, and am cc'ing the email address posted on that web page. If you wish to add anything, let me know. linas (talk) 18:38, 31 December 2007 (UTC)


 * Thanks a lot for your cc-letter! I have sent a reply to you from that e-mail; have you obtained this message? I.Kaporin


 * Yes, I've been too busy to reply...linas (talk) 04:07, 25 January 2008 (UTC)

Delimiting numbers
Linas: You’ll note that by delimiting with span control (as done with e here: 2.718 281 828 459 ), the gaps are slightly narrower, don't break (like a non-breaking space), and (most conveniently) can also be copied and pasted into Excel, where they are treated as numeric values.

We've got a small group of people discussing a plan to create a template-like parser function to make it easier to delimit numbers like this. It would work similarly to how produces $1/undefined$ and produces $1/undefined$. This new template would be powerful and convenient. It would allow an editor to type only in order to obtain the following: 6.022 464 79(30) × 1023 kg. As it currently stands, one must code all of the following to accomplish this: ×

Although highly capable and feature-rich, this template/parser function wouldn’t be cumbersom for simple tasks. For instance, one need only type to obtain 6.022 464 or they could type to obtain 1,579,800.298 728

The discussion was originally discussed at WT:MOSNUM but got disorderly and dysfunctional there so I and some other advocates picked up the discussion here on my talk page. We would appreciate the assistance of someone deeply involved in mathematics. Judging from you user page, it appears you also do not shy away from controversial issues if you think something is wrong and needs fixing. The link I provided takes you straight to a “nutshell” overview of the proposal. If you like what you see, please weigh in. Greg L (my talk) 23:47, 25 January 2008 (UTC)


 * Ehh? I don't shy away from telling people off if I think they're wrong. In this case, I'm not sure I have an opinion. Perhaps a good place to gain support would be on talk pages of WP:WPM and WP:WPPhys. linas (talk) 03:35, 27 January 2008 (UTC)

GnuCash
Hi there,

Is the GnuCash project still active? I read that you have always been paying the server's bill -- thanks for that, though I cannot access the www site at the moment.

Cheers, Benjamin. —Preceding unsigned comment added by 118.90.94.114 (talk) 11:48, 9 February 2008 (UTC)


 * A router between the gnucash server and the rest of the internet died on friday afternoon, and it wasn't replaced till monday morning. According to the internet provider, 24x7 only applies if the outage happens during business hours, and since it was less than 8 business hours between the outage and it's fix, the 24x7 sla terms don't apply. Impeccable logic, eh? linas (talk) 17:35, 12 February 2008 (UTC)

Hello, I'm not sure if this is related to the above, but all of the mailing list archives, developer areas, etc are down. I can access the main site so I'm not sure if this is an extension of that issue or not. In any case, I have been using GnuCash for a while now and would like to contribute to it. Is there development still ongoing for this project, and who would be the best person to contact about it? Thanks a lot for all the work to make it such a good program. --Jtrick (talk) 20:35, 13 February 2008 (UTC)


 * I haven't been actively involved in development in quite a long time. However, if you just look at the front page, you will notice that there are releases every month or two, sometimes more often, each with a dozen or two of fixes and enhancements. I believe that there is a core of 5 active developers, with another 10-15 part timers, all hacking away at it. You may find development daunting; last I counted (many years ago) there was 1/3 of a million of lines of code.  Printed out on proverbial sheets of paper, this  would be a whole library-shelf worth of paper, so "contributing to gnucash" is similar to contributing to a bookshelf-worth of existing books ...


 * See http://wiki.gnucash.org/wiki/Development for how to help. Try the irc channel for quicker answers. The devel mailing list is at https://lists.gnucash.org/mailman/listinfo/gnucash-devel and the mailing list archives look fine to me: e.g. this month: http://lists.gnucash.org/pipermail/gnucash-devel/2008-February/thread.html


 * There are also lists for gnucash users, and also french, german mailing lists. linas (talk) 22:59, 13 February 2008 (UTC)

Is meromorphic = regular?
Please see Talk:Meromorphic function. --Acepectif (talk) 19:16, 16 March 2008 (UTC)

Merge
In this edit, you said there's nothing there to merge. But there was a reference, and the target of the new redirect page has none. There was also a comment about the unified neutral theory of biodiversity, which is not in the target article. So at least two things were there to merge, which didn't get merged. Michael Hardy (talk) 21:30, 25 March 2008 (UTC)


 * Hi Michael, I didn't think that either the reference nor the comment were even vaguely appropriate to the article, and so I shot them on sight. After that, there was nothing left to merge. I'm having a hard time understanding why you would bother to defend such a thing. linas (talk) 22:05, 25 March 2008 (UTC)

Hello
I've seen your important contributions for the article Hypergeometric series. I'm looking for the general (non-iterative) non-trigonometric expression for the exact trigonometric constants of the form: $$\begin{align}\cos \frac{\pi}{2^n}\end{align}$$, when n is natural (and is not given in advance). Do you know of any such general (non-iterative) non-trigonometric expression? (note that any exponential-expression-over-the-imaginaries is also excluded since it's trivially equivalent to a real-trigonometric expression). Eliko (talk) 08:05, 31 March 2008 (UTC)
 * Let me explain: if we choose n=1 then the term $$\begin{align}\cos \frac{\pi}{2^n}\end{align}$$ becomes "0", which is a simple (non-trigonometric) constant. If we choose n=2 then the term $$\begin{align}\cos \frac{\pi}{2^n}\end{align}$$ becomes $$\begin{align}\frac{1}{\sqrt{2}}\end{align}$$, which is again a non-trigonometric expression. etc. etc. Generally, for every natural n, the term $$\begin{align}\cos \frac{\pi}{2^n}\end{align}$$ becomes a non-trigonometric expression. However, when n is not given in advance, then the very expression $$\begin{align}\cos \frac{\pi}{2^n}\end{align}$$ per se - is a trigonometric expression. I'm looking for the general (non-iterative) non-trigonometric expression equivalent to $$\begin{align}\cos \frac{\pi}{2^n}\end{align}$$, when n is not given in advance. If not for the cosine - then for the sine or the tangent or the cotangent.


 * OK, I think it would be a very good idea for you to work out the answers for n=3,4 and 5. You should be able to see that the the answers will contain nested square-roots, which go deeper and deeper. This is clearly an iteration on the square-root, spefically, the iteration of f(x) where
 * $$f(x)=\sqrt{\frac{x+1}{2}}$$
 * so I don't believe that a non-iterated solution even exists. You claim that it does, but you haven't actually shown that one does! Not to be obtuse, but:


 * $$\cos{\frac{\pi}{2^n}}= f^n(0)$$
 * which is the answer; but you claim you don't want it. Did I just solve your homework problem for you? You know that you only cheat yourself when you get others to do your homework...linas (talk) 18:01, 31 March 2008 (UTC)
 * "homework"? the last time I attended school was a few decades ago.
 * As you properly guessed, your response couldn't be helpful. Anyway, thank you for your response. By the way, any solution for my original question could have been helpful in answering other questions, like: finding a non-trigonometric proof for the following algebraic, non-trigonometric claim: Every real interval includes a point x having a natural number n such that $$\begin{align}(x+i)^n\end{align}$$ is a real number. Note that this is an algebraic, non-trigonometric claim, so one may naturally expect that it may be proven by a non-trigonometric proof. Have a nice day. Eliko (talk) 14:58, 1 April 2008 (UTC)

X-Ray
Hi, Linas. I was admiring some of your graphics the other day and wondered if you have seen this PDF.

X-Ray of Riemann's Zeta function by J. Arias-de-Reyna - it's a way of presenting a complex function that I've never seen before.

Virginia-American (talk) 20:51, 1 April 2008 (UTC)


 * Hi, sorry for the late reply, didn't see your post earlier. I had not seen that paper before, but have certainly looked at similar-looking graphics. Somewhere, I have an animation of the polylogarithm; the Riemann zeros occur when the polylog zeros pass through the point z=-1. There are also these clearly-visible contour lines, that dance around and reconnect as if they were magnetic field lines. I don't have anything as insightful as Arias-de-Reyna to say about it, thought. Here's the movie... polylog animation. Its a very slow download, though. linas (talk) 15:07, 17 April 2008 (UTC)

Inappropriate edit summaries
Re this edit: Using an edit summary of "rvv" on edits that clearly are not vandalism and on reversions that are clearly controversial is inappropriate. Use proper edit summaries. -- JHunterJ (talk) 12:01, 8 April 2008 (UTC)


 * ?Huh? Its clearly vandalism, I'd added the exact same text just a few weeks earlier, and someone had erased it. linas (talk) 14:13, 8 April 2008 (UTC)


 * Not all deletions of text are vandalism. In this case, the shorter intro line is stylistically appropriate. -- JHunterJ (talk) 23:53, 8 April 2008 (UTC)


 * And its also factually incorrect. linas (talk) 15:17, 9 April 2008 (UTC)
 * In which case, the line making the same "factually incorrect" statement on Entropy should be corrected first. -- JHunterJ (talk) 18:26, 13 April 2008 (UTC)

Entropy
The WP way is Bold-Revert-Discuss.

You've made a Bold change. I have Reverted it. Now we take it to the talk page and Discuss. Jheald (talk) 19:12, 9 April 2008 (UTC)


 * Oh and if you check the edit histories going way back, I think you'll find I have a lot of interest in these articles. So I'll thank you for not patronising me in future. Jheald (talk) 19:14, 9 April 2008 (UTC)


 * The topic got pounded to death on the talk pages. There was (I thought) a clear consensus for the need for a general article on the concept of entropy, that wasn't strongly focused on thermodynamics. The obvious path from here to there was blocked when the page move went awry. The next step is then to create the desired, factually correct article on the general concept of entropy, and when its completed, take another shot at installing it as the "main" article.


 * Sorry I bit, I've been reverted three times today, and I view the first few as spurious and clueless. I'm rather loosing my temper in resolving what should be an easy and clear problem, with an obvious and immediate direct way forward. linas (talk) 19:22, 9 April 2008 (UTC)


 * I see a different if not consensus then at least majority view. Namely that thermodynamic entropy, particularly classical thermodynamics, is the most common (and most commonly first met) usage.  Information-theoretic entropy is substantially an independent usage.  And the relations between the two are probably best treated as an advanced topic.  Especially when there are chemists like Denbigh who flat-out decry the idea of identifying physical entropy (to them objective) with an information entropy (subjective).  Okay, you and I may think that's a spurious objection; but the fact remains that you don't need to identify Gibbs entropy as a Shannon entropy to use it, and many don't.


 * Also, for your own future sanity, be very very careful using the word "disorder" in connection with entropy, because there are some chemistry educators, and their fervent followers, who are on almost a holy jihad against the "d" word. See eg Entropy (energy dispersal),  and Entropy (order and disorder).   Yes, I know both those pages are lousy; but I'd had enough of discussing with "entropy isn't disorder" zealots, and at least with pages of their own there isn't so much of it on the main thermodynamic entropy page any more.


 * Sigh... I'm sorry, I wasn't trying to do you down.  But I really think 1 redirect is about the most we can make people go through to get to a disambig page to find their article title.  I just don't think people will click through twice to get to the links your new page is offering; I do think that it is much easier for users to have those links remain on the first dab page.  Jheald (talk) 19:47, 9 April 2008 (UTC)


 * Heh. Chemists must be crazy. "Subjective"? Entropy is immediate and "in-your-face" when working with radio systems and transmission lines, you can watch it go up & down on a signal/noise meter when you turn the knob. By contrast, in chemistry, they still don't have a "entropy meter" that you could dip into a solution and just measure the entropy -- it always struck me as a difficult and hard-to-grasp concept in chemistry -- I almost failed college thermodynamics, and didn't "grok" entropy until I saw the info theory definition of it. The math definition is certainly far more concrete and "hands on" and manipulable than the slippery thermodynamic concept. At least, for me it is.


 * OK, I think I can agree that calling entropy "disorder" is incorrect; its an old force of habit in that when you heat a system, it gets, hotter, more disordered, and increases in entropy; but the subjective sense of "disorder" is a subjective by-product of shaking things up. linas (talk) 20:15, 9 April 2008 (UTC)

Zero Divergence
Linas, I was misrepresented by Steve on the issue of "div B = 0 follows from curl A = B, but not vica-versa". I merely pointed out that curl A = B is a less ambiguous equation because div B = 0 might also be interpreted as referring to an inverse square law.

We know of course that in the case of the inverse square law solution that the divergence will not be zero at the origin and so in the case of the magnetic B field, it is important that we emphasize that the divergence is zero everywhere.

In the case of curl A = B, the equation is unambiguous. My ultimate point was that the equation div B = 0 equation in Maxwell's equations is really referring to curl A = B and as such, the term 'Gauss's law for magnetism', while techniccally correct, is not the best name to use in the circumstance.George Smyth XI (talk) 00:16, 12 April 2008 (UTC)


 * Heh. OK. FWIW, at some point or another, physics students are told or realize that div B=0 is the same as "no magnetic monopoles", i.e. its not just zero somewhere, it has to be zero everywhere. This is a case of holonomy at play. But you are right; until you finally grok what the words "no magnetic monopoles" really really mean, (which may seem trite to a beginning student, but is actually rather subtle, because its a global condition, not a local condition: there are no magnetic monopoles anywhere in space, not just 'here'. Global conditions are a lot trickier, as they say things about *all* of space, which is why I say holonomy above, rather than just local space), so yes, curl A=B is (much) more correct (because it is a local condition). linas (talk) 22:35, 12 April 2008 (UTC)


 * I don't get what you're saying, Linas. In the physics literature at least, the statement "div X = 0" means that the divergence of the vector field X is "0", the "zero vector field", i.e. X is divergenceless everywhere. If a physicist means to say that the divergence of X is zero at a particular point or in a particular region, he would write "div X(r) = 0" or "div X = 0 in the region R". Is that not true in the math literature? Are there math papers that just write "div X=0", and expect the reader to guess what points this equation is supposed to hold at and which points it doesn't?? From what I've seen of mathematics, if an equation is written down it is understood to be a true equation at every point, unless otherwise stated.


 * If you believe that definition of "div X = 0", then of course you'll agree that if you have a vector field B in Euclidean space, the statement "there exists a vector field A with curl A = B" and the statement "div B = 0" are equivalent. This is Helmholtz decomposition.


 * As for local/global:
 * If there's a field A with curl A = B everywhere in the universe, then there are no monopoles everywhere in the universe. If there's a field A in a small region with curl A = B in that region, then there are no monopoles in that region.
 * If div B = 0 everywhere in the universe, then there are no monopoles everywhere in the universe. If there's a small region with div B = 0 in that region, then there are no monopoles in that region.
 * It seems to me like they're equally local-global. Could you please explain better what you're talking about? --Steve (talk) 00:38, 13 April 2008 (UTC)


 * I don't know if I can do it justice. Let me try a simple example: log z on the complex plane. This is analytic everywhere, except at z=0. To say a function is analytic means that the real and imaginary parts are related to each other by a certain pair of equations: they're differential eqns, they're local. The problem is at z=0; the logarithm has a non-trivial monodromy (its Z, the integers). So locally, ln z is analytic, but not globally; the only globally analytic function is a constant. The generalization of monodromy to higher dimensions, and more general cases is the holonomy. One of the simplest examples commonly given of holonomy is the Dirac string aka magnetic monopole.  In trite terms, you use Gauss's law on a monopole: if you integrate magnetic field over a closed surface, and its not zero, there's a (magnetic) charge inside.  Yet, we wanted to insist that div B=0 everywhere, so how can that happen? It can happen if you glue together patches of a fiber bundle so that there's a twist in there: you can't flatten the result into a trivial fibration, but you also can't point to one location, and say "here's where that twist occurred." The Dirac string is this funny handwaving that says "div B=0 is still true everywhere because the magnetic flux sneaks out in a thin little solenoid tube that you can't see because you're looking in the wrong place cause its snaking around", and then appeals to the need for a single-valued phase of a quantum mech. wave fn. to explain why the monopole charge is quantized. The Dirac string is this wacky argument (typical of physics!) that you use when you don't understand the underlying math ... but not so wacky, as it does give the right answer, at least for the charge of the monopole... just that the proper math is cleaner and prettier (oh, and the correct math doesn't require an apeal to QM!) . A similar holonomy effect is Berry phase, and my favorite is the Bohm-Aharonov effect (which, given the "correct math", is seen to be a classical effect, which only by accident needs QM to be observed experimentally).  (Other holonomies include precession of orbiting satellites, and movement of robot arms, which are purely classical mechanics effects, requiring no appeal to QM at all, yet the effect is there and very real!). More generally, in physics, where-ever you see solitons, the corresponding math will have this kind of glueing, some sort of non-trivial homology.  So, locally, you have differential equations, but what that means for global structure is .. subtle-- e.g. the KdV equation, which has topological solitons in its solution; you'd never guess from the fact that its a differential eqn. From there, the math spreads out to many interesting directions: Morse theory, for example, is used to make all sorts of statements about the global structure of plain-old classical Hamiltonian mechanics.  anomolies have a topological aspect to them.  Sigh. I guess I wrote up a catalogue of topics that are interesting to me, I wish I had the time to study them in greater depth; I'm quite foolish in how I allocate my time. linas (talk) 02:17, 13 April 2008 (UTC)


 * Thanks for that reply. I'm not sure that you're remembering the monopole argument correctly. The starting point of everything, at least when I learned it, is $$B(r)\propto \hat r/r^2$$, which certainly does has a divergence at zero. (Likewise, the 't Hooft-Polyakov monopole has the same magnetic field, see equation (31)).) The "gluing-together" argument is what you do to define A, not B. As you know, you can't have a single A--even if it doesn't have to be defined at the origin--so you have to glue together multiple pieces to get something whose curl can be B all around. When you say "div B=0 is still true everywhere", that's not right, div B is a delta-function at the origin; otherwise it wouldn't be a monopole. I think you mean to say "curl A=B is still true everywhere on the sphere", which is indeed only possible because you use a twist or glue or what-have-you. --Steve (talk) 05:49, 13 April 2008 (UTC)


 * If you built, say, a real-life solenoid, say 10m long and 1cm diameter, wouldn't divB=0 everywhere? Yet, if you drew a sphere around one end of the solenoid, say 50cm or 1m out, wouldn't the B field look like $$B(r)\propto \hat r/r^2$$, more or less? I think the point of the argument is not that magnetic monopoles might exist, its that *if* they exist, then their charge *must* be quantized.  There is no corresponding argument for the electric field; the electric charge can be anything. (the electric charge is quantized, but apparently for different reasons, I think its a root system but that's not commonly accepted.) You can write E = curl K too wherever there's no electric charge, but this looses once you have to deal with charges. And yes, I commonly mis-remember things. Don't take my word as gospel, rather take it as inspirational; that's all I ever use it for :-) linas (talk) 12:09, 13 April 2008 (UTC)


 * Well, you're definitely remembering the quantization wrong :-). Dirac's argument (or its equivalents) shows that the product of any electric charge and any magnetic charge must be an integer (in the appropriate units). So it immediately follows that if there's any monopole in the universe, electric charge must be quantized, and vice versa. See Jackson sections 6.11-12, for example.


 * If you say that you have a physical model of a magnetic monopole, and then say that the divergence of B is zero everywhere including at the alleged location of the monopole, a physicist, at least, would find this contradictory. Nonzero divergence of B is the defining signature of a magnetic monopole, even in an advanced theory like for the 't Hooft-Polyakov monopole. Of course with an actual solenoid, there is no monopole, and div B=0 at the origin because there's a solenoid tube in some direction; on the sphere, B points uniformly outward, except where the tube passes through, where it's huge and inward. In Dirac's string argument, you have the actual magnetic field which is divergenceless, and then you subtract off the field inside the string to get the hypothetical monopole field which is not divergenceless (see Jackson p279). Again, everything that might involve holonomy (or glue or twisting or whatever) is involved in the attempt to define A in the presence of a monopole; the field B is always simple vector field (indeed, it's experimentally measurable), and in this case you can write it down explicitly. More generally, the divergence of B at a given point is the monopole density at that point; I don't see how saying "div B=0 means no monopoles" is anything less than 100% correct.


 * Again, though, I've never taken a math course that discussed monopoles or a physics course that discussed holonomy (as such), so there's no guarantee that I'm filling in the details correctly. --Steve (talk) 17:34, 13 April 2008 (UTC)


 * I'm not remembering it wrong. I think you misunderstood most of what I was trying to say. I tried to be clear. Oh well. linas (talk) 23:16, 13 April 2008 (UTC)


 * Hmm. Sorry. Is there a book on the math of monopoles that you can recommend? --Steve (talk) 23:55, 13 April 2008 (UTC)


 * There's a nice, short pdf on line bundles that is referenced from the Maxwell's eqns article. There is a longer book Richard Montgomery, A Tour of Subriemannian Geometries, Their Geodesics and Applications (Mathematical Surveys and Monographs, Volume 91), (2002) American Mathematical Society, ISBN 0-8218-1391-9. which covers many of the basic ideas. I don't remember if either covers monopoles explicitly, but if not, they at least lay the foundations. Here's another, that I haven't read, but probably should provide an excellent overview: Michael Atiyah and Hitchin N.J., The geometry and dynamics of magnetic monopoles, M. B. Porter Lectures. Princeton University Press, Princeton, NJ, 1988. ISBN 0-691-08480-7 linas (talk) 02:25, 14 April 2008 (UTC)

Thanks! One more thing: Can you just indicate which of the following statements you agree or disagree with, so I know where the disagreement is as I try to sort this out?
 * (1) If there is a monopole anywhere in the universe, then it would follow (from Dirac's argument) that all electric charge must be quantized. Likewise, if a monopole exists, its magnetic charge would have to be quantized, (since we know that there are electric charges). Or as Shnir says in his (mathematically-oriented) book Magnetic Monopoles (p28), "If there is a monopole somewhere in the universe, even one such object placed anywhere would be enough to explain the quantization of electric charges".
 * (2) In classical physics, B is a well-defined vector field, and its divergence, at a given point, is the monopole density at that point.
 * (3) If you saw the equation $$\nabla \cdot X = 0$$ written down in a book or paper, where X is a vector field, with no further in-text explanation, then you would interpret that equation to mean that the divergence of X is zero at every point, and you could justly conclude that X is a solenoidal vector field. You would not interpret that equation to mean that the divergence of X is "zero except possibly at isolated points".

Thanks!! --Steve (talk) 18:25, 14 April 2008 (UTC)


 * You appear to be working extremely hard to pick an argument of insidious intent.  This is very unpleasant, and it makes me dread seeing the "you have new messages" banner at the top of my browser.  I see why George Smythe reacted badly to your editorial efforts. I have no particular desire to discuss monopoles, as my current interests lie elsewhere. I suppose that someday, you will gain your PhD, and you will be asked to referee papers. PLEASE do NOT bring this attitude that you currently carry to that task! linas (talk) 19:44, 14 April 2008 (UTC)


 * Well, you picked up on the fact that my 3 questions were not innocent curiosity. I have, indeed, had a primary agenda, namely to prove that I'm right, which here was partially superceding my secondary agenda, to understand what you're saying about monopoles. My endless back-and-forth with George has left me increasingly impatient and embittered, and I swear that my usual persona -- in my research, wikipedia, and elsewhere -- is much more humble and lighthearted. Anyway, I appreciate the time you did spend writing the above posts, I'm very sorry to have bothered you, and I won't again.


 * PS: Gnucash is great, I use it religiously. --Steve (talk) 02:10, 15 April 2008 (UTC)


 * If you trust in your own abilities, then you would know if you are actually right or not, and wouldn't need social support. You wouldn't need to prove it to anyone, except maybe the person offering you a job. And if you're not certain, try not to go out on a limb. Healthier for the ego that way. I suppose the broad answer to all three of your questions is "yes", although I wonder about the first one, because I don't remember the details: doesn't it only demonstrate that the product of the magnetic and electric charges is quantized? If not, if it shows that the electric charge alone is quantized, then why doesn't it also give a precise, physical value for the electric charge? linas (talk) 14:09, 15 April 2008 (UTC)


 * p.s. thanks for the compliment about gnucash. Its what I created as a way of procrastinating, instead of actually doing something about my finances. I'm still disappointed to admit that my finances are in poor condition. I ain't gonna retire rich, it seems. linas (talk) 14:12, 15 April 2008 (UTC)

Linas - looking for your advise on continued fractions
http://mathworld.wolfram.com/ContinuedFraction.html A "general" continued fraction representation of a real number x is  one of the form x=a_0+(b_0)/(a_1+(b_1)/(a_2+(b_2)/(a_3+...))), 1) Let c_1, c_2, ... be the coefficients of the Engel expansion of Pi-3. Now form the continued fraction [a_0 = 0; a_1=a_2=...=a_n=1; b_0=1, b_1=c_1, b_2=c_2, ..., b_n=c_n]

2) Let d_1, d_2, ... be the coefficients of the Engel expansion of Champernowne constant ( 0.1234567891011... ) Now form the continued fraction [a_0 = 0; a_1=a_2=...=a_n=1; b_1=d_1, b_2=d_2, ..., b_n=d_n]

3) Let f_1, f_2, ... be the coefficients of the Engel expansion of Thue-Morse constant (0.412454033640107597783361368258455283089478374455769557573379415348793592365...) Now form the continued fraction [a_0 = 0; a_1=a_2=...=a_n=1; b_1=f_1, b_2=f_2, ..., b_n=f_n]

1)for the cf based on Engel expansion for (Pi-3) accumulation points values are:

0.380584284572662371562639165710.... and 0.1592660038318136438508348696366....

Note also that 0.15926 (coincidetally ?) coinsides with digits in Pi ...

2)for the cf based on Engel expansion for the Champernowne constant( 0.1234567891011... ) accumulation points values are:

0.2759798024459901336871649381947 and 0.2739011310597606967841981731833

3)for the cf based on Engel expansion for the Thue-Morse constant (0.412454033640107597783361368258455283089478374455769557573379415348793592365...) accumulation points values are:

0. 5274057404032596816148597487462.... and 0.4271272989773713616643261534074

A) I wonder if it is coincidental (or not) that for (Pi-3) and Champernowne constant, which have "close" to each other decimal values, the sums of theirs "low" + "high" accumulation points are also relatively close to each other:

0.2739011310597606967841981731833 + 0.2759798024459901336871649381947 ~= 0.549880934...

0.1592660038318136438508348696366 + 0.380584284572662371562639165710 ~= 0.539850288... ?

B) Also may be the size of the "delta" between "high" and "low" accumulation points represents the relative "degree of randomness" between those 3 transcendental numbers (constants) with Pi (Pi-3) being the "most" random and the Champernowne constant being the "least" (due to initial plateaus ?) ?

At this point I got *stuck* in my research related to the summation of specially constructed continued fractions, which are based on using coefficients from engel expansion of mathematical irrational constants as quotients of the constructed continued fraction) yielding two accumulation points.

The last *step* I took was (just for comparison) exploiting the fact that five terms of OEIS's A135405 (starting from n=1) coincide with the coefficients of Engel's expansion of Pi-2[1, 8, 8, 17, 19, 300, 1991, 2492, 7236, 10586, 34588...]

a(A135405 n=1,30)=[1, 8, 8, 17, 19, 30, 34, 47, 53, 68, 76, 93, 103, 122, 134, 155, 169, 192, 208, 233, 251, 278, 298, 327, 349, 380, 404, 437, 463, 498, 526, 563, 593, 632, 664, 705, 739, 782, 818]

OEIS's A135405 could be described by the close form formula a(n)=(n+2)*(n+1)/2+2*(-1)^n or recurrently a(n)=(a(n-1)+ 2*(-1)^n)*(n+2)/n + 2*(-1)^n

One could see that (which is quite obvious) that engel summation using terms of A135405 (for n>=1) as engel's coffcts  and adding to the resulting sum value of  2 yields Pi's upper bound: 3.141594153201641657740748196

gp > 2 + suminf(i=1,prod(n=1,i,1/((n+2)*(n+1)/2 + 2*(-1)^n))) %1 = 3.141594153201641657740748196

So then I applied already described above procedure of constructing (summation) of continued fraction

1/(1+8)= 0.111111111 1/(1+8/(1+8))= 0.529411765 1/(1+8/(1+8/(1+17)))=0.152941176 1/(1+8/(1+8/(1+17/(1+19))))=0.39959432 1/(1+8/(1+8/(1+17/(1+19/(1+30))))=0.174682639 1/(1+8/(1+8/(1+17/(1+19/(1+30/(1+34))))=0.343323641 1/(1+8/(1+8/(1+17/(1+19/(1+30/(1+34/(1+47))))))=0.187900121 1/(1+8/(1+8/(1+17/(1+19/(1+30/(1+34/(1+47/(1+53)))))))=0.313167456 1/(1+8/(1+8/(1+17/(1+19/(1+30/(1+34/(1+47/(1+53/(1+68))))))= 0.196725643 1/(1+8/(1+8/(1+17/(1+19/(1+30/(1+34/(1+47/(1+53/(1+68/(1+76)))))))= 0.294843049 1/(1+8/(1+8/(1+17/(1+19/(1+30/(1+34/(1+47/(1+53/(1+68/(1+76/(1+93))))))))= 0.203000989 1/(1+8/(1+8/(1+17/(1+19/(1+30/(1+34/(1+47/(1+53/(1+68/(1+76/(1+93/(1+103)))))))))= 0.282738859 1/(1+8/(1+8/(1+17/(1+19/(1+30/(1+34/(1+47/(1+53/(1+68/(1+76/(1+93/(1+103/(1+122))))))))))= 0.207669807 1/(1+8/(1+8/(1+17/(1+19/(1+30/(1+34/(1+47/(1+53/(1+68/(1+76/(1+93/(1+103/(1+122/(1+134))))))))))= 0.274252624 1/(1+8/(1+8/(1+17/(1+19/(1+30/(1+34/(1+47/(1+53/(1+68/(1+76/(1+93/(1+103/(1+122/(1+134/(1+155)))))))))))= 0.211264587 1/(1+8/(1+8/(1+17/(1+19/(1+30/(1+34/(1+47/(1+53/(1+68/(1+76/(1+93/(1+103/(1+122/(1+134/(1+155/(1+169))))))))))))= 0.268031118

Below is PARI program allowing to automate above *hand-work* calculations

alex(n) = {local(result=0,denom=1, a = vector(n)); for(m=1,n, a[m]=(m+2)*(m+1)/2+2*(-1)^m; ); /* print1("a="a " \n"); */ for(m=1,n-1, denom = 1.0 + (a[n+1-m])/denom; /* print1("denom="denom" \n"); */ ); result=1/denom; print1("result="result" \n"); }

Using above one could see that above summation is converging to ONE value

(10:13) gp > alex(100000) result=0.2349099499071107134123929950 (10:13) gp > alex(100001) result=0.2349102611437880794655424316

Thanks, Best Regards, Alexander R. Povolotsky e-mail pevnev@juno.com

Apovolot (talk) 23:10, 16 April 2008 (UTC)

Hi Alexander .. I can't quite figure out what you are trying to say. Since I am too lazy/busy to try to carry out some of these experiments, I would like to ask you some questions. Lets start with 1)
 * ''Let c_1, c_2, ... be the coefficients of the Engel expansion of Pi-3.

Now form the continued fraction f_n = [a_0 = 0; a_1=a_2=...=a_n=a_{n+1}=1; b_0=1, b_1=c_1, b_2=c_2, ..., b_n=c_n, b_{n+1}=0...]''

Can you tell me what f_n is equal to?

AP> f_n = [a_0 = 0; a_1=a_2=...=a_n=a_{n+1}=1; AP> b[0-...]={from Engel expansion of Pi-3 - see OEIS's A006784 - without first 3 termas which are ones: AP> 8, 8, 17, 19, 300, 1991, 2492, 7236, 10586, 34588, 63403, 70637, 1236467, 5417668, 5515697, 5633167, 7458122, 9637848, AP> 9805775, 41840855, 58408380, 213130873, 424342175, 2366457522, 4109464489, 21846713216, 27803071890 }]

What are the first few numerical values?

AP>1 1/(1+8)= AP> 0.11111111111111111111111111111111111111111111111111111111111111111111111111111111

AP>2 1/(1+8/(1+8))=

AP> 0.52941176470588235294117647058823529411764705882352941176470588235294117647058823

AP>3 1/(1+8/(1+8/(1+17)))=

AP> 0.15294117647058823529411764705882352941176470588235294117647058823529411764705882

AP>4 1/(1+8/(1+8/(1+17/(1+19))))=

AP> 0.39959432048681541582150101419878296146044624746450304259634888438133874239350913

AP>5 0.15530266249781523702250791369700736022371972889518963742644631309109976113258113

AP>6 0.38741752625220704395502276740079918223213455998513149335563609329987919338351454

AP>7 0.15715640008568141104556090787785655339875627129762715864478527279160940631483874

AP>8 0.38350077630687712559115823752278956970346898888213408211259160123091558402064981

AP>....

Is it converging? What is it converging to? AP>....

AP>293 0.15926600383181364385083486963661663550819115460698528732118270187985569813520793

AP>294 0.38058428457266237156263916571038194516158517253665616696154895943399980536463022

AP>295 0.15926600383181364385083486963661663550819115460698528732118270188914429526816027

AP>296 0.38058428457266237156263916571038194516158517253665616694642107026071715503000734

AP>297 0.15926600383181364385083486963661663550819115460698528732118270189616592367456096

AP>....

AP>So those two above values I call accumulation points.

AP>Same happens (with other values for accumulation points) for Champernowne constant and Thue-Morse constant

AP>but when (it looks like always) Engel coefficients (of some number, which is uniquely corresponding to those coefficients)

AP>could be generatable via alorithmic (only close ?) formula - then I am getting full convergence (instead of two accumulation points).

AP> I don't have answers for the rest of your questions - sorry

I have two more questions, then. In general, let
 * $$h_n=\frac{1}{c_1}+\frac{1}{c_1 c_2}+ \cdots + \frac{1}{c_1 c_2 \cdots c_n}$$

be an Engel expansion, with the c_k being integers. Then consider the continued fraction
 * $$g_n=\frac{1}{1+\frac{c_1}{1+\frac{c_2}{1+\cdots}}} =

[0;1,1,1,1,\cdots, 1; 1; c_1, c_2, \cdots, c_n]$$

Then my four questions are:
 * Does the equation $$h_n=g_n$$ have any solutions (i.e. integer solutions, where the c_k are integers)? Clearly, for n=2, we have
 * $$c_2 = \frac {c_1+1}{c_1 -1}$$
 * which cannot have any (positive) integer solutions. Are there any solutions for higher n?


 * Do you have examples of such solutions?
 * Does the equation
 * $$\lim_{n\to\infty} h_n = \lim_{n\to\infty} g_n$$
 * have any integer solutions?


 * Do you have any examples of these?

AP> I do not think that we are seeking equality between h_n and g_n

AP> Such equality (I think) is not achievable at any n.

AP>Perhaps that instead you are really thinking in the lines of what wikipedia (on Engel expansion) says:

"Kraaikamp and Wu (2004) observe that an Engel expansion can also be written as an ascending variant of a continued fraction: They claim that ascending continued fractions such as this have been studied as early as Fibonacci's Liber Abaci (1202). This claim appears to refer to Fibonacci's compound fraction notation in which a sequence of numerators and denominators sharing the same fraction bar represents an ascending continued fraction"

AP>BTW, who wrote above comment ? —Preceding unsigned comment added by 63.241.174.129 (talk) 16:20, 17 April 2008 (UTC)

Four more questions: same as above, but for p-adics?

linas (talk) 14:35, 17 April 2008 (UTC)


 * Seems to me that any geometrically increasing series of integers will have accumulation points like this.

>AP Thanks !

>AP Is above statement of yours described someplace for "common" (like wikipedia, etc) consumption ?


 * No I'm just guessing. Just write down, for example b_n = z^n and plug it in, and make the approximation that z^n >> 1 and you get a sum that converges quickly. Add to that the well-known fact that alternate terms of a continued fraction lie above and below the convergent point, and that seems to describe most of what you are seeing.

>AP I presume that at least it is described in the special literature (would you have reference to it ?) ?


 * Have you read at least Khinchin's "Continued fractins"? Its thin, its a dover reprint, its cheap, you can probably find it for a few dollars.

>AP Would it be in "Continued fractions" related special literature (where it applies)?

>AP What is known on how to calculate those values analytically (I presume it should be function of original integer and

>AP the "effective" "factor" of the "geometrical progression" (and/or "effective" "exponential growth degree" ?) ?


 * Not that I know of. Relating continued fractions to anything analytic is quite very hard. It has something to do with the Minkowski question mark function, I think if that were controlled, the rest would be a lot easier.

>AP ... and how this fact could be "brought back" to the "original number", from which such

>AP "geometrically increasing series of integers" was generated from - as Engel's expansion coefficients ?


 * I seriously doubt there's any magic formula to do this; I thought perhaps you had discoveredevidence for this, but if I understand you properly, you don't have any evidence for this?

>AP Could this serve as the qualitive measure of that "original number" irrationality ?


 * Unless you have some fancy theory, I don't see how. You'd need to discover some special cases, and see if they generalize.

> Is there something special about these two particular accumulation points?


 * I can't imagine any reason why the accumulation points should be special. Other than that, for ordinary continued fractions, they would both eventually converge to the same point. I guess if your b_n series does not grow rapidly enough, perhaps these two points could stay apart indefinitely. But that's just a guess.

>AP probably not, except that 4 digits of the "smaller" acummulation point (1592) are "strangely enough"

>AP coincide with digits 3,4,5,6 of Pi-3.


 * 4-digit accidental overlaps are extremely common. If you had a 10 or 20 digit lineup, that might be odd. Just play with Plouffe's inverter, you'll get tens of thousands of strings with those digits in them. To publish something interesting, you'd need to find 100  or 1000 digit accidental overlaps; that would attract attention.


 * You should also spend some time learnig how to indent properly in wikipedia. linas (talk) 04:24, 18 April 2008 (UTC)

I don't recognize their values as being anything in particular.

linas (talk) 18:00, 17 April 2008 (UTC)

Apovolot (talk)apovolot

>AP I am guessing the opposite ...

>AP for "constants"->Engel coefficients->CFAccumulationPts

>AP - the data has no evidence in converging between 2 pts

>AP and the reason for such non-convergence is exactly

>AP the question I sought you might have the answer for ?


 * Please look at the mathworld article, formula 18. Divide both sides by Q_n Q_{n-1}. That formula gives an exact value for the numerical values you are seeing. As long as the product of the b_k is always larger than Q_n Q_{n-1} then the continued fraction will not converge. Whether it *will* converge to a pair of points, that is not clear to me.  It might bounce around without converging, e.g. by eventually wandering off to infinity. If it does wander off to infinity, after a period where it tries to converge, that might be interesting... linas (talk) 17:08, 18 April 2008 (UTC)


 * See also eqns 43, 44. linas (talk) 17:20, 18 April 2008 (UTC)

WikiProject History of Science newsletter : Issue IV - May 2008
A new May 2008 issue of the WikiProject History of Science newsletter is hot off the virtual presses. Please feel free to make corrections or add news about any project-related content you've been working on. You're receiving this because you are a participant in the History of Science WikiProject. You may read the newsletter or unsubscribe from this notification by following the link. Yours in discourse--ragesoss (talk) 23:10, 2 May 2008 (UTC)

Dyadic monoid and modular group article
Fabulous article - thank you!

I'd certainly enjoy reading anything more you add to it - it's provided me with a real education!

Is the definition of the dyadic group on the Modular_group page not quite right, as it does not appear to allow T as an element of the monoid?

All the best!

Julian Gilbey (talk) 23:59, 25 May 2008 (UTC)


 * Thank you for the complement. Its the "dyadic monoid", not group, and that's a subtle but important point. Also, there are many ways in which it can be embedded into the modular group, and this too can be a source of confusion. Perhaps one "easy" way to understand this is outlined at http://www.linas.org/math/rotations.pdf Scroll to the end of that (rather long, but mostly simple) text, and look at the graphs. The upper half-plane, divided into fundamental domains, is represented by the three binary trees joined together. The dyadic monoid can be associated with just one single binary tree -- but its your pick as to which of the infinite possibilities this is. linas (talk) 14:41, 27 May 2008 (UTC)

Minkowski question mark function
Labas, Linai. Can You please add an additional reference to the reference list concerning Minkowski question mark function? Here I've tried to collect as exhaustive bibliography as possible. Thanks! Labai ačiū, ir sėkmės. Giedrius Alkauskas http://www.maths.nott.ac.uk/personal/pmxga2/minkowski.htm —Preceding unsigned comment added by 128.243.220.41 (talk) 15:00, 29 May 2008 (UTC)


 * Labas,
 * I have seen very little in the way of literature talking about it. Most everything I know about it is stuff I've had to work out for myself.


 * Three or four references fly to mind. John Conway's "On Numbers and Games", has a section in chapter 2 or 3 that discuses this function from his "Dedekind cut"-like approach to defining numbers. (he lays a vaguely ZFC-like axiomatic basis for numbers so covering alternate number systems is where he spends a fair amount of energy)


 * Another ref is Dieter H. Mayer. Ergodic Theory, Symbolic Dynamics and Hyperbolic Spaces, chapter 7 - Continued fractions and related transformations, pages 175–222. Oxford University Press, 1991. where he claims without proof, handwaving his way through, that the measure for the one-dimensional Ising model is the question-mark function. I had to read this paper over and over carefully before I realized that he wasn't able to provide a proof, and that he was just suggesting a connection.  I think he's wrong, I tried real hard to find the formal connection, to finish what he didn't, but was unable to do so. This is detailed in my own (unfinished, and not-polished, not-yet-edited-for-full-correctness) paper: "Modular fractal measures" at http://linas.org/math/fdist.pdf   I am very interested in defining d?(x)/dx as I think this will provide an important tool for analyzing the GKW operator, and I think its a tool that will be applicable to the Riemann hypothesis. My conjecture is that the zeroes of the Riemann zeta are the eigenvalues of the Ruelle-Frobenius-Perron operator version of d?(x)/dx, properly formulated.  So I think this is a critical object of study.


 * I saw something by Mandelbrot, written by him circa 2001-2004 (i.e. his current, latest work) about fractal measures, where he drew pictures of what looked like d?(x)/dx but he seemed to be utterly unaware of its connection to the ?(x) and so that surprised me. Actually, there seems to be a lot of literature on "fractal measures" that seems to be unaware of any connection to ?(x).


 * You should write to Lieven Le Bruyn, he may have something to add; maybe? So would Lagerias, as someone who is really really smart *and* has written a lot about RH :-)  Also, ask Philippe Flajolet, Philippe also knows RH and senses some connection to ?(x).


 * I have before me a paper "Continued fractions, Modular Symbols and non-commutative geometry" by Yuri Manin and Mathilde Marcoli. I haven't read it. It does talk about the GKW operator. They may be on to something as well. More than once, when I tried to pursue some line of reasoning about ?(x), I found myself reading about quantum groups.  There's something about the Yang-Baxter equations that somehow connects into all of this, but I don't know what. I guess the tie is through the Ising model, but this may be somewhat hallucinatory, it goes back to Dieter Meyer's unfounded claims and the fact that the resulting functions look similar but are not the same (one is a "trivial" by-product of the mapping of the state space to the real number line).


 * If I may blather like an idiot ... the Berry conjecture is that the zeroes of RH are the eigenvalues of some suitable quantization of the operator px where p is momentum and x is position; the 1/2+iy axis follows from the Heisenberg uncertainty principle [p,x] = 1. (I think he's right) I think the needed quantization has to have d?(x)/dx as the thing connecting position to momentum: the position is in dyadic fractions, while the momentum is in continued fractions, or something like that. So ... how to perform this quantization? one way is to try to find a hyperbolic analogue of the simple harmonic oscillator. One needs a state space and so one needs to look at dynamical flows on some small dimensional manifold, and this manifold has to be hyperbolic, in order to get the connection to modular forms and chaotic systems and etc. But none of the usual, simple hyperbolic manifolds seem to offer the right ingredients.  Most modular forms have eigenvalue spectrums that have one or two discrete eigenvalues, and then a continuous spectrum starting at some low value like 1 or 2. See for example "spectral methods of automorphic forms," H. Iwaniec. This is utterly unlike the Riemann zeros, and I can't see how to cure it, so this seems like a dead end.


 * So instead, one looks for a Hilbert space with appropriate properties. One needs self similarity and shift invariance. The simplest such space is just the strings of binary digits, and the most basic problem posed on strings of binary digits is the Ising model. So now one has a measure, and one has a Hilbert space. The dieter meyer angle shows how ?(x) seems to somehow "naturally" occur on this space. But its not the whole story; there's something wrong. The shift operator is the momentum p, but its not enough. So one wants to propose a mapping from the binary strings to the farey fractions; this is ?(x). Naively, this is d?(x)/dx and so the Berry conjecture becomes "the quantized version of xd?(x)/dx gives the zeroes of RH". This seems to have the correct spectral properties, but I get sort of stuck right around there. I really really wish I had more time to think about this .. its exciting.


 * That's all that I've got ... Sekmes! linas (talk) 17:26, 29 May 2008 (UTC)

List of Baryons
I'm currently in the process of getting the list of baryons to be a featured list. I see you have a Ph.D in particle physics, so I thought you might want to help.

So far I've detailed all known and predicted JP =$10/11$+ and $10/11$+, as well as all reported pentaquarks, fully referenced through the PDG Review of Particle Physics 2006 and some other articles (more recent = better, new data).

I've also given an explanation of concepts such as isospin, spin, flavour quantum numbers, and the rules of particle classification, and I believe I've covered these topic accurately. However, I am not a particle physicist, and considering the list of baryons sums what I've self-taught myself in a month and a half using pretty much only the PDG Review and a cryptic handout (I couldn't find a single book that could explain isospin or lie algebra starting from a fundamental reality or a fundamental concept, so I threw them all out and did it myself), it could use some fact checking and a few references (if you have some).

Comments and feedback are appreciated. Headbomb (ταλκ · κοντριβς) 06:59, 30 May 2008 (UTC)


 * I replied on the talk page there. linas (talk) 02:51, 31 May 2008 (UTC)

Just so you know, I've given the list an overhaul (expanded overview, with parity, fixed typos and inconsistencies, explained what symmetry meant, and generally took your comments into account). I believe it to be more factually accurate. Reply here or talk page.Headbomb (ταλκ · κοντριβς) 23:30, 31 May 2008 (UTC)

Aitken method
Hi, I had some problems in recognizing my contribution to Sequence transformations, and after searching for quite a while, I found what I actually had written, which is here: http://en.wikipedia.org/w/index.php?title=Aitken%27s_delta-squared_process&oldid=81071114

I am really unhappy about what has happened to my article, a bit too much "cleanup" for my taste. I think this method merits a page on its own, going beyond what I created, mentioning its inventor, references, analysis, explanations, examples, counter-examples. IMHO it is a very bad idea to stuff more than 2-3 lines about in into a section in Sequence transformations. There should be a link "see main article..." and just a very brief summary. Apart from that, I prefer not writing out what I think about stuff like "defined by the mapping A : (s_n,s_{n+1},s_{n+2}) \to s'_n  with" which is complete nonsense.&mdash; MFH:Talk 13:28, 2 June 2008 (UTC)

Minimum polynomial extrapolation
This is an automated message from CorenSearchBot. I have performed a web search with the contents of Minimum polynomial extrapolation, and it appears to include a substantial copy of http://www.destinationscience.com/search/Sequence_transformations. For legal reasons, we cannot accept copyrighted text or images borrowed from other web sites or printed material; such additions will be deleted. You may use external websites as a source of information, but not as a source of sentences.

This message was placed automatically, and it is possible that the bot is confused and found similarity where none actually exists. If that is the case, you can remove the tag from the article and it would be appreciated if you could drop a note on the maintainer's talk page. CorenSearchBot (talk) 18:08, 2 June 2008 (UTC)


 * Pfft. the identified text is a mirror of WP. linas (talk) 18:09, 2 June 2008 (UTC)

WikiProject Physics participation
You received this message because your were on the old list of WikiProject Physics participants.

On 2008-06-25, the WikiProject Physics participant list was rewritten from scratch as a way to remove all inactive participants, and to facilitate the coordination of WikiProject Physics efforts. The list now contains more information, is easier to browse, is visually more appealing, and will be maintained up to date.

If you still are an active participant of WikiProject Physics, please add yourself to the current list of WikiProject Physics participants. Headbomb {ταλκ – WP Physics: PotW} 15:49, 25 June 2008 (UTC)


 * Hey there. I hope you don't mind me bargin' in Wikiproject Physics all guns blazing, going from rewriting entire sections to reformatting the main page. Headbomb {ταλκ – WP Physics: PotW} 01:52, 26 June 2008 (UTC)
 * No problem. Its been re-written at least several times since I started. As long as, ahem, you've taken the Hippocratic Oath: "First, do no harm...". linas (talk) 01:57, 26 June 2008 (UTC)