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User talk:AJim

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Re: Faraday effect diagram, you're right about the direction of rotation (I checked in Hecht). I drew it that way because it was easier, not for any good reason. Actually I'm not too happy with how the diagram came out (I don't think it shows the rotation very well),so I'll try altering it and see how it looks. Oh, and welcome to Wikipedia! -- DrBob 18:59, 12 Mar 2004 (UTC)

Thanks, this is a lot of fun for me. I like the look of the diagram, especially the color and shading. I would have an easier time if it could show the direction of propagation. For my money, showing the sense of the current flow, rather than the coil itself, as Hecht does, would make it easier to apply the right hand rule to get the B field. Am I right in thinking that the polarizers Hecht shows are essential to the isolator design? That the 45 degree rotation (he implies) means that the reflected wave will be rotated 90 degrees and thus be blocked by the entrance polarizer? The microwave isolator also includes an absorber for the relected wave. In the optical isolator is the dissipation achieved with a dichroic polarizer? AJim 21:13, 12 Mar 2004 (UTC)

 * I'll see what I can do with adding the propagation direction to the diagram - I might make it more like Hecht's diagram to make the polarization direction clearer. I think the coil would obscure things too much, though. Yes, polarizers are necessary for optical isolators, usually polarizing beam splitters on either end. They have adjustments to fiddle with the magnet spacing to get exactly 45° rotation at the wavelength you want. -- DrBob 17:56, 15 Mar 2004 (UTC)

It does not make so pretty a picture, but what I drew to check myself was an end-on view, looking into the source, no coil, just I circulating ccw. The right-hand rule then said that B was in the propagation direction. Then the rotation had to be ccw also from the definition. Perhaps the end-on picture might be a supplement? There is something mnemonic about the current and the rotation being in the same sense anyway. I found the coil distracting myself. I think there is a potential for confusion by introducing the coil helix sense. The real fun, for me, is that I now understand Faraday isolators better. Thank you for that. AJim 22:57, 16 Mar 2004 (UTC)

Thanks for the compliment :) And you're right, that Sellmeier graph should be micrometres rather than nanometres along the x-axis. I've uploaded a corrected version. Thanks for spotting that. -- DrBob 16:15, 18 Mar 2004 (UTC)

Tip (re [[wild pointer). If you're writing code blocks, you don't need to use pre and nowiki wrappers. If you start a line with a single, leading space, you get the effect (edit this page to see the markup) int main(void) {   return 0; }

which has the advantage of looking much easier to read in the markup. By the way, K&R-style function declarations might look nice, but we might want to stick to ANSI C ;)

HTH Dysprosia 06:17, 22 Sep 2004 (UTC)

Pageboy
At Page Boy (which I've renamed to the apparently more standard Pageboy), you say:
 * The Page Boy is a hair style named after a drawing of a woman dressed as an English page boy.

I'm wondering what your source for this is. Do you mean a particular drawing by a particular artist? Also, as far as I know, the style was actually popularized in the 1920s by the actress Louise Brooks. I would expand the article a bit, but I'd like to drop the reference to the drawing unless there's a source. - dcljr 18:40, 23 Mar 2005 (UTC)

Cotton effect
Hi there! You seem to be the guy to ask. Can you do anything with Cotton effect and with Optical rotatory dispersion? I have just added them as stubs. Jeff Knaggs|Talk 11:08, 20 October 2005 (UTC)

Deuterium arc lamp update
Hello, I've justed started editing Wikipedia and I've started by expanding the stub on deuterium arc lighting. I've noticed that you have made some contributions to the Xenon arc lamp page and I was wondering if you might be able to give me some advice on things I could add, change, etc. Thanks!

eggles37 —Preceding unsigned comment added by Eggles37 (talk • contribs) 00:17, 8 October 2007 (UTC)

Sheeran
I deleted the reference on the Sheeran page in response to a single purpose account linked to the book's publisher who was spamming links to a website promoting the book. If I remember correctly, the link is pretty promotional, but if you think it should be in, it's unlikely I'll visit the page or try to delete it again. --Steven J. Anderson (talk) 08:08, 24 December 2008 (UTC)

 * Maybe I am confused. I think what was removed and put back was a simple citation of the book, not the link you mention. In fact, I think the link was, and still is, there. I agree the link is pretty promotional. I think the book ought to be cited as the source of the information about Sheeran. I do not think referencing the book gives undue weight. --AJim (talk) 08:26, 24 December 2008 (UTC)
 * Without looking closely at what I did, I'll say you're probably right. I was kinda ticked at the SPA when I was reverting him and might have gotten a little over-exuberant. Apologies. --Steven J. Anderson (talk) 01:29, 25 December 2008 (UTC)

Re: question about "fix structure"
Oh my gosh, a user from 2004? I'm not worthy... :)

I've only seen section headings with 2d level, 3d level and so forth, not 1st level. I've read plenty of Wikipedia articles, and that's what I always see. When I saw your article, I said to myself "This doesn't look right" and I changed it. I did not intend to change anything substantive, only to reduce 1st-level headings to 2d-level, and 2d-level headings to 3d-level. I hope I did more good than harm. Crystal whacker (talk) 05:35, 2 January 2009 (UTC)

Lead poisoning
I replied at Wikipedia_talk:WikiProject_Medicine/Dermatology_task_force. Thank you for your question. kilbad (talk) 02:33, 3 February 2009 (UTC)

Cary Instruments
Hi. I saw your notes on Talk:Cary Instruments. I started the article, and wrote from my experiences working in a lab that used a Cary-60 (we also had a Beckman Model-E, but that's another story). It sounds like you know more about these instruments than I do, and I encourage you to improve the article, especially if you can find reliable, citable references. -- RoySmith (talk) 20:42, 6 February 2009 (UTC)

Proposed deletion of Unittest

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Speedy deletion nomination of Gunnar Berndtson
A tag has been placed on Gunnar Berndtson requesting that it be speedily deleted from Wikipedia. This has been done under section A7 of the criteria for speedy deletion, because the article appears to be about a person or group of people, but it does not indicate how or why the subject is important or significant: that is, why an article about that subject should be included in an encyclopedia. Under the criteria for speedy deletion, such articles may be deleted at any time. Please see the guidelines for what is generally accepted as notable, as well as our subject-specific notability guideline for biographies. You may also wish to consider using a Wizard to help you create articles - see the Article Wizard.

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Artificial altitude
Hi, AJim. I went to the pages you linked to at the See Also section of Altitude training. None of the applications of those technologies involved people breathing the nitrogen-enriched air. They did not sound like appropriate links. Do you want to research people breathing their output? —hike395 (talk) 23:48, 16 June 2010 (UTC)

notes to myself
Combinatorial_number_system, first step of Gosper's Hack is extract rightmost bit, 1972 HACKMEM 175 uses it. Therefore the rightmost bit trick is older than that. But I knew that, since Eric Jensen taught it to me around 1969. --AJim (talk) 22:13, 6 June 2011 (UTC)

having notes in an article as well as references: e.g. place in the text, then: ==Notes== Some sources gives... --AJim (talk) 03:11, 14 July 2011 (UTC)

Book references at Switching circuit theory
Thanks! --Wtshymanski (talk) 18:17, 23 June 2011 (UTC)

Fessenden oscillator
Hi there. Is there a chance your group can get a picture of the Fessenden hydrophone you are working on? One picture would be worth many words in the article. --Wtshymanski (talk) 17:59, 28 June 2011 (UTC)
 * At the present time the supposed transducer is housed in a vertical cylinder installed near the midline on the inner hull in the forward magazines section of the ship. Access is difficult. The cylinder has a mechanism on top that appears to allow height adjustment of something inside. This cylinder has not been opened, to anyone's knowledge, in many years. It is not clear if the top of the cylinder is above the water line. We are very curious. We are proceeding cautiously. We have no current photo.

 * In the meantime, there appears to be a historic photo available from the NOAA Photo Library: Reginald Fessenden and his electric oscillator. From the NOAA Library about page it appears that this photo is available for use. The photo shows an object of the approximate size and shape expected from the patent description and the article that this picture is used in. My problem is that I do not have the time to read and try to cope with the current Wikipedia image upload rules. If you are familiar with the rules, and can navigate them(!), and think the image really is available, please go ahead and add it. The photo caption says the source is: "Submarine Signaling," Scientific American Supplement, No. 2071, pp. 168-170, Sept. 11, 1915.

 * By the way, the NOAA article is an excellent source for other references. The article also includes, for instance, a transcript of the Report of Captain J.H. Quinan of the U.S.R.C Miami on the sea trials conducted on 27 April 1914, as reported in the Hydrographic Office Bulletin of May 13, 1914.

 * The key patent, which describes the construction of the transducer, is, "Dynamo-Electric Machinery" – issued 4 January, 1916 (original application filed on 29 Jan 1913). It would also be useful to add Figure 1 from the patent to the article, or possibly a cleaned-up version. Again, I am not prepared to cope with this. However, the operation section of the current article has not yet captured some of the key design details, such as the opposed armature windings used to drive the copper tube in the opposite air gaps, and the claimed advantages of this design. It would be helpful to show the field orientations to help make it clear that the opposed windings ultimately produce forces that add. I can confirm that the net inductance of the two armature windings nearly cancels (< 1 mH, compared to 670 mH for the field winding, which is about the same resistance and apparent wire size).

--AJim (talk) 23:05, 28 June 2011 (UTC)

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Circular polarization
My understanding is that "mixed polarization" as described above is not allowed by the FCC -- only either Horizontal, Circular or Elliptical. Slant polarization, with H and V at different powers but in phase, is not allowed for FM or TV. See: CFR 47 § 73.316. I believe the entire paragraph should be removed.

-- algocu (talk) 19:01, 10 September 2012 (UTC)

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CD figure
Tables are boring. I put some colours ;) Check it out and tell me what do you think. Masur (talk) 08:08, 28 November 2014 (UTC)

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Thank for your advice,you help me to become a better wikipedian and let me know how to better deliver and distribute knowledge!I will add several lines on the photo you confuse about to show how light goes throw it.The change will happen about a week after,because presently I am not convenient for making change on wiki(I am in China,the tech equipments here are not properly set up).Thank you again! Warren Leywon (talk) 08:20, 27 March 2017 (UTC)

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title: User talk:AJim


text:
Talk:Almost complex manifold

Incorrect statement
"An ordinary almost complex structure is a choice of a half-dimensional subspace of each fiber of the complexified tangent bundle TM." -- This is not correct.

i somewhat dislike this sentence: "equipped with a structure that defines a multiplication by i on each tangent space". the almost complex structure is an object defined on a real manifold, whose tangent spaces are real vector spaces. what meaning does multiplication by i have on such a space? perhaps it is a good description to give an intuitive understanding, perhaps it just needs a qualifier?Lethe

also, i am starting to think that it may be better if almost complex structure and almost complex manifold were separate articles. what do you think? (you=whoever is reading this)Lethe

I was just trying to give a informal introductory sentence or two on what an almost complex structure is before defining it. Technically, the introduction of an almost complex structure, does allow for multiplication by i on TpM via
 * i v = Jpv

This turns Tp from a 2n dimensional real vector space into a n dimensional complex vector space.

At this point, I don't see a reason to separate the manifold page from the structure that defines it. It seems the two go hand in hand. Perhaps if the article gets overly large. -- Fropuff 20:32, 2004 Mar 12 (UTC)

I believe that the discussion of almost complex structure should be replaced with a link to the "almost complex" page. That page is well-developed and the discussion here is now superfluous. Lost-n-translation (talk) 23:26, 1 March 2010 (UTC)

Complex/Almost complex
This part seems to be missleading, instead of talking about ... for complex structure on an almost complex mnfld you did just test for almost complex structure to be non complex

I'm not a specialist, so can not fix it correctly but maybe you can, or at least clearify it.

''existence of an almost complex structure on a real manifold is a necessary, but not sufficient, condition for the existence of a complex structure. The obstruction to the existence of a complex structure can be codified in a rank (1,2) tensor, called the Nijenhuis tensor.''

again it is not an obstruction, one can take a complex structure and deform it locally and get an almost complex and it should be noted that not any almcomplex mnfld are of that type

Also it showld be a couple of words on simplectic-Riemannian connection with use of almost complex structure...

Tosha 01:30, 13 Mar 2004 (UTC)

 * In the formal definition section, the following appeared:
 * dω=0 if and only if ∇J=0
 * I'm not sure what the gradient of an almost-complex structure is, so I'm moving it here (after writing the requested subsuming section on compatible triples) hoping someone would see this and explain. Orthografer 03:14, 11 October 2006 (UTC)

Tosha - your text makes it seem like vanishing Nijenhuis tensor is the definition of an integrable almost complex structure. i think a more appropriate definition is closure of the Lie bracket on the space of holomorphic vector fields. this is what the word integrable means. then the vanishing of the Nijenhuis tensor is a useful theorem about integrable almost complex structures, rather than a definition. in your text, you seem to want to give two different definitions of integrable.

i think the most clear way to convey the situation is to state these equivalent conditions: 1. an almost complex manifold admits a complex structure (which is then uniquely determined by the almost complex structure), 2. the Nijenhuis tensor vanishes, 3. the almost complex structure is integrable (the Lie bracket closes on holomorphic vectors) 4. the almost complex structure is covariantly constant

(i think sometimes 4 is taken as the definition of integrable instead of 3)

also, you deleted my comment about how the existence of a J tensor implies that the tangent space (and therefore also the manifold) must be even dimensional. - Lethe

I partly agree, change it if you want, it is not exacly my text, I just compactified an older one. For me integrable means that it defines complex structure, I think it should be the very base definition and it stated in the first par. of the subsection Tosha 15:20, 13 May 2004 (UTC)

I tend to agree Lethe. We should be sure of what the word integrable actually means in this context. Although, I think point (4) in your list should be excluded. It applies only after the manifold is equipped with a metric. -- Fropuff 16:31, 2004 May 13 (UTC)

I think integrable used to mean the integrability condition - but a long time ago (eg Weil's book on Kahler manifolds, from the 1950s). So perhaps now things have moved on. Charles Matthews 16:54, 13 May 2004 (UTC)

I am a little uncomfortable with the phrase "linear map J: TM→TM". a tangent bundle is not a vector space. it shouldn't have linear maps. maybe it should read "bundle morphism" instead, or else we should just say "restricts to linear map Jp: TpM→TpM at each point p in M"?Lethe

OK, i have added some stuff:

 * complex structure on a vector space. i was thinking about having this for a while, this is why i thought it might be a good idea to separate complex structures from complex manifolds, in a discussion with Fropuff at the top of this page.  because you can have a complex structure on other spaces than manifolds.  but there isn't really enough about complex structures on vector spaces alone to warrant its own article, plus that has limited (none?) applicability outside of complex manifolds.

 * i changed the stuff about J being a linear map to it being a bundle morphism, as per my above comment.

 * i tried to make clear in precisely what sense J can be thought of as "multiplication by i". i.e. only on the holomorphic sector of the complexified space, not on the antiholomorphic sector, nor on the underlying real space.

 * i added a section called "differential topology". right now this section is kinda rough, but i actually have to get some work done today, so i will have to come back later.  i don't feel like my explanation of $$\Omega^{m,n}(M)$$ is very lucid.  nor my explanation of the Doubeault operators.

 * i added another equivalent condition to the list of conditions which determine when an almost complex structure is induced by a complex structure.   namely,  $$d=\partial+\overline{\partial}$$.  this implies that $$\partial^2=0$$ and so the almost complex structure is integrable.

 * the problem with my explanation of the Doubeault operators is that it assumes an integrable almost complex structure (so that $$d=\partial+\overline{\partial}$$), whereas the Doubeault operators can be defined for a nonintegrable almost complex structure. my definition, in addition to being unclear, assumes (without stating) integrability.  this should be fixed.

Holomorphic?
The section on differential topology defines holomorphic and antiholomorphic vector fields. Holomorphic vector bundles are usually defined only over complex manifolds. It seems wrong to me to call TM+ a holomorphic vector bundle when M is only an almost complex manifold. In what sense in the projection map TM+ → M holomorphic? Is this naming standard? -- Fropuff 07:05, 2005 Feb 27 (UTC)

 * An almost differential structure is all that is required to define a locally holomorphic vector field. It's built in to the definition. Lethe | Talk 08:58, Mar 1, 2005 (UTC)

I know these vector fields exist (although I'm not sure what you mean by a locally holomorphic vector field). My concern is whether or not they should be called holomorphic/antiholomorphic unless J is integrable. Better to call them vector fields of type (1,0) and type (0,1) or something similiar. I have found one author who calls them J-holomorphic vector fields. -- Fropuff Fropuff 02:38, 2005 Mar 3 (UTC)
 * 1) I guess what I meant by "locally holomorphic vector field" was redundant. I just meant an eigenvector of J.
 * 2) My previous reply was kinda drunken. Isn't there a rule about using wikipedia while drunk?
 * 3) I'm not averse to a different choice of definition of holomorphicity. But I can provide references which use the convention in this article, I believe.  Let's take a survey of sources.
 * 4) I like that you moved the linear complex structure to a different article. That's actually what I had in mind a year ago, about splitting the article, which you advised against, I think because I didn't state very clearly that that's what I had in mind.
 * So, for a year now, I've left this article in a half finished state, including knowingly leaving false information! Shame on me!  I should get back to work on my wikipedia tasks, I've become lazy about those.  I think I had more momentum to write wikipedia articles when I was new.  But that's a pretty lousy excuse.
 * 1) I think I'm going to have trouble tracking down one of the books that I was using as a reference when I wrote some of this stuff. There are plenty of other books though.  I guess I'll be OK.
 * -Lethe | Talk 02:51, Mar 3, 2005 (UTC)

[I must confess to drinking and editing myself — funny how it all seemed so clear at the time.] Of the sources I've checked (mostly symplectic topology books) I've only seen Nakahara call these fields holomorphic, and I've never really trusted that book when it comes to rigor. I've been thinking about almost complex structures lately so I may do some more work on this article. I'll try to fix the problems you mentioned above (assuming you don't get around to it first). -- Fropuff 03:36, 2005 Mar 3 (UTC)

what does integrable mean?
I'm currently fixing this article (about time). But I'm having a problem. I think we've discussed this in the past, but we have to decide what "integrable" actually means. Right now, the article states that an almost complex structure is integrable if it arises from a bona fide complex structure, a view that Tosha was advocating a long time ago. But with that convention, the Newlander-Nirenberg theorem becomes tautologous sounding (an almost complex structure is integrable iff it is integrable). I've seen different books use different definitions of integrable: Nijenhuis vanishes, Lie bracket closes, exterior derivative decomposes. I propose that the first of those be taken as the def. -Lethe | Talk 20:41, July 14, 2005 (UTC)

 * I strongly advocate returning to the definition of integrable that I had before. That is, an almost complex structure is integrable iff one can find local holomorphic coordinates around every point. The almost complex structure then arises from an honest complex structure. It is a simple and intuitive definition which captures the essence of what it means to be integrable: an almost complex structure is integrable iff it comes from an honest complex structure. One doesn't need any extra constructions or definitions.

 * The constructions like the Nijenhuis tensor are superfluous as far as the basic idea is concerned. Trying to define integrability in terms of this tensor and then claiming (without proof) that the vanishing of this tensor guarentees the existence of an honest complex structure seems very circuitous and unnecessarily complicated. The Newlander-Nirenberg theorem states that J is integrable (i.e. local holomorphic coordinates exist) if and only if the Nijenhuis tensor vanishes. This to me is very non-obvious statement and not the least bit tautological.

 * Take an analogy with symplectic manifolds. Every vector space with a nondegenerate 2-form is a symplectic vector space. However, a manifold with a global nondegenerate 2-form ω is not always a symplectic manifold, but rather an almost symplectic manifold. Every almost symplectic manifold looks like a symplectic vector space at a point (i.e. the tangent space is symplectic). But in order to be an honest symplectic manifold ω must be closed. One can't determine if ω is closed merely by looking at it at a point, one must look at ω in a neighborhood of that point. This is the essence of integrability: if one can move from the infinitesimal description to the local description. An almost complex manifold looks like a complex vector space at a point. It is a complex manifold iff it looks like a complex vector space in every neighborhood of that point. This is integrability.

 * -- Fropuff 22:57, 14 July 2005 (UTC)

 * Fropuff, I agree that the weird looking constructions like the Nijenhuis tensor hide what it means to be integrable. The reason for my change is that I'm having a hard time stating the N-N theorem without having some local differential definition of integrability.  That's what makes N-N a deep theorem, right?  The local differential equation yields global equivalence to a complex structure? -Lethe

 * I'm not entirely sure what you mean. The construction of introducing complex analytic coordinates is a purely local one. The N-N theorem should by stated as
 * An almost complex structure is integrable if and only if the Nijenhuis tensor vanishes.
 * I should point out that Newlander & Nirenberg defined integrability in their original article just as I have: in terms of the existence of local holomorphic coordinates (see the first paragraph of that article). See also McDuff's book Introduction to Symplectic Topology (searchable on Amazon). Again both integrability and the statement of N-N are as I propose. -- Fropuff 04:30, 15 July 2005 (UTC)

 * OK, I could get on board. -Lethe | Talk 17:34, July 15, 2005 (UTC)

This doesn't seem right
This statement:
 * "Any even dimensional vector space always admits a complex structure. Therefore an even dimensional manifold always locally admits a (1,1) rank tensor (which is just a linear transformation on each tangent space) such that Jp2 = −1 at each point p. Only when this local tensor can be patched together to be defined globally does the almost complex structure yield a complex structure, which is then uniquely determined."

doesn't seem right to me. For example, any almost complex manifold has a globally defined J tensor, while it may not be complex. See current revision for a better attempt. It's quite possible that I wrote that wrong statement. -Lethe | Talk 20:48, July 14, 2005 (UTC)

 * That statement is clearly not true. I'm can't figure out why I wrote it. I must have been typing something different than I was actually thinking. Thanks. -- Fropuff 23:04, 14 July 2005 (UTC)
 * I was surprised to see you taking the blame for a mistake which I was sure must be mine, but after careful inspection, I see that it really was in fact me, as expected, who made the egregious statement. So stop taking credit for my blunders! -Lethe | Talk 03:31, July 15, 2005 (UTC)

 * Oh good, I thought I was losing my mind. I guess I checked the history too quickly. Anyway, it doesn't really matter so long as it's fixed. However, the uniqueness part of the statement still bothers me. Even on an vector space the complex structure is not unique (not even up to homotopy). Do you mean something I'm not grasping? -- Fropuff 04:35, 15 July 2005 (UTC)

rewrite
OK, I've rewritten a bunch of stuff, and I believe I have removed all the stuff that was known to be incorrect. -Lethe | Talk 20:49, July 14, 2005 (UTC)

Remaining errors
The page is still not correct. It gives two different definitions of integrable (as the existence of local holomorphic coordinates, and then as the vanishing of the Nijenhuis tensor). I believe that simply removing the sentence "An almost complex structure is called integrable if this tensor vanishes for all smooth vector fields X and Y on M (here [·, ·] denotes the Lie bracket of vector fields)." will solve the problem. Then there is one definition of integrable, followed by the statement of the NN theorem.

As a matter of opinion, I strongly think that integrable should not be defined (as here) as the existence of local holomorphic coordinates. Rather, integrable means that the holomorphic tangent space is closed under the Lie bracket (as Charles Matthews says above). Then the NN theorem states that a manifold with an integrable almost complex structure admits local holomorphic coordinates. -Chris 3 April 2006

deletion of nonexample
prior to my edit, the page claimed that surfaces of general type failing to meet certain restrictions on their chern numbers furnish examples of almost complex manifolds with non-integrable complex structures on their tangent bundles. while it is probably true that one can deform the complex structure on the tangent bundle of such a surface given to you by the definition of complex surfaces to give a nonintegrable structure, that's not something that's going to be detected by the chern classes. bottom line is, the sentence is either flat out nonsense or the editor who wrote it made some sort of error in committing it to writing. since it's unclear what was meant and no one seems to know, seems like it ought to be removed. it would be nice to have some examples of the sort of thing the sentence tried to address, but i'm going to have to leave that to someone else. 140.254.93.115 (talk) 16:33, 15 October 2008 (UTC)

Changes
I made these changes: Given any linear map A on each tangent space of M; i.e., A is a tensor field of rank (1,1), then the Nijenhuis tensor is a tensor field of rank (1,2) given by
 * $$ N_A(X,Y) = -A^2[X,Y]+A([AX,Y]+[X,AY]) -[AX,AY].$$

The individual expressions on the right depend on the choice of the smooth vector fields X and Y, but the left side actually depends only on the pointwise values of X and Y, which is why N_A is a tensor. This is also clear from the component formula
 * $$ (N_A)_{ij}^k=A_i^m\partial_m A^k_j

-A_j^m\partial_mA^k_i-A^k_m(\partial_iA^m_j-\partial_jA^m_i).$$ In terms of the Frolicher-Nijenhuis bracket, which generalizes the Lie bracket of vector fields, the Nijenhuis tensor N_A is just one-half of [A,A].

First of all, the statement about smooth X and Y is not correct. The local values suffice: that is essential for the concept of a tensor. (Too bad, the page on tensors does not deal with this matter.) Second. the text I propose places the N tensor in the context of the F-N bracket. The page on the F-N bracket only casually mentions the relationship. MY10SOR (talk) 18:58, 13 September 2010 (UTC)  MY10SOR  (I am new here, please forgive the intrusion.)
 * The Lie bracket of vector fields is defined in terms of the Lie derivative, so the vector fields have to at least be differentiable. The point the article was making was that it's not immediately obvious that the RHS is tensorial. Its not even obvious that it transforms as a tensor under a change of basis. 67.198.37.16 (talk) 03:09, 8 May 2019 (UTC)

6-sphere
I started a discussion here that relates to this article as well... Franp9am (talk) 20:34, 23 December 2015 (UTC)

Higgs and spontaneous symmetry breaking
The above-mentioned link in the 6-sphere has this (to me) astounding paragraph:
 * A complex structure on a complex manifold can always be re-interpreted as a spontaneously broken classical vacuum solution in a non-linear version of a Yang-Mills-Higgs theory formulated on the underlying manifold. This is because an almost complex tensor field J is mathematically the same as a Higgs field Φ. Although the terminology comes from theoretical physics, the corresponding mathematical structures are well-defined.

I find this to be quite remarkable, and worth a detailed exposition here, or somewhere. I have a good-enough imagination that I can imagine how this might be true (plus I know what Higgs fields are, I'm unclear why J is the Higgs field, instead of being just the classical solution to one, or some patch near the minimum...) But I find that imagination is unsuitable, when something more explicit can be stated. — Preceding unsigned comment added by 67.198.37.16 (talk) 21:44, 7 May 2019 (UTC)

Worked examples needed.
This article could be improved by adding sections on the following: 67.198.37.16 (talk) 02:55, 8 May 2019 (UTC)
 * Examples of almost-complex structures that are not complex. The only one given is S^6 and that one is clearly too complicated, and too ambiguous.
 * A worked example of integrability, and an example where integrability fails.
 * A more precise definition of integrability. I imagine it means "smooth integral curves on a neighborhood" which works only if J is no-where degenerate in the neighborhood and satisfies a few other properties.... But this is just my imagination...
 * Articulation of the compatible triple in these example settings.

Link clarity
In the Jean’s complex manifold solution the reader was sent to explanations about manifolds and complex numbers at first, without much information how does it correspond to real manifolds. In the reader is immediately directed to the word “integrable”. Incnis Mrsi (talk) 11:57, 11 September 2019 (UTC)

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Talk:Almost complex manifold


text:
Gnome sort

Gnome sort (nicknamed stupid sort) is a variation of the insertion sort sorting algorithm that does not use nested loops. Gnome sort was originally proposed by Iranian computer scientist Hamid Sarbazi-Azad (professor of Computer Science and Engineering at Sharif University of Technology) in 2000. The sort was first called stupid sort (not to be confused with bogosort), and then later described by Dick Grune and named gnome sort.

Gnome sort performs at least as many comparisons as insertion sort and has the same asymptotic run time characteristics. Gnome sort works by building a sorted list one element at a time, getting each item to the proper place in a series of swaps. The average running time is O(n2) but tends towards O(n) if the list is initially almost sorted.

Dick Grune described the sorting method with the following story:

"Gnome Sort is based on the technique used by the standard Dutch Garden Gnome (Du.: tuinkabouter).

Here is how a garden gnome sorts a line of flower pots.

Basically, he looks at the flower pot next to him and the previous one; if they are in the right order he steps one pot forward, otherwise, he swaps them and steps one pot backward.

Boundary conditions: if there is no previous pot, he steps forwards; if there is no pot next to him, he is done."

Pseudocode
Here is pseudocode for the gnome sort using a zero-based array:

procedure gnomeSort(a[]): pos := 0 while pos < length(a): if (pos == 0 or a[pos] >= a[pos-1]): pos := pos + 1 else: swap a[pos] and a[pos-1] pos := pos - 1

Example
Given an unsorted array, a = [5, 3, 2, 4], the gnome sort takes the following steps during the while loop. The current position is highlighted in bold and indicated as a value of the variable.

title: Gnome sort


text:
User talk:Fridaeklof

Hello Frida, and welcome to Wikipedia! Nice job on starting Student union and the addition to University of Newcastle upon Tyne. However your addition to Åre seems to have been copied from, and as this may be a copyright infringement the article has been reverted to the previous version. You can either use the information found in the text and write your own version or possibly use the Boilerplate request for permission to acquire permission to use it on Wikipedia. Cheers! -- Mic 19:02, Mar 12, 2004 (UTC)

authors: Template namespace initialisation script

title: User talk:Fridaeklof


text:
Protonation

In chemistry, protonation (or hydronation) is the adding of a proton (or hydron, or hydrogen cation), usually denoted by H+, to an atom, molecule, or ion, forming a conjugate acid. (The complementary process, when a proton is removed from a Brønsted–Lowry acid, is deprotonation.) Some examples include
 * The protonation of water by sulfuric acid:
 * H2SO4 + H2O H3O+ +
 * The protonation of isobutene in the formation of a carbocation:
 * (CH3)2C=CH2 + HBF4 (CH3)3C+ +
 * The protonation of ammonia in the formation of ammonium chloride from ammonia and hydrogen chloride:
 * NH3(g) + HCl(g) → NH4Cl(s)

Protonation is a fundamental chemical reaction and is a step in many stoichiometric and catalytic processes. Some ions and molecules can undergo more than one protonation and are labeled polybasic, which is true of many biological macromolecules. Protonation and deprotonation (removal of a proton) occur in most acid–base reactions; they are the core of most acid–base reaction theories. A Brønsted–Lowry acid is defined as a chemical substance that protonates another substance. Upon protonating a substrate, the mass and the charge of the species each increase by one unit, making it an essential step in certain analytical procedures such as electrospray mass spectrometry. Protonating or deprotonating a molecule or ion can change many other chemical properties, not just the charge and mass, for example solubility, hydrophilicity, reduction potential or oxidation potential, and optical properties can change.

Rates
Protonations are often rapid, partly because of the high mobility of protons in many solvents. The rate of protonation is related to the acidity of the protonating species: protonation by weak acids is slower than protonation of the same base by strong acids. The rates of protonation and deprotonation can be especially slow when protonation induces significant structural changes.

Reversibility and catalysis
Protonation is usually reversible, and the structure and bonding of the conjugate base are normally unchanged on protonation. In some cases, however, protonation induces isomerization, for example cis-alkenes can be converted to trans-alkenes using a catalytic amount of protonating agent. Many enzymes, such as the serine hydrolases, operate by mechanisms that involve reversible protonation of substrates.

title: Protonation


text:
Talk:Zeta (operating system)

I am proud to be listed here in that amazing project. Thx a lot in the name of all members of yellowTAB

Bernd T. Korz CVO yellowTAB GmbH

title: Talk:Zeta (operating system)


text:
Talk:Xanadu (film)

Image
1980? Then why was it not until about 1983 that the movie was released? Maybe I have this wrong.. I had the LP for a year befor ethe movie was released in Texas, --Rofthorax 07:38, 15 August 2005 (UTC)

I removed the image that was attached to the article. It was dismal in quality and had no information as to source or declaration of use. Using such an image is usually fair use, but it was such poor quality and cropped so badly that it was more a blotch than an asset to the article (stub or not). If someone wants to replace it with a decent image (with appropriate sourcing), please do so. —Frecklefoot 19:10, Mar 12, 2004 (UTC)

This article read like a commentary. Scientology and the film genre preferences of the gay are community are generaized. 199.212.129.2 12:54, 30 December 2005 (UTC) McKell

Separate section for Bway musical version
I am removing the reference to the broadway production under development and creating a new section, which can then be moved into a daughter article should the production open and take on its own relevance as a separate article.NYDCSP 06:45, 29 January 2007 (UTC)

--The production begins previews this weekend, and it opens at the end of June. I think it might be a good idea to beging a page on the musical. Thanks--Broadwaygal 18:39, 23 May 2007 (UTC)

 * There is an article on the musical at Xanadu (musical). -- Ssilvers 23:08, 23 May 2007 (UTC)

--Thanks, I guess I should have done more research. I was looking at the discussion page and thought that it hadn't happened yet. --Broadwaygal 23:10, 23 May 2007 (UTC)

External links modified
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Twice the budget?
"The movie was a box office failure, making only $23 million against a reported $20 million budget (to be profitable, movies must gross at least twice their budgets after all costs are taken into account).[2]" The cited source makes no claim that a movie must gross at least twice its budget, and this seems absurd on its face. Indy (talk) 14:55, 10 December 2022 (UTC)

title: Talk:Xanadu (film)


text:
Josaphat

Josaphat can refer to:

People

 * Jehoshaphat, in the Bible, fourth king of the Kingdom of Judah
 * Josaphat, a Christian saint of India, appearing in the legend of Barlaam and Josaphat
 * Giosafat Barbaro (1413–94), Venetian explorer and diplomat
 * Josephat T. Benoit (1900–76), mayor of Manchester, New Hampshire
 * Josaphat Celestin, Haitian-American politician, North Miami's first black mayor
 * Josaphat Chichkov, Bulgarian priest, rector and teacher
 * J.-J. Gagnier (Jean-Josaphat Gagnier), Canadian conductor, composer, and musician
 * Josaphata Hordashevska, Ukrainian Greek-Catholic nun
 * Blessed Josaphat Kotsylovsky (Йосафат Йосиф Коциловський), Ukrainian Greek Catholic bishop and martyr
 * Josaphat Kuntsevych (c.1580 – 1623), Belarusian martyr and saint of the Ruthenian Catholic Church
 * Josaphat-Robert Large, Haitian-American poet, novelist and art critic
 * Yuz Asaf, a sage buried at Srinagar, Kashmir, whose name is interpreted as "Josaphat" by Christians
 * Israel Beer Josaphat, the birth name of Paul Reuter
 * Josaphat (Metropolis) is a main character in Fritz Lang's Metropolis (1927 film)

Churches

 * St. Josaphat Cathedral (disambiguation)
 * St. Josaphat's Church (disambiguation)

Other places

 * Valley of Josaphat, mentioned in the biblical book of Joel
 * Josaphat Park, a public park located in the municipality of Schaerbeek in Brussels
 * Planet Nine, which has been called by its original proponents: "Jehoshaphat", "George", and "Phattie".

Other uses

 * Barlaam and Josaphat (book), several books dealing with the lives of Saints Barlaam and Josaphat
 * Order of Saint Basil the Great, also known as the Basilian Order of Saint Josaphat, monastic religious order of the Greek Catholic Church
 * Priestly Society of Saint Josaphat
 * Ukrainian Catholic Eparchy of Saint Josaphat in Parma, a diocese of the Ukrainian Greek Catholic Church

title: Josaphat


text:
Talk:Barlaam and Josaphat

Arabic version
I moved this page to make room for a page for Josaphat (king), one of the kings of Judah. --ESP 19:12, 12 Mar 2004 (UTC)

 * Would that be Jehoshaphat? At any rate, I strongly suspect that at some point, Jehoshaphat's name got confused with the name of this character. It's the only explanation I can think of for the weird change of initial consonant from B to J. User:Angr 15:16, 22 June 2006 (UTC)

 * It appears from the article that the change took place in the 8th century Arabic version. The change may have been prompted by the resemblance to the name Josaphat (Yehoshafat), but probably originated in the fact that the difference between initial Ba and initial Ya in Arabic is whether there is one or two dots below the letter. Ba has one dot, Ya has two.

BobGriffin-Nukraya 21:34, 18 August 2006 (UTC)

Josaphat is the Latin form of the Greek name Joasaph (Ioasaph). So, it has nothing to do with Johoshaphat. The Arabic name for the same character was Bilauhar. —Preceding unsigned comment added by 70.112.229.236 (talk) 17:50, 10 January 2009 (UTC)

 * "Josaphat" is a Latinization of the Hellenized form of "Jehoshaphat", "'Iosaphát" (Ιωσαφατ). This comes from the Septuagint (see the LXX translation of 1 Kgs. 22:41)-- several centuries before the earliest known date of the account of these two people, and about two centuries before the emergence of Christianity at all. Arsenic-03 (talk) 19:08, 6 July 2023 (UTC)

One small problem...
It doesn't at all seem similar to Gautama Buddha's story. I won't run down a list of bullet points but except for smatterings of "king" and "hermit" I see nothing substantially similar.

I tend to agree. That Josaphat means Bodhisatva is convincing. However, according to Buddhists, Bodhisatva is not Buddha. Bodhisatva is a person who is on the path to become a Buddha upon perfection. (In this story, a perfect Christian?). In order to dispel a common misconception, Buddhists believe that there are obvious (manifest) Buddhas like 'Gautama / Gotama' Buddha, the person we generally call Buddha, and silent / private (Pali: pase [pron. pa-say]) Buddhas as well, who are not skilled in teaching. Then, there can be many, perhaps thousands of (pasé) Buddhas right now, because the essence of Buddhism is being watchful of the way mind works that one could logically discover on their own. Please read Devadaha sutta, Brahmajala Sutta or most other suttas - JC (talk) 06:16, 13 December 2016 (UTC)

 * There are plenty of parallels. It would take too long to list them, but you might like to download this 450kB pdf which has a chapter about it. Scroll to chapter 9 (page 153 in the printed edition).--Shantavira 14:22, 22 June 2006 (UTC)

You should take a look at "Otras Inquisiciones", a book, published in 1952 by the great argentinean writer Jorge Luis Borges, where he describes, in a short article called, according to the portuguese translation, "Formas de uma Lenda" - in English could be "Variatons of a Legend" -, the same parallels. According to Borges, Buddha's legend, as the centuries went by and the oral transmission of the story from the East to the West provoked gradual modifications, became Barlaam and Josaphat's story and, given to that, Siddharta Gautama became a Saint for the Catholic Church.

 * With all due respect to Borges I don’t see any reason why to consider him the proper authority for this page. Besides, this article should be clearly labelled as non-NPOV, as the pro-Buddhist view clearly distort the objectivity of its message.
 * Ceplm (talk)
 * Ceplm (talk)

I see some small parallels, but nothing to draw a proving conclusion like this article does. It seems this stems from the movement that likes to portray Christianity as a collection of other religions. —Preceding unsigned comment added by 70.234.190.71 (talk) 23:58, 29 September 2008 (UTC)

I agree. From the text of this article as written, I don't see the connection of the story to Buddha, except for the possible etymology of Josaphat back to Bodhisattva. I'd like to see some more explication and citations to that effect. TuckerResearch (talk) 20:29, 22 April 2014 (UTC)

Pronunciation
Hello,

I'm curious if any of you know how to properly pronounce the name 'Barlaam'. Thanks

Adam s 11:48, 18 June 2006 (UTC)

In the movie "Repentance" ("Monanieba" was the Georgian title) directed by Tengiz Abuladze in 1987, the main character is "Varlam Aravidze". Since the Greek Beta softens to Veta, the pronunciation of Varlam's name is probably that of Barlaam, at least in Greek. Bar Lahm (lahm sounds like the first syllable of 'lama') or Var Lahm. BobGriffin-Nukraya 21:27, 18 August 2006 (UTC)

That explanation matches Sanskrit pronunciation guide that has single and double length vowels just as in Dutch. JC (talk) 06:22, 13 December 2016 (UTC)

John of Damascus versus Euthymius of Athos
John of Damascus died in the mid 8th century, while Euthymius died in the early 11th century. If John Damascene is the actual author/editor of the Barlaam and Ioasaph, then it must have entered the Christian world well before the time of Euthymius. It does appear from the name of the work by Euthymius, 'Sibrdzne Balavarisa (Wisdom of Balahvari)', that Euthymius' source was not the work attributed to John of Damascus (see wiki page for Euthymius of Athos). This brings into question my claim (above) that the Georgian name 'Varlam' is derived from Barlaam. BobGriffin-Nukraya 01:58, 19 August 2006 (UTC)
 * The article at project Gutenberg ( http://www.gutenberg.org/dirs/etext96/bioas10.txt ) states that the story is 'believed to have been translated into Greek (possible from a Georgian originial) sometime in the 11th century A.D.' Attribution to John of Damascus is traditional.  The title refers to one John the monk (Ο Ιωαννης Ο Μοναχος) from the monastery of Saint Sabas which would be the monastery of Mar Saba, where John of Damascus dwelt after leaving Damascus. Whether the title is intended to refer to John of Damascus or to someone else is not clear. However there is no tradition (known to me) of John of Damascus travelling from there.
 * So it looks like the evidence supports Euthymius of Athos as preceding the Greek translation 65.123.241.82 23:19, 21 August 2006 (UTC)

Move

 * The following discussion is an archived debate of the . Please do not modify it. Subsequent comments should be made in a new section on the talk page. No further edits should be made to this section.

PAGE MOVED per discussion below. -GTBacchus(talk) 23:26, 10 December 2006 (UTC)

Requested move
Saint Josaphat → Barlaam and Josaphat — reason for move Lima 13:43, 5 December 2006 (UTC)

Survey

 * ''Add  * Support   or   * Oppose   on a new line followed by a brief explanation, then sign your opinion using ~.

 * Support The article is more about the legend called "Barlaam and Josaphat" than about the alleged Saint Josaphat, who was never canonized; it is about both alleged saints, not one only; "Barlaam and Josaphat" is the name of the corresponding other-language Wikipedia articles and the articles in other encyclopedias Lima 13:52, 5 December 2006 (UTC)
 * Support per Lima. --Kober 15:21, 5 December 2006 (UTC)
 * Support. While the interest here is in the story, and it should be under that title, Lima's remarks are far too strong. Several third-century saints are apocryphal, and few were ever canonized. Septentrionalis 21:06, 5 December 2006 (UTC)

Discussion

 * ''Add any additional comments:
 * The above discussion is preserved as an archive of the debate. Please do not modify it. Subsequent comments should be made in a new section on this talk page. No further edits should be made to this section.

the seeking of inner peace

 * "The story of Josaphat was re-told as an exploration of free will and the seeking of inner peace Meditation in the 17th century."

This line is a little unclear. It can't stand as it is, so I'll assume it means "inner peace through meditation" and change it accordingly. Heavenlyblue (talk) 01:05, 2 February 2012 (UTC)

Also, can anyone add any further information on this subject? Heavenlyblue (talk) 01:12, 2 February 2012 (UTC)

few were ever canonized
From the Move discussion:
 * "Several third-century saints are apocryphal, and few were ever canonized. Septentrionalis "

Can anyone further clarify this point and incorporate this information into the text? It seems that this is a bit of a grey area. Heavenlyblue (talk) 01:19, 2 February 2012 (UTC)

Non-Georgian sources
It's clear that the available sources for the article are mostly from the Western scholarly tradition. This could be similar to the 72 virgins issue where an Islamic topic receives much more weight in Western tradition than it did in the actual Islamic tradition. First things first: does anybody know how these names are spelled in Arabic? Even if you can't read Arabic, if you (general you) know the names written in Arabic text, please copy paste them here. Myself and other Arabic-speaking Wikipedians can then take that and run a search to see where this topic is mentioned in Arabic sources; the next step would be sifting through such sources and finding ones which were translated into English first, as that would be less time consuming than producing original translations for Wikipedia (though that is also an option). MezzoMezzo (talk) 11:58, 24 February 2013 (UTC)
 * Mezzo thanks so much. It's perhaps more part of en.wp's WP:WORLDVIEW problem. Until I added Ibn Babuya etc. Buddhist and Islamic sources weren't even mentioned, this was entirely a Georgian Christianity article. I believe the commonest spelling is بوداساف but that is from memory. I did see the Arabic spelling embedded in brackets of one of the English texts but can't remember which. That spelling brings up a few hits on Google including http://www.berzinarchives.com/web/ar/archives/study/islam/general/historical_survey_knowledge.html In ictu oculi (talk) 17:58, 24 February 2013 (UTC)
 * Yes, that has "بيلاوهار وبوداساف" In ictu oculi (talk) 18:13, 24 February 2013 (UTC)
 * Unfortunately, there is some Egyptian pharmaceutical manufacturing a drug called بوداسيف that is skewing my search results. Let me try again... MezzoMezzo (talk) 03:51, 25 February 2013 (UTC)
 * Whoa, we're reporting live here! A thread on some forum titled, "Is the grave of the Messiah in Kashmir?" Unfortunately, the link is blocked at the server here in my office. MezzoMezzo (talk) 03:52, 25 February 2013 (UTC)
 * كتاب بلوهر وبوذاسف IYY (talk) 23:54, 18 October 2019 (UTC)

 * From this document, page 21:
 * "And the translation of this book by Ibn Muqfi' was dubbed Kalila wa Damna, and those are the names of two intelligent sons of Awa; one of them was called Karataka and the other Damanaka, and they're the subject of the first chapter of the book. And the book Kalila wa Damna resembles the book Bilawhar and Budasaf, which is often mistranslated as Yudasaf, and it's the story of an aescetic with many different forms revolving around the life of the Buddha."
 * This is from volume two of what appears to be an academic journal called Turath al-Islam or The Civilization/Culture of Islam. It has two authors: Dr. Hasan Nafi'ah and Clifford Bosworth (I know, I was also like wtf), translated by Drs. Hussein Mu`annas and Ihsan Sidqi al-'Umd and reviewed by Dr. Fouad Zakaria. The publishing date of this issue is June 1998; it's a part of some series called Series of Famous Civilizational Books Published by the National Council for Civilization, the Arts and Literature in Kuwait. This series was established in January 1978 under the supervision of Ahmad Mishari al-'Udwani who lived from 1923 to 1990. There is a big number 234 at the top of the cover page so I think that's the issue number. MezzoMezzo (talk) 04:07, 25 February 2013 (UTC)
 * Also from that Berzina Archives site which I can't access here at work is this thread about the relationship between Sufism and Buddhism which might be of interest on multiple articles if it mentions sources (or of interest to nothing at all if it's just chatting, I can't view it to find out). MezzoMezzo (talk) 04:10, 25 February 2013 (UTC)

Article lede
The lede is somewhat telegraphic and needs to be expanded per WP:LEDE to explain teh body better. One of you guys who knows more about the topic should do it, rather than myself - that way it will be done right. History2007 (talk) 17:32, 20 March 2013 (UTC)

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Proposed merge of Barlaams saga ok Jósafats into Barlaam and Josaphat
Covers a subset of the topics of the main article, the lead section is enough for a single section on the main article. AtlasDuane (talk) 11:39, 13 December 2020 (UTC)
 * After some thought, I agree with this proposal, AtlasDuane (and I have just added a wikilink from Barlaam and Josaphat to Barlaams saga ok Jósafats). Although I'd have thought that there's enough scholarship on many of these different versions of the Barlaam and Josaphat story for them usefully to develop Wikipedia entries of their own, this doesn't seem to be the case for Barlaams saga, which seems to be the object of few studies. Moreover, the entry for Barlaams saga ok Jósafats is, if the content is correct, actually about three independent translations into Old Norse, and there's no particular reason why they would need to be clustered into one Wikipedia entry. Alarichall (talk) 11:37, 17 January 2021 (UTC)
 * ✅ Klbrain (talk) 10:55, 22 June 2021 (UTC)

Article Improvement Propositions
I added a {disputed} tag to the article for three reasons:

1. The article itself doesn't describe how the account is supposed to be a Christianized legend of the Buddha despite also recording that the Catholic and at least some Orthodox churches venerate the two figures in question (i.e. they assert that these two not only existed, but that Josaphat was not and could not have been the Buddha). On this front, the article can be improved by describing the similarities noted by academics as well as being clear about the existing camps regarding the matter, as well as some of those that comprise them (it should also note if there's anything that can be called "scholarly consensus" on the matter).

1a. Additionally, as stated elsewhere in this talk page, "buddha" and "bodhisattva" are two different concepts (albeit on a continuum); at once, "Josaphat" is asserted to ultimately be a corruption of "Bodhisattva" but is also regarded as evidence that the account was in reference to the "Buddha" (who had been considered such for centuries). Meanwhile, the story says that Josaphat was named "Josaphat" (or "Ioasaph", rather), as if "he" (that is to say "Josaphat", that is to say "the Buddha" according to the aforementioned) was named "Bodhisattva" at his birth-- as opposed to, say, his known birth name "Siddhartha". On this front, evidence discussing this detail should be described and cited.

2. There's no evidence presented in the article that Sanskrit and Manichean predecessors to the Christian account exists. The article as written also casts doubt on the certainty of the penultimate source being a Manichean manuscript ("The origins of the story may be a Central Asian manuscript written in the Manichaean tradition.") even though it initially asserted this as fact in the "History" section. (For what it's worth, I haven't found evidence of either in my own searching, only allusions to such-- I've found images of the document of the account in Arabic, Persian, Greek, Old French, and Ge'ez, but I've never found any production of their proposed Manichean or Sanskrit prototypes). On this front, the article can be improved by citing the known Sanskrit/Manichean Persian documents/fragments.

3. "Josaphat" is a rather direct Latinization of Iwsafat ('Iosaphát, which the "Ioasaph" found in the account is only two letters off from), which is a Hellenization of the Hebrew "Jehoshaphat"; this is evidenced in the the Septuagint translation of 1 Kings 22:41 where the namesake is described. The translation predates Christianity by about two centuries, and the account of the two figures in question by several more. In contrast, citation #3 doesn't express certainty about its etymological assertion (writing anecdotally, "This is said to be a corruption of the original Joasaph...."). This detail is relevant and, at the least, contradictory to the explanation given in the "Names" section (as of April 16th, 2023) but also calls into question the premise of the article (that the tale is based on Gautama) in a way that impacts its entirety. This, along with any acknowledgement/refutation of this, should be noted-- if any exists. Arsenic-03 (talk) 23:10, 6 July 2023 (UTC)

 * The ultimate source is the original Buddhist accounts of the life of Buddha (as likely to be written in Pali as in Sanskrit); see Family of Gautama Buddha to start with. More generally, it's the role of Wikipedia to report on generally-accepted scholarship, not to try to refute such scholarship (which would be "original research").  As for dubious traditional saints, have you heard of Decanonization? AnonMoos (talk) 00:35, 10 July 2023 (UTC)

 * "[T]he original Buddhist accounts of the life of Buddha" isn't a specific source; the article as-is currently asserts "The tale derives from a second to fourth century Sanskrit Mahayana Buddhist text, via a Manichaean version...." and then cites a paywalled paper without excerpt (something that's done for citation #14, even if it's in untranslated French). There are no citations in this article that point to either of these proposed texts-- either the texts themselves or papers that materialize them.
 * The article later states "The origins of the story may be a Central Asian manuscript written in the Manichaean tradition.", which makes uncertain a claim that was earlier made in certainty.
 * I'm not  proposing that the scholarship be refuted in the article, as I'm outright saying that statements in this article ought to be sufficiently fleshed out with citations and more detail (which would also be cited). Before there were scholars that asserted that the account was based off a document of the life of Siddhartha Gautama, the figures were canonized and venerated in certain churches and communions that are absolutely not "extremely small minorities"-- and still are to this day. That this disagreement in fact exists (and, implicitly, that said parties wouldn't venerate figures they believe are both fictional and derivative of a figure outside their religion) needs to be reflected.
 * The name business needs citation to more in-depth material and better development: the claim in the present version of the article is that "Josaphat" is a corruption of "Bodhisattva", but the account itself says that Josaphat was named "Josaphat" at his birth. "Bodhisattva" isn't a forename, it's a title; it also wouldn't necessarily be how the Buddha would be referred to after centuries, absent specific context. It's the claim itself that doesn't make sense-- this seeming incongruence could be at least partly resolved through the production of the proposed Buddhist and Manichean texts (or citations to papers analyzing them), as well as discussion of those materials as they relate to this detail. Put another way: this account could still be a Christianized record of the life of Siddhartha Gautama even without this name detail-- but if that's said, then there has to be some context as to how it's the penultimate title that was corrupted into "Josaphat" (the given name of the Josaphat in question) and not the actual given name. It can make sense if the proposed ancestor texts refer to the Buddha as the Bodhisattva the entire time.
 * Nothing involved in addressing these matters would constitute "original research". Arsenic-03 (talk) 03:13, 10 July 2023 (UTC)

 * Whatever -- I know nothing about the specific Indic text which may have been the immediate source for the non-Indic versions, but there are multiple well-known Buddhist texts on the life of Buddha which you could easily find for yourself by paying a little attention to the Wikipedia articles about Buddha. I really don't know why this specific text is so critically important anyway, since it was merely the funnel for transmitting well-established Indian traditions about the life of Buddha to cultures west of India.  I really don't care too much about this "Manichean manuscript", since the important fact was that Buddhist traditions were spread to cultures without much knowledge of Buddhism, regardless of the particular way in which this was done.  AnonMoos (talk) 05:28, 10 July 2023 (UTC)

 * "I really don't know why this specific text is so critically important anyway, since it was merely the funnel for transmitting well-established Indian traditions about the life of Buddha to cultures west of India."
 * ...it's important because 1) the present article is alleging the existence of a specific Manichean text that's based off a specific Buddhist text, and 2) because these proposed texts are at the very root of the proposed transmission history. They are reportedly how these traditions spread.
 * These specific texts have to demonstrably exist for the statement "The tale derives from a second to fourth century Sanskrit Mahayana Buddhist text, via a Manichaean version...." to not be conjecture. Arsenic-03 (talk) 12:49, 10 July 2023 (UTC)

 * You seem to believe that unless we have a manuscript chain of transmission for every intermediate step along the way (like the provenance trail of an artistic object in a museum), then the hypothesis that Buddhist traditions were transmitted to cultures with little knowledge of Buddhism must fail. Unfortunately, that's simply not the way that it works.  The transmission of traditions is inferred from the similarities between the ultimate source (life of Buddha) and the destination result (Christian folklore tale of Barlaam and Josaphat).  In many somewhat parallel cases, we have very little idea about what most of the intermediate stages were.  If, in this particular case, we have some information about the actual text that was used to transfer the traditions from an Indic culture to a non-Indic culture, then that would be a definite bonus, but it would not be critical evidence for the fact of transmission.  In zooming in on the Manichean manuscript, you have given great importance to something which is not really that important (though of course it should be properly cited, and deleted from the article if it's impossible to properly cite it). AnonMoos (talk) 14:20, 10 July 2023 (UTC)
 * "You seem to believe that unless we have a manuscript chain of transmission for every intermediate step along the way (like the provenance trail of an artistic object in a museum), then the hypothesis that Buddhist traditions were transmitted to cultures with little knowledge of Buddhism must fail." I'm saying that "The tale derives from a second to fourth century Sanskrit Mahayana Buddhist text, via a Manichaean version...." either has to be substantiated with said texts or explicitly qualified as a hypothesis (unlike how it's presently presented in the article) because it alleges the existence of specific documents that have yet to be cited. The "Document Hypothesis" for the Pentateuch and the hypothetical "Q source" are thus considered specifically because they're postulations based on ultimately circumstantial evidence and patterns; its adherents have yet to produce these alleged ancestral documents.
 * If those documents can't be produced, then you aren't allowed to assume with certainty that the documents must have existed, because 1) that wouldn't be the only conceivable way that such information was transmitted, and 2) that wouldn't be the only conceivable pattern of transmission past the documents that have been demonstrated. Arsenic-03 (talk) 15:02, 10 July 2023 (UTC)

The case of Barlaam and Josaphat is simply not parallel to J, E, D, P, and/or Q in any meaningful way, because we have amply-attested Buddhist texts in the original languages recounting the life of Buddha (texts whose existence you've been carefully ignoring, to instead chase after the Manichean whatever which is simply not too important). These texts actually exist (those who know Pali etc can read them), and are not in the slightest degree hypothetical. AnonMoos (talk) 11:25, 12 July 2023 (UTC)

 * At no point have I ignored the existence of Buddhist texts recounting the life of the Buddha. However, Pali manuscripts aren't relevant to the very narrow matter at hand; my first response to you emphasized how the issue (well, one of them that I have) is that a claim about specific documents in specific languages has been made in order to substantiate a claim about the source and transmission of the account, but the alleged documents have yet to be cited. Whether you think this is "important" in light of the existence of other documents that aren't being cited as the texts in question is irrelevant-- you cannot even begin to claim with certainty that these Sanskrit and Manichean texts exist, and that an account derives from them, if you cannot produce those specific texts.
 * If they have as much known existence as J/E/D/P/Q, this statement definitely has to be phrased as a hypothesis as to how the story reached West Asia. The assertion "we have Buddhist texts in Pali" as evidence of derivation is an assertion of a different substance, and that relevance needs to be demonstrated in a way more involved than mere citation. Arsenic-03 (talk) 12:14, 12 July 2023 (UTC)

 * I've already explained what's worth explaining in my comment of "14:20, 10 July 2023" above. You say that you're not ignoring the amply-attested actually-existing Buddhist texts on the life of Buddha, yet in that very same July 12th comment of yours you go on to ignore them yet again (recidivist behavior)!  The Manichean manuscript is not given great emphasis in the article because it does not have great importance to the subject-matter of this article.  As I've already explained several times, it would have been a funnel by means of which the amply-attested and in no way hypothetical Buddhist traditions would have been transmitted from Indic cultures to a non-Indic culture.  We know what the Buddhist traditions on the life of Buddha are, and we know what the end-result is in the legend of Barlaam and Josaphat.  Since we have the end-points, knowing the exact nature of any intermediates is not too critical (though it would be a nice bonus).  J/E/D/P/Q are not intermediates between two known attested existing entities, but hypothetical ultimate sources, so your analogy is somewhat inept, and not too relevant.  AnonMoos (talk) 03:49, 15 July 2023 (UTC)
 * Firstly, who is this "we" that's aware of the relevant Buddhist traditions? You have no way to tell what any given person knows about this article topic, the matter of the life of the Buddha, and any interrelation between the two. In the first place, the purpose of this article at present is to accomplish informing about the third. You're projecting an undue expectation of prior knowledge onto an arbitrary readership that would obviate any informative value of this article with its present thesis.
 * Secondly, you've been repeatedly discussing "Buddhist texts" that the article itself does not cite for the claim it makes to substantiate the assertion for the origin of the text. I am not ignoring the Pali texts-- the current article is. Meanwhile, what's being invoked are specific texts in specific other languages and/or traditions that have yet to be produced either primarily or secondarily.
 * That these texts exist are in fact not just important because they propose how the account reached West Asia, but because it's also proposed that some of the names are corruptions of Sanskrit terms, which is part of what's used as evidence for this specific transmission claim. Overall, you're claiming that it's "not that important" despite the fact that the claim exists, answers a major question of transmission given the present article thesis, and is intertwined with other claims.
 * If the evidence of the transmission is not in the proposed chain, but in the idea that it resembles Buddhist traditions, then it's the cited similarities that need to be written as evidence (they currently aren't). Otherwise, there's no demonstration that these are endpoints, to begin with-- that would just be you assuming your conclusion.
 * On the other hand? If a claim of fact is being made about these documents being part of a definitive transmission chain, then those documents better well be produced, because if they don't exist, then the reality is that the transmission chain could be anything or nothing, and that lack of clarity (whether or not it's due to the sources being "bad") also needs to be remarked on in some way. This isn't a "nice-to-have", given what the article is-- it's necessary that either this happens or that at least some of the names of the people who make that claim are explicitly attached to it. At any rate, this article possibly also has a POV issue (as someone else here remarked), given that it's acknowledged that these two figures are canonized in certain communions but doesn't point out that they don't believe that they canonized the Buddha-- a later-quoted assertion.

 * On a side note: in asserting that the the J/E/P/D/Q sources are irrelevant, you assumed the certain existence of the intermediate Sanskrit and Manichean sources. Meanwhile, my point was that if they can't be produced, then they're as hypothetical as the J/E/P/D/Q sources and need to be remarked on as such-- the matter of whether they're intermediate or ultimate is dwarfed by the matter of whether they can be produced in the first place. Arsenic-03 (talk) 05:32, 15 July 2023 (UTC)

Conversion from what to Christianity?
The article seems to imply a conversion from Buddhism to Christianity, but doesn't clearly state it, and seems to require on the reader having prior knowledge of the story?

Is this ChatGPT summary correct?

'' The character of Josaphat in this narrative is loosely based on the historical figure of Siddhartha Gautama, who later became known as Gautama Buddha, the founder of Buddhism. In adapting the story, Christian authors transformed it into an allegory of conversion to Christianity from a non-Christian faith or a worldly life. Therefore, the specific religion or belief system Josaphat practiced before his conversion is left intentionally vague in the story. '' SalaDitman (talk) 07:16, 9 October 2023 (UTC)

 * I don't know or care about ChatGPT, but in the minds of Christian readers of the Barlaam and Josaphat story in the middle ages, the previous religion would have presumably been some form of Indian paganism which they knew nothing about. Few of them would even have known about Buddhism (though before the fall of the Roman empire, Clement of Alexandria had at least heard of it)... AnonMoos (talk) 22:44, 9 October 2023 (UTC)

Fictional premise
Greetings! Regarding Special:diff/1215885472, I was attempting to point out for readers unfamiliar with the chronology, that Christianity didn't exist during the lifetime of the Buddha, and so the idea that the Buddha or a character based on his life lived in a kingdom with Christians is clearly fictional. And the conflict over Christianity seems like a major part of the story. I didn't mean to imply the story wasn't based on the Buddha, or that the Buddha was fictional, or perhaps some other implication you're objecting to. Is there a better way to word this note? -- Beland (talk) 20:09, 27 March 2024 (UTC)
 * I'm afraid it did read that the story wasn't based on the life of the Buddha. Consider: "Their life story was based on the life of the Gautama Buddha, and tells of the conversion of Josaphat to Christianity. As the Buddha lived several centuries before Jesus, the premise is fictional".  The phrase "the premise is fictional" appears to refer to "their life story was based on the life of the Gautama Buddha" whereas that is believed to be the origin of the whole B&J story.  I do understand now why you inserted the sentence and can see why you thought the summary might be misleading, a common problem with summaries.  How would you feel about:

"Barlaam and Josaphat, also known as Bilawhar and Budhasaf, are Christian saints. Their life story was based on a retelling of the life of the Gautama Buddha who actually lived some centuries earlier. The legend tells of an Indian king who persecuted the Christian Church in his realm. After astrologers predicted that his own son would some day become a Christian, the king imprisoned the young prince Josaphat, who nevertheless met the hermit Saint Barlaam and converted to Christianity. After much tribulation the young prince's father accepted the Christian faith, turned over his throne to Josaphat, and retired to the desert to become a hermit. Josaphat himself later abdicated and went into seclusion with his old teacher Barlaam."

 * Please feel free to hack it about here, that's what talk pages are for! Kind Regards, Martin of Sheffield (talk) 21:37, 27 March 2024 (UTC)
 * Your suggestion was much clearer. I've tweaked the wording a bit and added it to the article. Feel free to continue revising if needed. Thanks! -- Beland (talk) 22:22, 27 March 2024 (UTC)

title: Talk:Barlaam and Josaphat


text:
Suntan

Suntan may refer to:

 * Sun tanning
 * Suntan (apple), a cultivar
 * Lockheed CL-400 Suntan, a concept aircraft
 * Suntan (1976 film), a Bollywood drama film
 * Suntan (2016 film), a Greek drama film
 * Project Suntan, a NASA project involving liquid nitrogen research

title: Suntan


text:
Heegner number

In number theory, a Heegner number (as termed by Conway and Guy) is a square-free positive integer d such that the imaginary quadratic field $$\Q\left[\sqrt{-d}\right]$$ has class number 1. Equivalently, the ring of algebraic integers of $$\Q\left[\sqrt{-d}\right]$$ has unique factorization.

The determination of such numbers is a special case of the class number problem, and they underlie several striking results in number theory.

According to the (Baker–)Stark–Heegner theorem there are precisely nine Heegner numbers:

This result was conjectured by Gauss and proved up to minor flaws by Kurt Heegner in 1952. Alan Baker and Harold Stark independently proved the result in 1966, and Stark further indicated the gap in Heegner's proof was minor.

Euler's prime-generating polynomial
Euler's prime-generating polynomial $$n^2 + n + 41,$$ which gives (distinct) primes for n = 0, ..., 39, is related to the Heegner number 163 = 4 · 41 − 1.

Rabinowitz proved that $$n^2 + n + p$$ gives primes for $$n=0,\dots,p-2$$ if and only if this quadratic's discriminant $$1-4p$$ is the negative of a Heegner number.

(Note that $$p-1$$ yields $$p^2$$, so $$p-2$$ is maximal.)

1, 2, and 3 are not of the required form, so the Heegner numbers that work are 7, 11, 19, 43, 67, 163, yielding prime generating functions of Euler's form for 2, 3, 5, 11, 17, 41; these latter numbers are called lucky numbers of Euler by F. Le Lionnais.

Almost integers and Ramanujan's constant
Ramanujan's constant is the transcendental number $$e^{\pi \sqrt{163}}$$, which is an almost integer, in that it is very close to an integer: $$e^{\pi \sqrt{163}} = 262\,537\,412\,640\,768\,743.999\,999\,999\,999\,25\ldots\approx 640\,320^3+744.$$

This number was discovered in 1859 by the mathematician Charles Hermite. In a 1975 April Fool article in Scientific American magazine, "Mathematical Games" columnist Martin Gardner made the hoax claim that the number was in fact an integer, and that the Indian mathematical genius Srinivasa Ramanujan had predicted it—hence its name.

This coincidence is explained by complex multiplication and the q-expansion of the j-invariant.

Detail
In what follows, j(z) denotes the j-invariant of the complex number z. Briefly, $$\textstyle j\left(\frac{1+\sqrt{-d}}{2}\right)$$ is an integer for d a Heegner number, and $$e^{\pi \sqrt{d}} \approx -j\left(\frac{1+\sqrt{-d}}{2}\right) + 744$$ via the q-expansion.

If $$\tau$$ is a quadratic irrational, then its j-invariant $$j(\tau)$$ is an algebraic integer of degree $$\left|\mathrm{Cl}\bigl(\mathbf{Q}(\tau)\bigr)\right|$$, the class number of $$\mathbf{Q}(\tau)$$ and the minimal (monic integral) polynomial it satisfies is called the 'Hilbert class polynomial'. Thus if the imaginary quadratic extension $$\mathbf{Q}(\tau)$$ has class number 1 (so d is a Heegner number), the j-invariant is an integer.

The q-expansion of j, with its Fourier series expansion written as a Laurent series in terms of $$q=e^{2 \pi i \tau}$$, begins as: $$j(\tau) = \frac{1}{q} + 744 + 196\,884 q + \cdots.$$

The coefficients $$c_n$$ asymptotically grow as $$\ln(c_n) \sim 4\pi \sqrt{n} + O\bigl(\ln(n)\bigr),$$ and the low order coefficients grow more slowly than $$200\,000^n$$, so for $$\textstyle q \ll \frac{1}{200\,000}$$, j is very well approximated by its first two terms. Setting $$\textstyle\tau = \frac{1+\sqrt{-163}}{2}$$ yields $$ q=-e^{-\pi \sqrt{163}} \quad\therefore\quad \frac{1}{q}=-e^{\pi \sqrt{163}}. $$ Now $$j\left(\frac{1+\sqrt{-163}}{2}\right)=\left(-640\,320\right)^3,$$ so, $$\left(-640\,320\right)^3=-e^{\pi \sqrt{163}}+744+O\left(e^{-\pi \sqrt{163}}\right).$$ Or, $$e^{\pi \sqrt{163}}=640\,320^3+744+O\left(e^{-\pi \sqrt{163}}\right)$$ where the linear term of the error is, $$\frac{-196\,884}{e^{\pi \sqrt{163}}} \approx \frac{-196\,884}{640\,320^3+744} \approx -0.000\,000\,000\,000\,75$$ explaining why $$e^{\pi \sqrt{163}}$$ is within approximately the above of being an integer.

Pi formulas
The Chudnovsky brothers found in 1987 that $$\frac{1}{\pi} = \frac{12}{640\,320^\frac32} \sum_{k=0}^\infty \frac{(6k)! (163 \cdot 3\,344\,418k + 13\,591\,409)}{(3k)!(k!)^3 (-640\,320)^{3k}},$$ a proof of which uses the fact that $$j\left(\frac{1+\sqrt{-163}}{2}\right) = -640\,320^3.$$ For similar formulas, see the Ramanujan–Sato series.

Other Heegner numbers
For the four largest Heegner numbers, the approximations one obtains are as follows. $$\begin{align} e^{\pi \sqrt{19}} &\approx {\color{white}000\,0}96^3+744-0.22\\ e^{\pi \sqrt{43}} &\approx {\color{white}000\,}960^3+744-0.000\,22\\ e^{\pi \sqrt{67}} &\approx {\color{white}00}5\,280^3+744-0.000\,0013\\ e^{\pi \sqrt{163}} &\approx 640\,320^3+744-0.000\,000\,000\,000\,75 \end{align} $$

Alternatively, $$\begin{align} e^{\pi \sqrt{19}} &\approx 12^3\left(3^2-1\right)^3{\color{white}00}+744-0.22\\ e^{\pi \sqrt{43}} &\approx 12^3\left(9^2-1\right)^3{\color{white}00}+744-0.000\,22\\ e^{\pi \sqrt{67}} &\approx 12^3\left(21^2-1\right)^3{\color{white}0}+744-0.000\,0013\\ e^{\pi \sqrt{163}} &\approx 12^3\left(231^2-1\right)^3+744-0.000\,000\,000\,000\,75 \end{align} $$ where the reason for the squares is due to certain Eisenstein series. For Heegner numbers $$d < 19$$, one does not obtain an almost integer; even $$d = 19$$ is not noteworthy. The integer j-invariants are highly factorisable, which follows from the form
 * $$12^3\left(n^2-1\right)^3=\left(2^2\cdot 3 \cdot (n-1) \cdot (n+1)\right)^3,$$

and factor as, $$\begin{align} j\left(\frac{1+\sqrt{-19}}{2}\right) &= {\color{white}000\,0}-96^3 = -\left(2^5 \cdot 3\right)^3\\ j\left(\frac{1+\sqrt{-43}}{2}\right) &= {\color{white}000\,}-960^3 = -\left(2^6 \cdot 3 \cdot 5\right)^3\\ j\left(\frac{1+\sqrt{-67}}{2}\right) &= {\color{white}00}-5\,280^3 = -\left(2^5 \cdot 3 \cdot 5 \cdot 11\right)^3\\ j\left(\frac{1+\sqrt{-163}}{2}\right)&= -640\,320^3 = -\left(2^6 \cdot 3 \cdot 5 \cdot 23 \cdot 29\right)^3. \end{align} $$

These transcendental numbers, in addition to being closely approximated by integers (which are simply algebraic numbers of degree 1), can be closely approximated by algebraic numbers of degree 3, $$\begin{align} e^{\pi \sqrt{19}} &\approx x^{24}-24.000\,31 ; & x^3-2x-2&=0\\ e^{\pi \sqrt{43}} &\approx x^{24}-24.000\,000\,31 ; & x^3-2x^2-2&=0\\ e^{\pi \sqrt{67}} &\approx x^{24}-24.000\,000\,0019 ; & x^3-2x^2-2x-2&=0\\ e^{\pi \sqrt{163}} &\approx x^{24}-24.000\,000\,000\,000\,0011 ; &\quad x^3-6x^2+4x-2&=0 \end{align} $$

The roots of the cubics can be exactly given by quotients of the Dedekind eta function η(τ), a modular function involving a 24th root, and which explains the 24 in the approximation. They can also be closely approximated by algebraic numbers of degree 4, $$\begin{align} e^{\pi \sqrt{19}} &\approx 3^5 \left(3-\sqrt{2\left(1- \tfrac{96}{24}+1\sqrt{3\cdot19}\right)} \right)^{-2}-12.000\,06\dots\\ e^{\pi \sqrt{43}} &\approx 3^5 \left(9-\sqrt{2\left(1- \tfrac{960}{24}+7\sqrt{3\cdot43}\right)} \right)^{-2}-12.000\,000\,061\dots\\ e^{\pi \sqrt{67}} &\approx 3^5 \left(21-\sqrt{2\left(1- \tfrac{5\,280}{24} +31\sqrt{3\cdot67}\right)} \right)^{-2}-12.000\,000\,000\,36\dots\\ e^{\pi \sqrt{163}} &\approx 3^5 \left(231-\sqrt{2\left(1- \tfrac{640\,320}{24}+2\,413\sqrt{3\cdot163}\right)} \right)^{-2}-12.000\,000\,000\,000\,000\,21\dots \end{align} $$

If $$x$$ denotes the expression within the parenthesis (e.g. $$x=3-\sqrt{2\left(1- \tfrac{96}{24}+1\sqrt{3\cdot19}\right)}$$), it satisfies respectively the quartic equations $$\begin{align} x^4 -{\color{white}00} 4\cdot 3 x^3 + {\color{white}000\,0}\tfrac23( 96 +3) x^2 - {\color{white}000\,000}\tfrac23\cdot3(96-6)x - 3&=0\\ x^4 -{\color{white}00} 4\cdot 9x^3 + {\color{white}000\,}\tfrac23( 960 +3) x^2 - {\color{white}000\,00}\tfrac23\cdot9(960-6)x - 3&=0\\ x^4 -{\color{white}0} 4\cdot 21x^3 + {\color{white}00}\tfrac23( 5\,280 +3) x^2 - {\color{white}000}\tfrac23\cdot21(5\,280-6)x - 3&=0\\ x^4 - 4\cdot 231x^3 + \tfrac23( 640\,320 +3) x^2 - \tfrac23\cdot231(640\,320-6)x - 3&=0\\ \end{align} $$

Note the reappearance of the integers $$n = 3, 9, 21, 231$$ as well as the fact that $$\begin{align} 2^6 \cdot 3\left(-\left(1- \tfrac{96}{24}\right)^2+ 1^2 \cdot3\cdot 19 \right) &= 96^2\\ 2^6 \cdot 3\left(-\left(1- \tfrac{960}{24}\right)^2+ 7^2\cdot3 \cdot 43 \right) &= 960^2\\ 2^6 \cdot 3\left(-\left(1- \tfrac{5\,280}{24}\right)^2+ 31^2 \cdot 3\cdot67 \right) &= 5\,280^2\\ 2^6 \cdot 3\left(-\left(1- \tfrac{640\,320}{24}\right)^2+ 2413^2\cdot 3 \cdot163 \right) &= 640\,320^2 \end{align} $$ which, with the appropriate fractional power, are precisely the j-invariants.

Similarly for algebraic numbers of degree 6, $$\begin{align} e^{\pi \sqrt{19}} &\approx \left(5x\right)^3-6.000\,010\dots\\ e^{\pi \sqrt{43}} &\approx \left(5x\right)^3-6.000\,000\,010\dots\\ e^{\pi \sqrt{67}} &\approx \left(5x\right)^3-6.000\,000\,000\,061\dots\\ e^{\pi \sqrt{163}} &\approx \left(5x\right)^3-6.000\,000\,000\,000\,000\,034\dots \end{align} $$

where the xs are given respectively by the appropriate root of the sextic equations, $$\begin{align} 5x^6-{\color{white}000\,0}96x^5-10x^3+1&=0\\ 5x^6-{\color{white}000\,}960x^5-10x^3+1&=0\\ 5x^6-{\color{white}00}5\,280x^5-10x^3+1&=0\\ 5x^6-640\,320x^5-10x^3+1&=0 \end{align} $$

with the j-invariants appearing again. These sextics are not only algebraic, they are also solvable in radicals as they factor into two cubics over the extension $$\Q\sqrt{5}$$ (with the first factoring further into two quadratics). These algebraic approximations can be exactly expressed in terms of Dedekind eta quotients. As an example, let $$\textstyle \tau = \frac{1+\sqrt{-163}}{2}$$, then, $$\begin{align} e^{\pi \sqrt{163}} &= \left( \frac{e^\frac{\pi i}{24} \eta(\tau)}{\eta(2\tau)} \right)^{24}-24.000\,000\,000\,000\,001\,05\dots\\ e^{\pi \sqrt{163}} &= \left( \frac{e^\frac{\pi i}{12} \eta(\tau)}{\eta(3\tau)} \right)^{12}-12.000\,000\,000\,000\,000\,21\dots\\ e^{\pi \sqrt{163}} &= \left( \frac{e^\frac{\pi i}{6} \eta(\tau)}{\eta(5\tau)} \right)^{6}-6.000\,000\,000\,000\,000\,034\dots \end{align} $$

where the eta quotients are the algebraic numbers given above.

Class 2 numbers
The three numbers 88, 148, 232, for which the imaginary quadratic field $$\Q\left[\sqrt{-d}\right]$$ has class number 2, are not Heegner numbers but have certain similar properties in terms of almost integers. For instance, $$\begin{align} e^{\pi \sqrt{88}} +8\,744 &\approx {\color{white}00\,00}2\,508\,952^2-0.077\dots\\ e^{\pi \sqrt{148}} +8\,744 &\approx {\color{white}00\,}199\,148\,648^2-0.000\,97\dots\\ e^{\pi \sqrt{232}} +8\,744 &\approx 24\,591\,257\,752^2-0.000\,0078\dots\\ \end{align} $$ and $$\begin{align} e^{\pi \sqrt{22}} -24 &\approx {\color{white}00}\left(6+4\sqrt{2}\right)^{6} +0.000\,11\dots\\ e^{\pi \sqrt{37}} +24 &\approx \left(12+ 2 \sqrt{37}\right)^6 -0.000\,0014\dots\\ e^{\pi \sqrt{58}} -24 &\approx \left(27 + 5 \sqrt{29}\right)^6 -0.000\,000\,0011\dots\\ \end{align} $$

Consecutive primes
Given an odd prime p, if one computes $$k^2 \mod p$$ for $$\textstyle k=0,1,\dots,\frac{p-1}{2}$$ (this is sufficient because $$\left(p-k\right)^2\equiv k^2\mod p$$), one gets consecutive composites, followed by consecutive primes, if and only if p is a Heegner number.

For details, see "Quadratic Polynomials Producing Consecutive Distinct Primes and Class Groups of Complex Quadratic Fields" by Richard Mollin.

title: Heegner number