User talk:Loraof

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Loraof, you are invited to the Teahouse![edit]

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Hi Loraof! Thanks for contributing to Wikipedia. Come join experienced editors at the Teahouse! The Teahouse is a space where new editors can get help from experienced editors. These editors have been around for a long time and have extensive knowledge about how Wikipedia works. Come share your experiences, ask questions, and get advice from experts. I hope to see you there! Writ Keeper (I'm a Teahouse host)

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Welcome +[edit]

Hi Laraof,

Welcome to this strange little community of Wikipedia math editors. I've been tracking some of your edits in elementary geometry and think that you are doing an excellent job of improving the articles – even when you've corrected some of my edits. Do keep up the good work. I thought that I would point out that in your recent edit of Similarity (geometry) you misinterpreted something that I did. I hid a statement about "|||" as a notation for similarity and commented that I had never seen that before - my remark was not about the standard notation. You are right in that I should have put this on the talk page, but I was being lazy and thought that if anyone was going to object to the removal, I could save a little cut-and-paste time by just leaving it hidden. I have no problem with the edits you made concerning this, although I would be a little hesitant about citing Posamentier and Lehmann since I have found them to be a little sloppy in their editing (not necessarily in this book, which I haven't seen, but in another of their monographs). Again, welcome aboard and happy editing. Bill Cherowitzo (talk) 18:51, 19 September 2014 (UTC)
Thanks, Bill. I see that I misunderstood what you had never seen before -- I interpreted the antecedent of "this" to be what preceded it (the standard notation) while you meant what followed "this". Incidentally, in line with your comment, I found three or four proofreading mishaps in Posamentier and Lehmann's current book. Cheers, Loraof (talk) 19:03, 19 September 2014 (UTC)

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September 2014[edit]

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Help[edit]

Hi Loraof,

As you may be aware of, I'm working on a merge of Internal and external angle into a section of Angle. User DVdm has asked for some basic references for this material on angles related to polygons and I am embarrassed to say that I can't find any in my personal library. I was wondering if you had access to some elementary geometry texts and could find some appropriate citations and add them to the page (either page would be fine). Thanks. Bill Cherowitzo (talk) 04:34, 7 October 2014 (UTC)

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October 2014[edit]

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Lists of topics[edit]

Apparently you recently created a new article titled Orthocentroidal circle. I have added it to the List of triangle topics and the List of circle topics. If you know of others that should be on those lists and are not, could you add those too? Michael Hardy (talk) 05:40, 5 November 2014 (UTC)

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Redirects[edit]

Just so you know, a redirect such as Rhombic chess (edit | talk | history | links | watch | logs) isn't necessary. If you were to type that in (and the redirect didn't exist) the software would have automatically taken you to the correct capitalization. G S Palmer (talkcontribs) 17:26, 5 January 2015 (UTC)

But I did, and it didn't. More specifically, I tried to put it in another article as a link but the preview page showed it as a red link. Loraof (talk) 17:30, 5 January 2015 (UTC)
True, it won't work for links. But you could always just use the normal linking process and use a pipe link. G S Palmer (talkcontribs) 23:38, 5 January 2015 (UTC)

Your experience with Wikipedia so far[edit]

Hello Loraof,

I am conducting research about newcomers to Wikipedia and I was hoping to ask you some questions. I’ve noticed you’ve had some good activity recently. Is there any chance you have time in the next month to speak with me? If you are interested or have any questions, please email me at gmugar [at] syr.edu or leave a message on my talk page.

I hope to be in touch soon,

Gabrielm199 (talk) 22:45, 6 January 2015 (UTC)

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January 2015[edit]

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Investment (macroeconomics)[edit]

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Reference errors on 5 May[edit]

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Napoleon's Theorem[edit]

Doesn't 1820 come before 1825? Clearly 1825 is not the earliest known reference to the result (although 1826 might be the earliest published proof); whoever set the examination paper in Dublin in 1820 had the idea before William Rutherford (for that matter, Napoleon only died in May, 1821). Similarly, 1867 comes before 1911, or doesn't an Encyclopedia count? (In any case, Faifofar has it in an earlier edition than 1911.) It might also be worth noting that James Thomson - father of William Thomson, Lord Kelvin - was based in Belfast, before moving to Glasgow.

It is very good of you to do a clean up of this entry - and all readers of Wikipedia stand in your debt in this regard - but why not do a proper job, and have things in chronological order? It would appear that much of what counted as 'history' is not quite correct. — Preceding unsigned comment added by 129.180.1.224 (talk) 21:03, 2 July 2015 (UTC)

Why don't you go ahead and do the proper job of integrating your additions into the article? Loraof (talk) 22:48, 2 July 2015 (UTC)

Sphere[edit]

Hi Loraof, I can use your help on sphere. Thanks, Bill Cherowitzo (talk) 19:44, 14 July 2015 (UTC)

Never mind, I think we've gotten this straightened out, but you might want to take a look to see if it needs any more tweaking. Bill Cherowitzo (talk) 22:48, 14 July 2015 (UTC)

edits[edit]

hello. The matter was already discussed and referenced. No "violation" of bold revert anything. Just bold elaboration that is sourced and obvious and clear. And what's done against it by you is frankly speaking, uptight impositions. Please stop accusing me of violations, etc, and assuming bad faith, because I won't tolerate it, as a lack of civility. And there are math teachers etc that clearly say that a sphere is "circular" because "round" MEANS "circular". There's a reference and I already went into heavy detail and points and the point again was that "round" is "circular" too in general, and there's definitely a "circularity" with spheres, in specific situations...webster says "round" means "circular". And sphere's are round, that's all. Good day. Gabby Merger (talk) 17:07, 16 July 2015 (UTC)

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Help desk request[edit]

I think I found what you were asking about and it should now be gone. -- GB fan 18:09, 8 September 2015 (UTC)

Nice theorem[edit]

https://vi.wikipedia.org/wiki/%C4%90%E1%BB%8Bnh_l%C3%BD_%C4%90%C3%A0o_(conic) — Preceding unsigned comment added by 117.6.86.31 (talk) 07:03, 9 November 2015 (UTC)

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Examples and applications section in Quadratic equation[edit]

Hi. I see that you are the major contributor to this section, which you inserted immediately after the lead. I hope you don't mind, but I thought it would be better if this section were pushed to a little bit later in the article. Some of the examples that you gave would be pretty obscure to what I envision would be the main target audiences of this article (1) middle school students and (2) parents trying to refresh their memories enough to help their kids with homework. Honestly now, how many frustrated parents are interested in Descartes' theorem or Fuss' theorem? Face-smile.svg 96.226.231.251 (talk) 08:56, 28 November 2015 (UTC)

December 2015[edit]

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Cubic function[edit]

Hi, Laraof.

I am disturbed by the fact that a large amount of material was deleted from the article on cubic functions and I have put most of it back there. Please take a look at the revision history of the article and give me your opinion. JCSantos (talk) 11:11, 7 June 2016 (UTC)

I deleted what you had written about “Cardano's formula, written unambiguously” and you decided to put it back. Since I don't take part in editing wars, I won't revert that. But that doesn't change the fact that you are wrong here. For reasons that I cannot understand, you seem to believe that the expression 3z is unambiguous, but it is ambiguous, since every non-zero complex number has 3 cube roots. Consider, for instance, z = 224 + 3603i. What is 3z then? Is it 8 + 23i? Or is it −7 + 33i? Or perhaps −1 − 53i? The answer, of course, is that all three of them are cube roots of z. JCSantos (talk) 11:09, 23 June 2016 (UTC)

@JCSantos: I don't think you read my amended version, nor my edit summary of it. The amended version answers your question. Cheers. Loraof (talk) 14:34, 23 June 2016 (UTC)
I suppose that you mean the passage “are defined to be any pair of complex conjugate cube roots”. Yes, I had not read it. I still don't like it (but it is not wrong) because what's written before applies to cubic equations with complex coefficients, but when you add the reference to “conjugate cube roots” then what is done here becomes valid only for real coefficients. I don't see any advantage in having this restriction. I will change one thing: I will replace ω with ξ (for consistence). I hope you won't mind. JCSantos (talk) 15:08, 23 June 2016 (UTC)
Actually, yes, you are wrong. Consider, for instance, the equation x3 − 2x + 4 = 0. In this case, p = −2 and q = 4; therefore, the square root of q2/4 + p3/27 is 10/93. Then, what you wrote, applied to this situation, was that one should consider a cube root of −2 + 10/93 and add it to a cube root of −2 − 10/93, with the cube roots chosen so that they are complex conjugate numbers. Problem: there are no such cube roots. JCSantos (talk) 13:57, 27 June 2016 (UTC)
Good catch—thanks. I think it looks right the way you've got it now. Loraof (talk) 18:29, 27 June 2016 (UTC)

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Your changes on Magic square[edit]

Could you rephrase your first sentence? I don't know what "Up to rotation and reflection," means. Does it mean "including rotations and reflections" or "excluding" them? Thanks in advance. Dhrm77 (talk) 21:19, 25 August 2016 (UTC)

This wording essentially repeated what already appeared earlier in the article, with a wikilink for up to. But I'll explain it in my sentence. (It means excluding them.) Loraof (talk) 21:25, 25 August 2016 (UTC)

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Nomination of Jacobi point for deletion[edit]

A discussion is taking place as to whether the article Jacobi point is suitable for inclusion in Wikipedia according to Wikipedia's policies and guidelines or whether it should be deleted.

The article will be discussed at Wikipedia:Articles for deletion/Jacobi point until a consensus is reached, and anyone is welcome to contribute to the discussion. The nomination will explain the policies and guidelines which are of concern. The discussion focuses on high-quality evidence and our policies and guidelines.

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Statistical stability[edit]

Hi. I wonder, what kind of clarification do you want to see in the first phrase of "Statistical stability"? Boris Tsirelson (talk) 17:44, 10 November 2016 (UTC)

Currently the first paragraph is
The phenomenon of statistical stability is one of the most surprising physical phenomena, comprising stability of statistics (i.e., functions of the sample), in particular relative frequencies (empirical probabilities) of mass events, sample means, and other sample moments.[clarification needed] This phenomenon is widespread and so can be regarded as a fundamental natural phenomenon.
  • "statistical stability ... comprising stability of statistics" -- this semantic tautology needs to be rewritten.
  • "stability of statistics (i.e., functions of the sample") -- "functions of the sample" presumably is an explanation of the word "statistics", but someone might read it as an explanation of "stability"
  • "relative frequencies ... of mass events, sample means, and ..." -- does this mean the relative frequencies of sample means? Probably not, but the wording gives rise to that interpretation.
In short, the grammatical structure of the sentence is unclear. Often when this sort of thing happens, it is helpful to break the sentence down into two or more sentences.
Also, "stability" needs to be defined early in the lead.
Loraof (talk) 19:46, 10 November 2016 (UTC)
Also, the lead needs a reference that uses the term "statistical stability" in this way. Loraof (talk) 20:37, 10 November 2016 (UTC)
Well, after looking at the corresponding article in Russian Wikipedia and the corresponding deletion debates there I see the problem. True, some interest to this topic (but maybe in different terms?) was shown by some great minds in the past. But now the topic is nearly "monopolized" by Professor Igor I. Gorban, and the majority of citations to his works are self-citations. Thus, the problem is probably not clarification but notability? Boris Tsirelson (talk) 21:47, 10 November 2016 (UTC)
Gorban insists that in the nature (in contrast to probability theory), usually, frequencies are stable (that is, as stable as predicted by the probability theory) for moderate sample size, but for very large sample size frequencies are less stable, and therefore not described by the probability theory. I'd say, this is a big claim, and it is better to wait for its confirmation by independent reliable sources, as reliable as big is the claim. Or, for now, describe it shortly, as a work in progress? Boris Tsirelson (talk) 22:03, 10 November 2016 (UTC)
My concerns are that the English grammar in the lead section makes it hard to understand, and that "stable" is never defined. But your point about notability is also valid. And it appears that Gorban not only is the main source for the topic, but also Gorbin is the one who created the article, which is a conflict of interest. Loraof (talk) 23:13, 10 November 2016 (UTC)
I see. Maybe this should be treated as alternative theoretical formulation. Boris Tsirelson (talk) 05:50, 11 November 2016 (UTC)

WP:REFERS[edit]

Although I follow WP:REFERS, it is a WP:Essay, not a WP:Policy or guideline, and, in many cases, it should not be applied to articles that are about words (WP:WORDISSUBJECT). That is why I reverted your edit with this edit. I also don't think your wording flowed well. Furthermore, how a term is defined is sometimes going to be in the leads of our articles about concepts. The main thing WP:REFERS is concerned with is not having the first sentence of a concept article begin with "is a term for" type of wording, but articles that are specifically about words are a different matter. Flyer22 Reborn (talk) 04:54, 21 November 2016 (UTC)

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Enough source to convert Dao's theorem on six circumcenters associadted with a cyclic hexagon From wiki Viet Nam to English?[edit]

Dear Mister Loraof,

Do you think Enough source to convert Dao's theorem on six circumcenters associadted with a cyclic hexagon From wiki Viet Nam to English?

https://vi.wikipedia.org/wiki/%C4%90%E1%BB%8Bnh_l%C3%BD_%C4%90%C3%A0o_v%E1%BB%81_s%C3%A1u_t%C3%A2m_%C4%91%C6%B0%E1%BB%9Dng_tr%C3%B2n

Main source

1. Dergiades, Nikolaos (2014). “Dao’s Theorem on Six Circumcenters associated with a Cyclic Hexagon”. Trong Yiu, Paul. Forum Geometricorum (PDF) 14. tr. 243–246. ISSN 1534-1178. Truy cập , MR3260500

2. Telv, Cohl (2014). “A Purely Synthetic Proof of Dao’s Theorem on Six Circumcenters Associated with a Cyclic Hexagon”. Trong Yiu, Paul. Forum Geometricorum (PDF) 14. tr. 261–264. ISSN 1534-1178. Truy cập , MR3267837

3. X(3649) = KS(INTOUCH TRIANGLE), http://faculty.evansville.edu/ck6/encyclopedia/ETCPart3.html#X3649

4. Grégoire Nicollier, Two six-circle theorems for cyclic pentagons, Forum Geometricorum, 16 (2016) 347--354.

5. http://www.cut-the-knot.org/m/Geometry/AnotherSevenCircles.shtml

6. Nguyen Ngoc Giang, The extension from a circle to a conic having center: The creative method of new theorems, International Journal of Computer Discovered Mathematics, pp.21-32.

7. Ngo, Quang Duong (2016). “Ngo Quang Duong, Some problems around the Dao's theorem on six circumcenters associated with a cyclic hexagon configuration”. Trong Deko, Dekov. International Journal of Computer Discovered Mathematics (PDF) 1. tr. 25–39. ISSN 2367-7775

8. http://mathoverflow.net/questions/234053/daos-theorem-on-six-circumcenters-associated-with-a-cyclic-hexagon — Preceding unsigned comment added by 222.252.34.231 (talk) 17:40, 29 December 2016 (UTC)

I cannot get into reference 3 because the website has a problem. Numbers 5 and 8 are not reliable sources. Number 4 barely mentions the case of hexagons. Numbers 6 and 7 are in a journal I'm not familiar with, and I don't know if it is a reliable source. References 1 and 2 look good. So I now have no opinion about the answer to your question.
I would mention that if you do create an article for the theorem, I don't think you should draw the six circumcircles as tangent to each other--I don't think that they are tangent in general. Loraof (talk) 20:22, 29 December 2016 (UTC)
But I got into reference 3; first I got only a part of that long page, and then, after waiting a minute, I got the whole page, with X(3649). Boris Tsirelson (talk) 20:32, 29 December 2016 (UTC)
I still get an error message saying "webpage crashed". It appears to be a page on a triangle center, but how does that relate to the point of concurrency in Dao's cyclic hexagon? Loraof (talk) 20:47, 29 December 2016 (UTC)

Thank to Dear Mister Loraof.

if cyclic hexagon ABCDEF degenerated to ACE (A=B, C=D, E = F) then AOa, BOb, COc are concurrent at X(3649).

if cyclic hexagon ABCDEF degenerated to ACE (A=B, C=D, E = F) then AOa, BOb, COc are concurrent at X(3649) so X(3649) of GHI = X(54) of ACE, so the theorem is a generalization of Kosnita theorem.

3 wrote[1]:

X(3649) = KS(INTOUCH TRIANGLE)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc(b + c)(2a + b + c)(a - b + c)(a + b - c)

                       = g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = (b + c)(2a + b + c)(a - b + c)(a + b - c)

Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b) Two proofs of Dao's theorem about cyclic hexagons were published in 2014: Nikolaos Dergiades's Dao's Theorem on Six Circumcenters Associated with a Cyclic Hexagon and Telv Cohl's A Purely Syntheitc Proof of Dao's Theorem on Six Circumcenters Associated with a Cyclic Hexagon. In the degenerate case that the hexagon is the intouch triangle A'B'C', the triangle of the circumcenters of AB'C', A'BC', A'B'C is perspective to A'B'C' with perspector X(3649). (Dao Thanh Oai, Francisco Javier Garcia Capitan, ADGEOM #1718, September 16, 2014)

X(3649) lies on these lines: 1,30 7,21 10,12 11,113 55,3487 57,191 140,3336 145,388 354,946 396,2306 429,1835 553,1125 940,1406 962,3303 1056,2098 1086,1193 1100,1839 1317,1365 1358,2795 1360,1367 1411,2647 1749,3337 1834,2650 1852,1870 1901,2294 3318,3324

There are two author ask me that, they give two new papers on this theorem will appear in April 2017. So we will waiting these papers. Thank to You very much for your helping. 113.189.226.37 (talk) 13:13, 8 March 2017 (UTC)

Dear Mister Loraof,

There are two new papers on the Dao's theorem on six circumceneters:

http://geometry-math-journal.ro/pdf/Volume6-Issue1/6.pdf

http://geometry-math-journal.ro/pdf/Volume6-Issue1/4.pdf

So now, the Theorem appear in three Journals and Kimberling center with some papers, with some good review, do you think it is enough reliable source to convert Dao's theorem on six circumcenters from wiki Viet Nam to English? — Preceding unsigned comment added by 27.72.247.152 (talk) 16:42, 31 March 2017 (UTC)

References

Generalization of the Simson line with nine proofs and many properties[edit]

Dear Mister Loraof,

[1]-Nguyen Van Linh, Another synthetic proof of Dao's generalization of the Simson line theorem, Forum Geometricorum, 16 (2016) 57--61. http://forumgeom.fau.edu/FG2016volume16/FG201608.pdf

[2]-Nguyen Le Phuoc and Nguyen Chuong Chi (2016). 100.24 A synthetic proof of Dao's generalisation of the Simson line theorem. The Mathematical Gazette, 100, pp 341-345. doi:10.1017/mag.2016.77. https://www.cambridge.org/core/journals/mathematical-gazette/article/div-classtitle10024-a-synthetic-proof-of-daoandaposs-generalisation-of-the-simson-line-theoremdiv/9CE7AA808F87FF05E290A7ED641FE9C4

[3]-Leo Giugiuc, A proof of Dao’s generalization of the Simson line theorem, tạp chí Global Journal of Advanced Research on Classical and Modern Geometries, ISSN: 2284-5569, Vol.5, (2016), Issue 1, page 30-32 http://geometry-math-journal.ro/pdf/Volume5-Issue1/4.pdf

[4]-Tran Thanh Lam, ANOTHER SYNTHETIC PROOF OF DAO’S GENERALIZATION OF THE SIMSON LINE THEOREM AND ITS CONVERSE, Global Journal of Advanced Research on Classical and Modern Geometries, ISSN: 2284-5569, Vol.5, (2016), Issue 2, page 89-92 http://geometry-math-journal.ro/pdf/Volume5-Issue2/3.pdf

[5]-Ngo Quang Duong, A generalization of the Simson line theorem, Forum Geometricorum, to appear in Forum Geometricorum.

[6]-Three other proof by Telv Cohl, Luis Gonzalez, Tran Thanh Lam, A Generalization of Simson Line http://www.cut-the-knot.org/m/Geometry/GeneralizationSimson.shtml

[7]-Another proof https://www.artofproblemsolving.com/community/c6h1075523p5181203 — Preceding unsigned comment added by 113.189.226.37 (talk) 16:51, 7 March 2017 (UTC)

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Determinacy[edit]

Hi Loraof,

The point of the text you changed is that determinacy is a field of study, not simply a mathematical property. I'm sure the text can be improved, but I would like it to focus on the field of study. Any suggestions? --Trovatore (talk) 22:05, 24 March 2017 (UTC)

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Project management[edit]

Sorry, I don't get it: how can the sentence "the system constraint for all projects is identified as are the resources" possibly be grammatically correct? IMO it should be either "the system constraint for all projects is identified as the resources" or "the system constraint for all projects are the resources". --Martin de la Iglesia (talk) 17:46, 23 April 2017 (UTC)

I would prefer to include a comma for clarity: "the system constraint for all projects is identified, as are the resources"; but some editors don't like to use as many commas as I do. The sentence as I have it means "the system constraint for all projects is identified, and the resources are identified." Note that the constraint is an equality or inequality, whereas the resources are physical entities or amounts. Loraof (talk) 17:54, 23 April 2017 (UTC)
OK, I get it now, but I'd strongly suggest to change the sentence so that its meaning becomes clearer. How about "...the system constraint for all projects as well as the resources are identified"? --Martin de la Iglesia (talk) 14:53, 24 April 2017 (UTC)
OK, but I would use commas around "as well as the resources" to give " the system constraint for all projects, as well as the resources, are identified" Loraof (talk) 15:08, 24 April 2017 (UTC)

Don't worry, it's not about you[edit]

A disciplinary discussion in which you may become involved, or may wish to comment, is at WP:ANI here regarding possible disciplinary action against User:The Banner for edit warring at Peacock Alley (restaurant). Akld guy (talk) 01:14, 19 July 2017 (UTC)

The only thing is that he misrepresents your stance, by claiming you said anything about "defunct restaurant" while you were responding on "former restaurant". The Banner talk 02:09, 19 July 2017 (UTC)

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Question about the Term "Bubble" in the Article Economic Bubble[edit]

Dear Loraof, I just read the article [[1]] and I'm not quite sure, but maybe you're the main author of the section "History and origin of term". Right? It says

The term "bubble", in reference to financial crisis, originated in the 1711–1720 British South Sea Bubble, and originally referred to the companies themselves, and their inflated stock, rather than to the crisis itself. This was one of the earliest modern financial crises; other episodes were referred to as "manias", as in the Dutch tulip mania. The metaphor indicated that the prices of the stock were inflated and fragile – expanded based on nothing but air, and vulnerable to a sudden burst, as in fact occurred.

Unfortunately there's no source and I'm curious who said that. If I get this correctly, the Term and concept of the "bubble" has been used since the 18th century, since the South Sea Bubble, in an economic context. is that correct? Or do we use this Term later for things like the Mississippi scandal and the crisis around the South Sea Company?

Thank you very much! :)

14:21, 24 October 2017 (UTC)Daktylo (talk)

Sorry, I didn’t write that and am not familiar with it. Loraof (talk) 15:16, 24 October 2017 (UTC)

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Your mail concerning my picture of correlation[edit]

Hi here, I'm currently on travel, and will be back at home at the beginning of New Year. Please let me look at your question at that time, and than decide what to do, ok? With regards --Qniemiec (talk) 18:24, 26 December 2017 (UTC)

Okay. Have a good holiday! Loraof (talk) 18:26, 26 December 2017 (UTC)
Hi there, I've just found some free moments to re-read your posting, and to comment it, so please take a look at what I wrote, ok? Maybe its a misunderstanding. However, of course, the diagram will look different if the data cloud will change its shape - in my picture it rather tends towards high values of , while being rather circular with tending towards zero, to diagram might change the way you proposed. Greetings --Qniemiec (talk) 14:31, 4 January 2018 (UTC)
Hi, please take a look at my answer to your last posting, and let me know if my current understanding of the issue is correct, ok? Greetings --Qniemiec (talk) 20:22, 4 January 2018 (UTC)
Hi, just uploaded the redrawn version of the diagram we spoke about. Hope that now everything is ok. Thank you for the advice. --Qniemiec (talk) 23:44, 4 January 2018 (UTC)
Hi, you wrote "I would note, though, that your yellow caution above is incorrect—on my screen the red “y=...” version shows as the steeper one. But on the page Pearson correlation coefficient the new version shows." I think this comes due to some delay in Wikimedia's/Wikipedia's effort to update all links to a given new file - e.g. after uploading a new version of a picture or diagram it often takes some time until the updated version finally appears on screen, so in the beginning I often panicked thinking that I've uploaded the old version again. And as far as it concerns thanking, I should also thank you for this valuable hint. So have a good time and New Year! --Qniemiec (talk) 00:03, 5 January 2018 (UTC)
PS. I seem to be right with my delay hypothesis: Just took a look at the French Wikipedia article on regression, where my picture also appears, and when first opening the page, still the old image appeared. So I reloaded the page once more with my browser, and not earlier than after that reload the new version showed up. Greetings --Qniemiec (talk) 00:20, 5 January 2018 (UTC)
PS². Hi, it's me once more. Provided that , this leads - as you explained - to , i.e. the slope of the x-versus-y correlation line must always be steeper than the one of the y-versus-x line. Ok, at least in case of positive correlations. But what about negative ones? In that case the 2nd inequation would be wrong, so - in order to be more exact - shouldn't we better work with absolute values then? I.e. . Although this won't immediately affect my drawing, I'd like to know whether my way of understanding is correct. Greetings --Qniemiec (talk) 22:09, 5 January 2018 (UTC)
Yes, that looks correct. A few months ago I augmented a passage in Pearson correlation coefficient#Geometric interpretation to say
”For uncentered data, there is a relation between the correlation coefficient and the angle φ between the two regression lines, y = gx(x) and x = gy(y), obtained by regressing y on x and x on y respectively. (Here φ is measured counterclockwise within the first quadrant formed around the lines' intersection point if r > 0, or counterclockwise from the fourth to the second quadrant if r < 0.”
So the rho < 0 case has both curves negatively sloped in the second and fourth quadrants, with the “y=...” line flatter. But I doubt that adding another drawing to make this point would be insightful. Loraof (talk) 22:39, 5 January 2018 (UTC)

Linear differential equations[edit]

In the article linear differential equations I think it was you who edited the current version of the first paragraph, in which I found the phrase: "They equate 0 to a polynomial that is linear in the value and various derivatives of a variable;". Is this phrase correct? I honestly cannot understand what is meant here. Thanks. Conjugado (talk) 20:12, 25 January 2018 (UTC)

The value 0 (zero) equals the value of a polynomial which is a constant plus (another constant times the value y of a variable) plus (another constant times the derivative of y with respect to an independent variable) plus possibly other terms involving second , third, etc. derivatives of y with respect to the independent variable. In this context y refers both to a function y(t) of the independent variable t and to the value of that function.
I’ll think about the wording and rewrite the sentence in the article if I can improve it. Thanks for bringing it to my attention. Loraof (talk) 23:22, 25 January 2018 (UTC)
I came here to look for the background of your edits at linear differential equations and noticed the misplaced ref. I apologize for my efforts to repair this. You are certainly free to revert all my edits on your TP. Purgy (talk) 13:13, 26 January 2018 (UTC)

Empirical distribution function[edit]

Hi, regarding the graph, have you tried to draw it on Octave? Often the whole code from Matlab works, but you could need to install a package (to get empirical cdf). There's also Scilab that has a Matlab-Scilab converter. If you generate new random samples, that could suffice. Ideally the author of the graph would save it with a better resolution, like nagual wrote. But having graphs drawn with free softwares could be a better alternative than matlab, more accessible (easier to perform later fixes by others). Yaḥyā ‎ (talk) 16:44, 28 January 2018 (UTC)

Thanks, @Yahya Talatin:. Unfortunately, I have zero graphing skills, so I have to leave graphing to others. Loraof (talk) 17:15, 28 January 2018 (UTC)
Alright, I'll do it if there is consensus. Just to be sure that I understand what you want; you want a new set (of 20) that represent what we would usually expect (last value <3). Just a word of caution though, it could be that the author of the picture chose a set where the last value is unusually that far to make it more obvious that there is less clustering on these regions. If you, nagual or other users agree on the ranges... I could provide a couple of randomized sets (that end within that range), and you could decide the one that is more acceptable for you all. Yaḥyā ‎ (talk) 19:11, 28 January 2018 (UTC)
After running it several time, I couldn't reproduce (on Octave) a value of 3.5; it usually does not go beyond this. [2] I guess this suggest that the author run it continuously until he finds that one for a purpose (to highlight the clustering variation). Yaḥyā ‎ (talk) 20:11, 28 January 2018 (UTC)
The purpose of this graph, which I presume was made specifically for the article Empirical distribution, is not to show clustering – it’s just to show the relation of the empirical step function to the population distribution function (namely that the former approximates the latter). So I think the picture you just linked looks good. Can you replace the old one with this one? Thanks. Loraof (talk) 20:38, 28 January 2018 (UTC)
OK, how's this? [3] Yaḥyā ‎ (talk) 23:44, 28 January 2018 (UTC)
Looks great! Can you make the curve a more vivid color? Loraof (talk) 00:02, 29 January 2018 (UTC)
I will ask nagual for that, stuff that concern personal tastes (like colors) I prefer leaving it to someone else. Yaḥyā ‎ (talk) 00:11, 29 January 2018 (UTC)

New graph[edit]

Previous image
New image

In my opinion, the new graph lacks the randomness of the current graph. Re-running the code until you find one that 'looks random' is more akin to a sort of subjective pseudorandomness. The original complaint with the image was that the curve appears to reach (or 'touch') y=1. However, this is also true of the two graphs linked to above. Now you've decided that there shouldn't be a sample at x>3 in order to satisfy your own expectations, which seems a bit misguided to me. I'm sure one of the mathematicians among us can calculate the exact probability of finding a sample >3 in a sample size of 20, and it isn't 0. While you're focussing your attention on the top end of the graph you don't seem the least bit bothered by the fact that the low end has the converse 'problem' – that none of the graphs have a sample at x<-2, making all of them subjectively a little right heavy. Of course, they're not, they're just random. That's the thing about randomness.

Anyway, I've been asked to colour the graphs provided and I'll do that. I assume everyone's happy with the pale green curve? But I would suggest that you link to this discussion from WT:MATH and get a broader consensus before changing the graph. I'll be back here shortly when I've finished the new graph... nagualdesign 10:05, 29 January 2018 (UTC)

I've just spent the last hour redrawing the graph and Photoshop crashed! I'll do it again in a while, after I've calmed down. nagualdesign 11:12, 29 January 2018 (UTC)
Hi nagual, I'd say the probability of having above 3 is pretty much rare. See here: 68–95–99.7_rule#Table_of_numerical_values, assuming variance of 1. I might be missing something, but I wasn't able to reproduce the original samples range. Yaḥyā ‎ (talk) 12:58, 29 January 2018 (UTC)
Well, of course it's pretty rare, you can see that by looking at the line. My point was that it's non-zero, but I'll leave it to others to decide which graph is more appropriate. Compared to the old graph I think it looks contrived, but the new graph is much higher resolution at least. Now that I have a Photoshop file with everything on separate layers I can make any changes to colour etc. if necessary. (By the way, I credited you in the file description, Yaḥyā. Face-wink.svg) nagualdesign 14:23, 29 January 2018 (UTC)
I've left a note at WT:MATH. At this point altering the samples, as well as the step function, would be very straightforward, as would changing any of the colours. nagualdesign 14:34, 29 January 2018 (UTC)

But as I mentioned above, this graph exists for only one reason: to give a visual impression of how an empirical frequency distribution can approximate a population distribution function. All we need is a generic-looking population function and a logically possible empirical function that approximates it.

This means (1) the empirical function must not have any horizontal values greater than the value (if there exists one) at which the population curve hits (and for visual clarity, appears to hit) a height of 1, and (2) the empirical function must not have any horizontal values below the value (if there exists one) at which the population function hits (and for visual clarity, appears to hit) a height of 0. These are logical requirements, and don’t involve merely satisfying my own expectations. And there’s nothing innately relevant about x=–2 or x=3 here (and it doesn’t matter whether there’s a zero or positive probability of x > 0).

Also, both the graphs shown above do look very random. Maybe it would look nicer if the whole image would be stretched horizontally in both directions.

Note further that in between the lowest and highest values of the empirical curve, the empirical curve should be sometimes above and sometimes below the smooth population curve, because any well-designed sampling process will have that feature. So that feature of the old graph needs to be restored. Loraof (talk) 18:04, 29 January 2018 (UTC)

Thus, what remains to do is: (a) make the curves intersect each other in at least several places, (b) stretch the whole diagram horizontally, and (c) re-color more vividly. Thanks, both of you! Loraof (talk) 18:11, 29 January 2018 (UTC)

When I say that the new graph doesn't look random I'm referring to the fact that the space between samples, from left to right, gets progressively smaller until x=0, then gets progressively larger, with only maybe 3 data points (out of 20) where this isn't the case. That is the opposite of random. I may as well have faked it by making every step cross the curve.
These are logical requirements, and don’t involve merely satisfying my own expectations. No they are not logical requirements, as pure logic would dictate that samples could exist anywhere within the range of the graph and have nothing to do with how it looks (subjectively). And these graphs have indeed been made to satisfy your specific requirements. You expressed your expectations to Yaḥyā who ran the code repeatedly until he got one that would satisfy you.
Moreover, you're now asking for a set of samples where the step function crosses the curve a few times. I can see why you'd want that but, again, that isn't random. If you asked me to toss a coin ten times and it came up H,H,H,H,H,H,T,H,T,H you'd probably think that wasn't quintessentially random, but it's actually just as likely as H,T,T,H,T,H,H,H,T,H or any other combination. Human perception of randomness is typically rather flawed and involves considerable bias, as you have amply demonstrated. If you want to avoid bias it's better to just run the code once and be satisfied with whatever it spits out. What would be even better is if the step function was generated on the fly each time the article is visited, but I'm not sure that's possible.
I'd like to hear more opinions from other editors before I make further changes to the images. nagualdesign 19:03, 29 January 2018 (UTC)

OK, I will wait and do what is decided. I can run (in another software though) a conditional randomization, but I don't think nagual will like that. :) For what it is worth nagual, some statistics course books also seem to do what he wants (like choosing the good apple from a couple of bad ones...). Here we have an expression for that: Arrangé avec le gars des vues. :) Yaḥyā ‎ (talk) 19:37, 29 January 2018 (UTC)

Thinking of it, seems similar to a photographer who takes a couple of pictures and pick the one that seems more appealing. Yaḥyā ‎ (talk) 20:15, 29 January 2018 (UTC)
  • @Nagualdesign: Any distribution function that is not a single point mass describes a random variable. So of course it is random in the sense of portraying a random variable. Your statement that the observations are farther apart far to the left and far to the right is not “the opposite of random”, as you put it – that spacing just reflects the feature that is uni-modal. Lots of random variables have cdf’s that look this way. For example, a random variable with a normal distribution, the most used distribution, is unimodal, and a typical sample will have observations bunched more near the center. It appears to me that you are thinking about a uniform distribution, which has no tendency to bunch; you appear to be thinking incorrectly that that is the only thing that is a random variable.
I suspect that part of nagual’s confusion lies in the fact that laypeople sometimes use the word “random” for a specifically uniform distribution, while in statistics, such as here, the word “random” pretty much only arises in the context of “random variable”.
Yes, the relation I mentioned above between the two curves is a logical requirement, for the simple reason that it is impossible to observe a sample unit drawn from the population and have that sample unit be a larger value than any value in the population. There is no speculation here, and frankly this should be obvious to anyone who has ever learned about cdf’s. Loraof (talk) 20:40, 29 January 2018 (UTC)
Well you assumed wrongly. I understand the mathematics; that this is a normal distribution. I've been running numbers on Wolfram Alpha and other websites all afternoon. Generally speaking, the samples should bunch in the middle, but you often get 2 or more clusters, as we see in the original graph. The new graph, as I said, hardly deviates from the curve at all. In such a small sample (20) you normally get more bona fide randomness. nagualdesign 20:53, 29 January 2018 (UTC)
(edit conflict) Maybe we’ve been talking past each other. In any event, we don’t need the example to be normally distributed. If it is normal, then as long as the curve is drawn thin enough, it will be clear, from what the reader can see at a glance, that the empirical cdf hits a height of 1 whereas the population cdf never does, so it’s fine, and it will be clear that going leftward the empirical cdf hits 0 whereas the population cdf never does. But we don’t need the normal cdf – any valid cdf will do for the purposes of the article that this graph is for. It might or might not be an easier graphing task to use a bounded one, in which case as I’ve said above, we have to be careful at both ends. Loraof (talk) 22:38, 29 January 2018 (UTC)

Problem?[edit]

I see a problem. The continuous green function should be closer to the discontinuous graph, and at no point diverge at the point of a vertical jump. Limit-theorem (talk) 20:42, 2 February 2018 (UTC)
Not differing at any point where there is a jump would be true if the continuous green curve was supposed to be a smooth approximation to the blue one. But actually the green one is the fixed standard normal cdf, while the blue one is the cdf of a set of sample points that could take any real values. As an extreme example for simplicity, suppose your randomly obtained sample of size 20 happened to include no values below x = 0, one value at x = 0, and 19 values greater than 0. Then the green curve at x = 0 has a height of 1/2 (because the standard normal is symmetric around 0); and the blue one has at x = 0 a height of 1/20.Loraof (talk) 21:25, 2 February 2018 (UTC)
Mathematically, if the smooth curve "dominates", then it may be improperly parametrized or your sample size is too small.Limit-theorem (talk) 13:12, 3 February 2018 (UTC)

Random graph[edit]

100 randomly generated ECDF graphs, one of which is picked at random (below)
The blue step function graph shows an empirical distribution function. The grey bars represent the observations in the sample corresponding to the sample’s empirical distribution function, and the green curve, which asymptotically approaches a height of 1 without reaching it, is the true cumulative distribution function. (Click here to load a new graph.)
The blue step function graph shows an empirical distribution function. The grey bars represent the observations in the sample corresponding to the sample’s empirical distribution function, and the green curve, which asymptotically approaches a height of 1 without reaching it, is the true cumulative distribution function. (Click here to load a new graph.)

There's some further discussion taking place on my talk page about creating a set of images and a function that picks one of those at random. nagualdesign 22:27, 29 January 2018 (UTC)

This morning I wrote some JavaScript that randomly generates a 'sheet' of 100 graphs, which I've saved as a single PNG file and uploaded (see right). User:MjolnirPants is going to write a script that will randomly select one of those graphs to display as a thumbnail, which will look something like the image below that.
Take a look at the graphs and let me know what you think. Also, please check the gaussian() function (below) to see if I've coded it correctly, if you can. In JavaScript, Math.random() gives you random number between 0 and 1 with uniform distribution, and you have to do a bit of mathematical jiggery-pokery to get a normal distribution, if that makes sense. nagualdesign 12:09, 30 January 2018 (UTC)
function gaussian() {
  // Generate a random number, mean=0, standard deviation=1
  return Math.sqrt(-2*Math.log(Math.random()))*Math.cos((2*Math.PI)*Math.random());    
}
MjolnirPants has finished the code, which you can now see to the right, and I've added it to the article. Do let us know what you think. Cheers. nagualdesign 16:15, 30 January 2018 (UTC)
Thanks to all of you for the work you’ve put in on this! The graph that’s now in the article is easy to read. My only concern is that a reader seeing it might get the false impression that the blue one has to be below the green one until the last step. If a substantial number of your 100 graphs show at least one crossing that is not near the ends, I think we ought to go with one of those. We do not have to use a graph that was obtained by random selection from these 100 – we just need one that gives a good generic idea without leaving any false impressions for the casual reader. Loraof (talk) 17:36, 30 January 2018 (UTC)
They all give a good generic idea without leaving any false impressions, since they're the real deal. The whole point of this endeavour is that cherry picking one that you (or anyone else) think is somehow quintessential, and exemplifies the right sort of step function more that any other, is completely misguided. As you can see by looking at File:ECDF-100.png, very few of them conform to the aesthetic ideals that you were aiming for with this graph, which is hardly random at all.
Looking at the first couple of dozen graphs I notice that as many of them have a step function that's above the curve as below the curve, which is exactly what we'd expect. The idea of being able to Click here to load a new graph is specifically to dispel any misapprehensions that any single graph on its own might otherwise impart, but in a way that removes any possibility of bias on our part.
There's no reason to treat casual readers as laypeople who ought to be handled using "lies-to-children", especially when it comes to topics like empirical distribution functions. Similarly, we don't dumb down articles on quantum wave functions and suchlike, as the people who read those articles are likely to have a keen sense of comprehension. I expect most of them will be in higher education. nagualdesign 18:10, 30 January 2018 (UTC)
If I were to pick one graph that somehow exemplifies the step function it would be the one on the third row, sixth from the left. It's got a nice s-curve to it and it crosses the green line multiple times. The one next to it (fifth from the left) isn't bad either. However, these two graphs only constitute 2% of those available, and the other 98% don't look very much like them, so they're not really representative at all. Do you see what I mean? nagualdesign 18:35, 30 January 2018 (UTC)

How about the one in row 3, column 7 – does that look more representative? It’s got places where it’s easy to see at a glance (even on a small computer screen) that the blue curve is sometimes above and sometimes below the green one, so it satisfies my concern. Loraof (talk) 18:50, 30 January 2018 (UTC)

ECDF crossings[edit]

I thought this might interest you; I altered the code so that instead of drawing graphs it just ran the numbers, so I was able to get some reasonable statistics based on a large number of runs:

Extended content

Samples: 10
Crossings: 9,7,3,5,7,1,9,11,5,5,5,7,5,13,3,3,3,7,7,5,5,5,5,9,7,13,5,5,9,13,9,7,5,7,3,7,3,11,9,5,9,1,9,7,5,5,7,5,9...
Average: 6.324
Average/samples: 0.6324

Samples: 20
Crossings: 5,13,7,13,1,7,11,19,11,11,13,5,13,1,15,17,13,17,15,13,7,5,9,9,1,3,11,7,7,7,5,3,5,1,19,3,11,7,5,1,5,13,17,19,9,7,17,11,5...
Average: 9.478
Average/samples: 0.4739

Samples: 30
Crossings: 25,21,5,11,5,7,1,5,7,1,27,19,9,25,11,11,13,15,11,9,11,13,3,3,25,27,13,15,21,9,13,13,19,11,25,3,13,21,3,7,23,5,23,17,15,17,11,11,11...
Average: 11.888
Average/samples: 0.39626666666666665

Samples: 40
Crossings: 5,13,21,19,9,9,13,9,19,19,11,7,23,5,7,13,11,11,31,21,9,25,7,11,13,25,5,13,11,17,19,11,9,7,19,11,29,19,7,9,15,17,25,11,23,11,15,9,13...
Average: 14.28
Average/samples: 0.357

Samples: 50
Crossings: 35,7,3,13,7,11,3,15,11,17,27,7,11,9,19,9,19,25,13,27,1,9,11,7,19,25,17,11,5,27,13,15,23,17,21,13,11,13,15,11,25,7,17,23,19,13,15,17,17...
Average: 15.956
Average/samples: 0.31912

Samples: 60
Crossings: 21,13,21,15,3,47,35,19,13,23,17,23,11,25,11,15,11,31,7,19,13,23,17,11,27,27,15,13,31,7,25,17,1,27,37,13,3,39,7,9,1,15,27,7,5,17,9,9,13...
Average: 17.198
Average/samples: 0.28663333333333335

Samples: 70
Crossings: 21,23,19,7,9,25,25,7,25,23,5,15,5,11,33,19,25,27,5,19,25,5,13,13,9,25,33,25,27,15,7,19,29,23,11,25,33,7,21,13,41,13,9,37,35,21,31,7,7...
Average: 19.754
Average/samples: 0.2822

Samples: 80
Crossings: 13,39,3,53,11,27,13,29,31,29,41,25,21,33,27,37,23,25,19,39,9,19,27,15,5,15,45,17,5,11,27,15,27,39,17,33,29,13,27,13,25,23,7,31,5,23,5,15,19...
Average: 21.562
Average/samples: 0.269525

Samples: 90
Crossings: 15,13,19,33,23,27,33,33,29,13,37,19,31,31,37,13,17,41,7,39,27,29,19,27,17,41,31,19,17,11,23,5,21,17,49,1,31,9,31,33,19,27,27,17,21,13,3,33,45...
Average: 22.154
Average/samples: 0.24615555555555554

Samples: 100
Crossings: 37,41,37,15,11,19,23,45,7,33,17,17,5,5,19,45,23,35,7,13,25,13,21,21,51,33,9,3,31,9,15,15,33,25,13,23,17,25,15,39,13,21,31,31,25,23,3,29,7...
Average: 23.58
Average/samples: 0.23579999999999998

Samples: 200
Crossings: 39,43,31,51,31,57,39,35,47,55,27,47,67,7,23,43,55,29,47,7,17,41,33,25,31,25,59,43,31,17,35,9,9,39,45,1,39,47,25,37,13,49,25,9,15,39,13,47,15...
Average: 32.28
Average/samples: 0.16140000000000001

Samples: 300
Crossings: 41,27,43,29,71,19,33,31,17,23,27,37,27,79,31,53,59,43,55,33,15,21,49,57,25,9,83,27,71,27,45,39,53,27,35,17,61,49,71,45,59,49,31,49,25,27,15,41,33...
Average: 39.2
Average/samples: 0.13066666666666668

Samples: 400
Crossings: 43,23,31,19,33,21,53,45,65,29,51,53,57,59,105,51,13,23,71,11,27,79,59,65,21,39,47,67,59,69,17,63,35,63,31,41,63,23,49,31,51,33,61,53,55,71,33,49,17...
Average: 45.72
Average/samples: 0.1143

Samples: 500
Crossings: 43,63,21,35,33,27,59,9,45,11,29,33,5,37,41,105,59,21,57,41,61,29,29,51,51,65,73,49,15,55,73,45,77,63,43,39,87,75,15,23,71,59,69,63,23,11,39,87,43...
Average: 46.56
Average/samples: 0.09312000000000001

Samples: 600
Crossings: 29,57,29,51,105,117,25,7,15,3,17,61,61,35,25,39,57,93,77,53,85,41,25,45,53,27,91,27,25,49,63,91,65,43,61,39,87,43,21,15,47,59,25,77,29,71,79,83,49...
Average: 53.18
Average/samples: 0.08863333333333333

Samples: 700
Crossings: 65,99,65,53,71,17,121,23,65,75,73,83,79,79,23,63,97,67,13,13,59,65,49,83,59,111,97,67,67,9,47,63,33,77,115,57,55,69,47,33,25,55,61,37,65,47,55,23,59...
Average: 59.78
Average/samples: 0.0854

Samples: 800
Crossings: 13,77,83,55,109,69,41,63,33,73,77,47,35,117,111,43,27,67,93,81,41,107,127,97,91,45,69,29,61,47,79,75,53,37,57,117,79,39,63,63,127,47,83,81,51,81,45,53,25...
Average: 68.76
Average/samples: 0.08595000000000001

Samples: 900
Crossings: 103,81,101,87,15,61,77,23,69,143,13,69,63,55,35,47,57,25,39,79,85,87,55,69,81,75,19,75,79,131,43,59,103,99,31,55,85,35,81,131,77,67,65,89,141,61,45,11,59...
Average: 71.34
Average/samples: 0.07926666666666666

Samples: 1000
Crossings: 109,107,49,77,77,119,101,139,61,43,109,39,55,23,69,21,81,69,31,45,59,75,97,57,81,111,101,23,89,109,51,35,95,59,75,49,27,91,65,63,57,67,31,75,53,59,45,51,95...
Average: 72.72
Average/samples: 0.07271999999999999

Samples: 2000
Crossings: 101,121,143,129,131,107,143,73,61,145,127,73,89,119,151,137,111,119,145,55,141,89,125,75,115,113,67,153,115,143,115,129,125,101,125,123,141,25,137,17,99,45,39,107,53,105,115,115,127...
Average: 105.58
Average/samples: 0.05279

Samples: 3000
Crossings: 99,153,95,143,137,125,163,125,153,189,169,195,141,73,99,115,127,97,89,115,159,133,109,127,93,195,141,121,153,147,51,171,135,187,93,153,141,57,119,137,119,113,147,161,107,91,87,145,183...
Average: 123.7
Average/samples: 0.04123333333333334

Samples: 4000
Crossings: 37,125,185,183,101,127,119,203,173,161,159,159,125,131,175,161,103,177,177,127,137,163,97,217,87,111,155,141,197,189,187,149,103,173,181,109,171,179,207,217,121,145,183,169,113,135,47,201,139...
Average: 145.38
Average/samples: 0.036345

Samples: 5000
Crossings: 125,207,177,191,141,209,185,161,175,123,139,147,153,193,221,145,175,183,125,133,191,223,215,113,157,229,173,151,171,115,155,69,185,187,213,75,201,69,131,161,113,177,169,185,47,175,147,239,135...
Average: 159.16
Average/samples: 0.031832

Samples: 6000
Crossings: 99,167,211,161,157,215,141,217,197,135,155,131,123,159,203,175,193,173,183,187,169,149,165,189,159,23,165,189,161,179,207,159,173,181,135,177,189,137,163,211,171,195,269,201,133,71,145,175,137...
Average: 168.86
Average/samples: 0.028143333333333336

Samples: 7000
Crossings: 187,225,221,141,189,181,149,193,205,127,185,239,219,161,231,163,151,209,209,69,185,193,187,135,179,183,265,121,209,199,181,183,185,229,143,177,163,217,207,159,243,119,155,215,135,123,131,171,217...
Average: 179.72
Average/samples: 0.025674285714285715

Samples: 8000
Crossings: 191,143,157,139,121,137,143,201,215,171,195,139,227,223,231,221,219,127,165,197,183,233,225,77,183,153,175,223,149,177,183,191,195,183,151,187,233,193,193,105,247,199,167,175,149,215,173,151,171...
Average: 187.42
Average/samples: 0.023427499999999997

Samples: 9000
Crossings: 255,199,205,231,285,169,179,125,207,237,181,217,237,229,205,235,215,133,209,249,239,241,203,201,247,209,251,113,223,241,203,139,247,187,195,191,207,157,153,161,229,225,243,119,205,153,219,187,201...
Average: 200.96
Average/samples: 0.02232888888888889

Samples: 10000
Crossings: 231,205,237,229,187,191,227,237,185,227,147,137,209,257,127,161,215,133,209,127,187,99,115,255,199,283,177,209,131,169,137,253,203,169,175,179,159,197,217,233,219,181,261,215,199,111,267,197,193...
Average: 198.02
Average/samples: 0.019802

The calculations were done in 3 batches. The first batch (10s) were based on 1000 graphs each. The second and third batches (100s and 1000s) were based on 100 graphs each. The final batch took a minute and a half to compute! I'm not sure what it means. I thought the Average/samples would converge on a number but they just seem to keep dropping. nagualdesign 00:41, 6 February 2018 (UTC)

It’s interesting that the average number of crossings as a ratio to the sample size decreases monotonically as the sample size increases. Loraof (talk) 02:42, 6 February 2018 (UTC)

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