Wikipedia:Reference desk/Archives/Mathematics/2024 June 9

= June 9 =

Source for Langmuir-Blodgett, Langmuir-Boguslavski, and weird Rayleigh equations?
While cleaning up List of nonlinear ordinary differential equations and citing all the ones listed, there were three that puzzled me to no end. The first was listed as the Langmuir-Blodgett equation:

$$\sqrt{y}\frac{d^2y}{dx^2}=e^x$$

The next was listed as the Langmuir-Boguslavski equation:

$$\frac{d}{dx}\left(x^n\frac{dy}{dx}\right) = \frac{1}{\sqrt{y}}$$

Finally, there was an equation listed as the Rayleigh equation:

$$\frac{d^2y}{dx^2} + k \frac{dy}{dx} + m \left(\frac{dy}{dx}\right)^3 + n^2 y =0$$

I just want to know if anyone recognizes these or has sources for them. The first two I could only find mention of in a footnote of an old edition of a differential equations handbook, which itself cited no sources for these and they do not appear in the more recent edition of the handbook as far as I can tell, and the last one looks neither like the regular Rayleigh equation (which is notably linear) or the variant of the Van der Pol equation which is sometimes called the Rayleigh equation (and both of these drown out any search results for this equation). The two equations named after Langmuir I also checked in plasma physics textbooks for, as I vaguely recall that Langmuir worked on plasma, but I could not find mention in the two introductory books I checked. The closest I could get were sources like this one but I can't seem to tell if the given equation is equivalent, and the source they cite is O. V. Kozlov, An Electrical Probe in a Plasma, which I cannot find online. (There's also Langmuir-Blodgett film but no differential equation is mentioned in that article.) These have been plaguing me, and the editor who added them hasn't edited in six years so no dice there. Any help would be appreciated! Nerd1a4i (they/them) (talk) 19:57, 9 June 2024 (UTC)


 * Differential equations with the names Langmuir-Blodgett and Langmuir-Boguslavski are given here, without further explanation of reference. The origin of the former is possibly an equation presented in a joint publication by Langmuir and Blodgett many years before the technique was developed to make Langmuir-Blodgett films. --Lambiam 06:31, 10 June 2024 (UTC)
 * Yes, that was the handbook that was the one other source I saw - the other edition of the same handbook I was referencing didn't mention these. Nerd1a4i (they/them) (talk) 01:46, 11 June 2024 (UTC)
 * @Nerd1a4i, I don't have time to delve into the details, but the first two might have been invented on Wikipedia. Check out the mention at List of citogenesis incidents. —Kusma (talk) 08:20, 10 June 2024 (UTC)
 * That was me adding it to the list of citogenesis incidents as that was what I believed to be true at the time. I then realized I should ask here. Nerd1a4i (they/them) (talk) 01:45, 11 June 2024 (UTC)
 * The Langmuir-Blodgett equation may come from this paper or the "previous papers" cited in footnote 1. --Lambiam 11:09, 10 June 2024 (UTC)
 * Thanks, I'll try to go through that paper and see if it's got the right equation. Much appreciated for finding a fresh starting point! I don't suppose you have any leads for the other two? Nerd1a4i (they/them) (talk) 01:47, 11 June 2024 (UTC)
 * Here is an unresolved lead. In the authors refer to "", which is not a differential equation but is called a  – it has a factor $$\Delta V_t^{3/2}.$$ In a second note,, the same authors call this a law that "". The abstract mentions "", and a later note by partially the same authors, , mentions "", two equations that were solved jointly.
 * Our article Debye sheath has subsections 2.2 The Bohm sheath criterion and 2.3 The Child–Langmuir law, while Child–Langmuir law redirects to . It remains unclear where Boguslavsky (or Boguslavski) enters the picture and how this relates to the differential equation. --Lambiam 05:11, 11 June 2024 (UTC)

What is the largest number satisfying this condition?
Numbers which contain no repeating number substring, i.e. does not contain “xx” for any nonempty string x (of the digits 0~9), i.e. does not contain 00, 11, 22, 33, 44, 55, 66, 77, 88, 99, 0101, 0202, 0303, 0404, 0505, 0606, 0707, 0808, 0909, 1010, 1212, 1313, 1414, 1515, …, 9797, 9898, 012012, 013013, 014014, …, 102102, 103103, 104104, … as substring. Are there infinitely many such numbers? If no, what is the largest such number? 2402:7500:92C:2EC4:C50:24C1:2841:C6B5 (talk) 23:25, 9 June 2024 (UTC)


 * Off the top of my head, I think there are an infinite number of them. Bubba73 You talkin' to me? 00:23, 10 June 2024 (UTC)


 * The decimal representation of a natural number is a word in the regular language $A*$ over the alphabet $A = {0, 1, 2, 3, 4, 5, 6, 7, <tt>8</tt>, <tt>9</tt>}.$ The numbers satisfying the condition that their decimal representation avoids the pattern XX correspond to the square-free words of that language. As you can read in the article, there are even infinitely long square-free words. --Lambiam 05:56, 10 June 2024 (UTC)
 * Another question: Are there infinitely many such numbers which are primes? 118.170.47.29 (talk) 07:25, 12 June 2024 (UTC)
 * Since there is no discernible logical relation between being non-repeating and being prime, the answer is almost certainly yes, although it may be difficult or impossible to prove this. The number of non-repeating numbers up to $$N$$ is $$O(N^\beta),$$ where $$\beta=\log_{10}\alpha\approx\log_{10}8.8874856\approx0.9487789$$. One can expect a fraction of $$\tfrac1{\ln N}$$ to be prime. --Lambiam 08:42, 12 June 2024 (UTC)