June 9[edit]

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:

The next was listed as the Langmuir-Boguslavski equation:

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

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^[1] 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)[reply]

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)[reply]

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)[reply]

@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 Wikipedia:List of citogenesis incidents. —Kusma (talk) 08:20, 10 June 2024 (UTC)[reply]

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)[reply]

The Langmuir-Blodgett equation may come from this paper or the "previous papers" cited in footnote 1.  --Lambiam 11:09, 10 June 2024 (UTC)[reply]

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)[reply]

Here is an unresolved lead. In doi:10.1063/1.4948923 the authors refer to "the Boguslavsky-Langmuir equation for a cylindrical probe under floating potential", which is not a differential equation but is called a “3/2 power” law – it has a factor In a second note, doi:10.1063/1.4960396 , the same authors call this a law that "for cylindrical probe under floating potential corresponds to the Child- Boguslavsky-Langmuir (CBL) equation". The abstract mentions "the Child-Boguslavsky-Langmuir (CBL) probe sheath model", and a later note by partially the same authors, doi:10.1063/1.5022236, mentions "the Bohm and Child–Langmuir–Boguslavsky (CLB) equations for cylindrical Langmuir probes", 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 Space charge § In vacuum (Child's law). 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)[reply]

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)[reply]

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)[reply]

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, 8, 9}. 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)[reply]

Another question: Are there infinitely many such numbers which are primes? 118.170.47.29 (talk) 07:25, 12 June 2024 (UTC)[reply]

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 is where . One can expect a fraction of to be prime.  --Lambiam 08:42, 12 June 2024 (UTC)[reply]

June 20[edit]

I wanted to ask you for help in better understanding two concepts that leave me a little perplexed regarding the first fundamental form:

1) The first fundamental form was defined to me as follows: given a parameterization ϕ(u,v) , the metric A is found with the dot products like ϕu​⋅ϕv​ (through the 4 permutations of u,v), where the dot products are restricted to the tangent spaces induced by R3. It was then explained that quantities dependent on E,F,G (such as the Christoffel symbols, and therefore the curvature via Gauss's Theorema Egregium since it shows that they depend only on the Christoffel symbols) are intrinsic quantities, meaning they do not depend on how the surface is immersed in R3, but are intrinsic to the object itself.

My confusion revolves around this: if I define the first fundamental form in this way, I note that ϕ(u,v) is the map ϕ:U→R3, so what comes out of this map is precisely the figure I have as a surface in R3R3, that is, the shape it takes. In fact, the map gives me the coordinates (x,y,z). Now, ϕ​ and ϕv​ are the tangent vectors to that figure, so they indeed depend on the shape realized in R3. Then I define the matrix A through the dot products of ϕu​ and ϕv​, so what is intrinsic here? I am using tangent vectors to a figure that has a shape given by ϕ:U→R3, so I would say it indeed depends on the immersion and how the figure is geometrically realized in it.

What seems to be suggested by the explanation is this: if I immerse the "abstract concept" of a sphere in R3, I have different realizable figures. I do not understand this concept given the considerations above: curvature depends only on E,F,G through the Christoffel symbols, but E,F,G depend on the dot products of tangent vectors to a shape of the surface, so on an immersion of it in R3.

2) The second question is this: if I change parameterization, I will have new ϕi'​,i∈u,v which differ from the initial ones, so I will find E',F',G'. However, for an object (let's take the usual sphere), the curvature is fixed and depends only on E,F,G right? Well, if I have changed parameterization and have E',F',G', why wouldn't the curvature change? They are different values. It seems to me that by changing parameterization, the first fundamental form changes and therefore the curvature should change as well (but it shouldn't obviously be so).

Could you help me with these two questions? Thank you. --151.36.108.141 (talk) 15:42, 20 June 2024 (UTC)[reply]

E,F,G are relative to a particular tangent space and the values will of course change if one changes the tangent space. One has to get rid of the tangent space dependency to get things like the curvature. The determinant gives a measure of the area of dudv and dividing the determinant of the second fundamental form by that of the first fundamental form gives something independent of dudv - and which in fact is the Gaussian curvature. NadVolum (talk) 15:09, 21 June 2024 (UTC)[reply]

June 22[edit]

I noticed that 1/7 in base that 3,6 & 9 are missing from the expansion in base 10, but it appears that there is a pattern.

• k = 2, 1/3 in base 5 is .13(rep), 2/3 in base 5 is .31 (no digit which is a multiple of 2 appears in the pentary expansion, neither 2 or 4 occur
• k = 3, 1/7 in base 10 is .142857(rep) (no digit which is a multiple of 3 occurs in the decimal expansion, none of 3, 6 or 9 occur
• k = 4, 1/13 (1/C) in base 17 is .153FBD(rep) (not only doesn't any digit which is a multiple of 4 occur (4,8,C,G), it appears that digits which are a multiple of 2 occur.
• k = 5, 1/26 (1/L) in base 26 is .164OJL(rep) (no digit which is a multiple of 5 occurs (5,A,F,K,P)

Any idea for a proof or extension of this? (I can't find an easy calculation in base 37)Naraht (talk) 02:11, 22 June 2024 (UTC) .[reply]

There seems to be a typo in the k = 5 case; (k-1)k+1=21, not 26, and 1/26 base 26 is just .1. It might be easier to use a different notation for large bases, say by inserting a comma between each digit and leaving the digits in base 10. So 1/21 = .1,6,4,24,19,21(rep). In this notation 1/31 = (base 37) .1,7,5,35,29,31. I think you can probably see what's going on by looking at k=10 and k=100: 1/91 = (base 101) .1,11,9,99,89,91(rep), and 1/9901 = (base 10001) .1,101,99,9999,9899,9901(rep). The pattern is 1/(k²-k+1) = (base k²+1) .1,k+1,k-1,k²-1,k²-k-1,k²-k+1, which shouldn't be too hard to verify. There is almost certainly python code to compute fractions in large bases to do further experimentation. k²-k+1 and k²+1 are both cyclotomic polynomials, and there are probably similar patterns when you look at 1/Phi_m(k) base Phi_n(k) for various m and n. --RDBury (talk) 15:15, 22 June 2024 (UTC)[reply]

PS. The fact that the denominator is a cyclotomic polynomial is the part of makes this work, the other part isn't that the base is a cyclotomic polynomial in k, but that it's equal to ±k mod the denominator. In this case the denominator is Φ₆(k) and the denominator is Φ₆(k)+k. A generalization is that if P(x) is congruent to x mod Φ_n(x) (taken as polynomials in x) then for a 1/Φ_n(k) has period dividing n base P(k). For example Φ₁₀(x)=x⁴-x³+x²-x+1 and P(x)=x⁸+x⁴+x²+1 is congruent to x mod Φ₁₀(x). So the generalization states that Φ₁₀(k) has period dividing 10 base P(k). For k=10 we get 1/9091 = (base 100010101) .11001,110010,1100100,11001001,9999909,99999099,99900090,98910000,89009099,90010191(rep) and this indeed has period 10. If P(x) is congruent to -x mod Φ_n(x) the rule is that the period divides 2n. For n=3, k=3, 1/13 = .076923(rep) base 10; for k=4, 1/21 = .0,13,12,16,3,4(rep) base 17; for k=10, 1/111=.0,91,90,100,9,10; and in general 1/(k²+k+1) = .0,k²-k+1,k²-k,k²,k-1,k(rep). There are probably additional tweaks and generalizations you can make on this. --RDBury (talk) 16:03, 23 June 2024 (UTC)[reply]

June 23[edit]