Wikipedia:Reference desk/Mathematics: Difference between revisions
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= June 7 = |
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== Reduced Dedekind Phi function == |
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[[Euler totient function]] <math>\phi(n)</math> {{OEIS|A000010}} has a reduced function: [[Carmichael lambda function]] <math>\lambda(n)</math> {{OEIS|A002322}}, and <math>\lambda(xy) = lcm(\lambda(x),\lambda(y))</math> instead of <math>\lambda(x)\lambda(y)</math> for coprime x, y, while <math>\lambda(p^r)=\phi(p^r)=(p-1)p^{r-1}</math> if p is prime and r>=1, and a composite number n is [[Carmichael number]] if and only if <math>\lambda(n)</math> divides <math>n-1</math>, and [[Dedekind psi function]] <math>\psi(n)</math> {{OEIS|A001615}} should also have a reduced function: <math>\eta(n)</math>, and <math>\eta(xy) = lcm(\eta(x),\eta(y))</math> instead of <math>\eta(x)\eta(y)</math> for coprime x, y, while <math>\eta(p^r)=\psi(p^r)=(p+1)p^{r-1}</math> if p is prime and r>=1, and a composite number n is [[Lucas-Carmichael number]] if and only if <math>\eta(n)</math> divides <math>n+1</math>, but I cannot even find the function <math>\eta(n)</math> in OEIS (it should start with (start from n=1) 1, 3, 4, 6, 6, 12, 8, 12, 12, 6, 12, 12, 14, 24, 12, 24, 18, 12, 20, 6, 8, 12, 24, 12, …). [[Special:Contributions/125.230.9.88|125.230.9.88]] ([[User talk:125.230.9.88|talk]]) 02:38, 7 June 2024 (UTC) |
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:<math>\eta(399)=\text{lcm}(\eta(3),\eta(7),\eta(19))=\text{lcm}(4,8,20)=40,</math> yet <math>40\not\mid398.</math> Did you mean to write "if and only if <math>\eta(n)</math> divides <math>n+1</math>"? But for any prime <math>p,</math> <math>\eta(p)=p+1.</math> The modified criterion requires <math>n</math> to be composite. --[[User talk:Lambiam#top|Lambiam]] 06:55, 7 June 2024 (UTC) |
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::Sorry, typo, <math>n+1</math> instead of <math>n-1</math>, fixed. [[Special:Contributions/2402:7500:92C:A1B7:8906:5B5F:EA02:8E54|2402:7500:92C:A1B7:8906:5B5F:EA02:8E54]] ([[User talk:2402:7500:92C:A1B7:8906:5B5F:EA02:8E54|talk]]) 15:02, 7 June 2024 (UTC) |
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::It is also true for Carmichael numbers, n is a number n is [[Carmichael number]] if and only if <math>\lambda(n)</math> divides <math>n-1</math>, but for any prime p, <math>\lambda(p)=p-1</math>, thus both require that n is composite … I fixed it. [[Special:Contributions/2402:7500:92C:A1B7:8906:5B5F:EA02:8E54|2402:7500:92C:A1B7:8906:5B5F:EA02:8E54]] ([[User talk:2402:7500:92C:A1B7:8906:5B5F:EA02:8E54|talk]]) 15:06, 7 June 2024 (UTC) |
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:Also, you state above that <math>\lambda(p^r)=\phi(p^r),</math> yet, according to both [[Carmichael function]] and A002322, <math>\lambda(2^3)=2</math> while <math>\phi(2^3)=4.</math> "Your" <math>\lambda</math> function is given as {{OEIS el|A011773|Variant of Carmichael's lambda function}}. --[[User talk:Lambiam#top|Lambiam]] 13:44, 7 June 2024 (UTC) |
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::Well, I forgot that it is only worked for odd prime p, and not worked for p=2 and r>=3, thus this function is in fact {{OEIS|A011773}}. [[Special:Contributions/2402:7500:92C:A1B7:8906:5B5F:EA02:8E54|2402:7500:92C:A1B7:8906:5B5F:EA02:8E54]] ([[User talk:2402:7500:92C:A1B7:8906:5B5F:EA02:8E54|talk]]) 15:04, 7 June 2024 (UTC) |
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== Choosing a point inside a quadrilateral == |
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Is it possible to choose a point strictly inside a simple quadrilateral such that all four sides (and hence the whole interior) are visible from the point? The chosen point is required to be a smooth function of the the vertices, reasonably middling and free from any nastiness (such as depending on the labeling of the vertices). Thanks, [[User:Catslash|catslash]] ([[User talk:Catslash|talk]]) 16:58, 7 June 2024 (UTC) |
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: All four sides would obviously be visible from any random point inside a quadrilateral that isn't concave. I suspect your question is more complex that that but you haven't expressed it optimally. -- [[User:JackofOz|<span style="font-family: Papyrus;">Jack of Oz</span>]] [[User talk:JackofOz#top|<span style="font-size:85%; font-family: Verdana;"><sup>[pleasantries]</sup></span>]] 19:25, 7 June 2024 (UTC) |
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:If you take each of the four edges and consider the half-planes on the interior sides of each edge, then the intersection of all half-planes is the precise set of points which can "see" all sides. Although I haven't formalized a proof, I'm confident that this set is always nonempty (and also a convex quadrilateral) if the original quadrilateral is non-trivial. |
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:As for a smooth function, I'm sure that there is some definition of the center of a convex quadrilateral that, applied to this set of points, would suffice. [[User:GalacticShoe|GalacticShoe]] ([[User talk:GalacticShoe|talk]]) 20:02, 7 June 2024 (UTC) |
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::On reflection, since convex quadrilaterals can only take one specific form, with one concave vertex, two "wingtips", and a "head", it's very easy to describe this shape. It's just the quadrilateral formed from the concave vertex, the head, and the two intersections of opposite sides. As for the center, you can use the intersection of diagonals of this internal quadrilateral. [[User:GalacticShoe|GalacticShoe]] ([[User talk:GalacticShoe|talk]]) 22:34, 7 June 2024 (UTC) |
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:::When just one angle of the original quadrilateral is very close to 180°, then so is one angle of the convex inner quadrilateral, at the same vertex, and the intersection of its diagonals is very close to that vertex. In the degenerate case that the angle is equal to 180°, as when a concave quadrilateral is continuously transformed into a convex quadrilateral, the point of intersection coincides with the vertex. Using the [[Centre (geometry)#General polygons|vertex centroid]] or [[Centroid#Of a polygon|area centroid]] of the convex inner quadrilateral as the chosen point avoids this. --[[User talk:Lambiam#top|Lambiam]] 07:01, 8 June 2024 (UTC) |
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::In general, for any area {{math|''A''}} enclosed by a [[Jordan curve]], the set {{math|''P''(''A'')}} of panoptic points is convex. It is the same as {{math|''A''}} iff {{math|''A''}} is convex. If the boundary is a polygon, {{math|''P''(''A'')}} is empty, a single point, a line segment, or the area enclosed by a polygon. --[[User talk:Lambiam#top|Lambiam]] 15:43, 8 June 2024 (UTC) |
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:<s>I think</s> I thought – but I was wrong – that the midpoint of the shortest diagonal satisfies the visibility criterion, but it is not a continuous function of the locations of the vertices. I conjecture that there is a continuous function w from these locations, ordered cyclically, to the unit interval, where w(v<sub>2</sub>, v<sub>3</sub>, v<sub>4</sub>, v<sub>1</sub>) = 1 − w(v<sub>1</sub>, v<sub>2</sub>, v<sub>3</sub>, v<sub>4</sub>), invariant under similarity transformations, such that the weighted mean of the diagonal midpoints (v<sub>1</sub> + v<sub>3</sub>) / 2 and (v<sub>2</sub> + v<sub>4</sub>) / 2, with weights w and 1 − w, satisfies both requirements. For any configuration, there is an interval [w<sub>lo</sub>, w<sub>hi</sub>] of w-values that guarantee visibility. If we can prove that both endpoints vary continuously with the vertices, a solution is provided by w = (w<sub>lo</sub> + w<sub>hi</sub>) / 2. --[[User talk:Lambiam#top|Lambiam]] 21:15, 7 June 2024 (UTC) |
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::The midpoint of the shortest diagonal does not work. Consider (0, 0), (1, 4), (0, 3), (-1, 4) taken in that order. The diagonal (1, 4), (-1, 4) is shorter than (0, 0), (0, 3), but its midpoint lies outside the quadrilateral. Of course if both diagonals lie inside the quadrilateral then it's convex and any interior point will do. --[[User:RDBury|RDBury]] ([[User talk:RDBury|talk]]) 02:06, 8 June 2024 (UTC) |
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:::The vertex centroid of the (convex) panoptic quadrilateral seems to do the job. Many thanks. If there is a practical way of choosing a point on the [[Newton line]] of the original quadrilateral, that would also be very interesting. [[User:Catslash|catslash]] ([[User talk:Catslash|talk]]) 12:57, 8 June 2024 (UTC) |
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::::At least one of the midpoints of the two diagonals is contained in the panoptic quad, so the panoptic part of the Newton line is nonempty. It should be easy to determine the endpoints of the (possibly degenerate) panoptic segment of this line and take its midpoint. If the original quadrilateral is convex, this is a trivial exercise. Otherwise, you have to determine where the extended sides adjacent to the concave vertex cut the segment connecting the midpoints short. As with many problems in [[computational geometry]], the hardest part is not the computations themselves but to get the case distinctions correct and complete. |
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::::I think though the result will in extreme cases be markedly less "middling" than the vertex centroid. --[[User talk:Lambiam#top|Lambiam]] 15:26, 8 June 2024 (UTC) |
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:I hoped to find a relevant theorem in [[Star-shaped polygon]], but no. [[User:Tamfang|—Tamfang]] ([[User talk:Tamfang|talk]]) 18:35, 19 June 2024 (UTC) |
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::''Kernel'' is less descriptive and perhaps duller than ''panoptic region''. [[User:Catslash|catslash]] ([[User talk:Catslash|talk]]) 09:09, 20 June 2024 (UTC) |
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= June 9 = |
= June 9 = |
Revision as of 00:37, 22 June 2024
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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:
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)
- 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 Wikipedia: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 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)
- 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)
References
- ^ Masherov, P. E.; Riaby, V. A.; Abgaryan, V. K. (2016-08-01). "Note: Refined possibilities for plasma probe diagnostics". Review of Scientific Instruments. 87 (8). doi:10.1063/1.4960396. ISSN 0034-6748.
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, 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)
- 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 is where . One can expect a fraction of to be prime. --Lambiam 08:42, 12 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)
June 20
First fundamental form and curvature
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)
- 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)