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April 26

duality vs. conjugacy

I noticed that Isbell conjugacy and Isbell duality have the same meaning. So, I would like to know the difference in meaning between duality and conjugacy in mathematics. Also, I found Category:Duality theories, but what is the field of mathematics called Duality theory? However, since Baez (2022) said that the Isbell conjugacy is an adjoints rather than a duality of the category, so I changed the category to Category:Adjoint functors. Thank you, SilverMatsu (talk) 03:54, 26 April 2024 (UTC)

The original duality occurs in projective geometry, see Duality (projective geometry). At some point people noticed that the axioms of the projective plane where the same (or equivalent) if you swapped the undefined terms "point" and "line". So any theorem in projective geometry can be transformed to a dual theorem by changing the roles of points and lines. The new theorem may simply be a restatement of the original theorem as in Desargues's theorem, but sometimes it's not as in Pappus's hexagon theorem. The result is that you often get two theorems for the price of one proof. You can define a dual category for a given category by reversing the arrows, but category theory was invented long after projective geometry so that's not the original meaning. You can also define the dual curve of a plane curve, the dual space of a vector space, the dual polytope of a polytope, etc. As far as I know there is no all-encompassing "theory of duality", just the custom of using "dual" to describe when mathematical objects seem to occur in pairs in some way. Calling something a dual usually implies that the dual of the dual is in some way identifiable with the original object, but this is not always required. For example the dual of a dual vector space is not identifiable with the original vector space unless it's finite dimensional. Duality does not always exist, for example there doesn't seem to be a useful concept for the dual of a finite group, though you can define one for abelian groups. And sometimes there is a duality that's not called that, for example cohomology can be viewed as the dual of homology. I don't think there is a formal distinction between a "dual" and a "conjugate", but usually a conjugate is the the result of applying an automorphism of order two. For example a complex conjugate is the result of applying the automorphism a+bi → a-bi. Again, this is more of a naming custom than a formal mathematical concept, and there is (apparently) some overlap. I'd say a "conjugate" is usually used when the two objects live inside the same structure, and "dual" is used when you're talking about two different structures. For example the dual of a plane curve lives in the dual of the plane in which the original curve lives. Category theory blurs the distinction between an object and a structure so I can see how the distinction is rather meaningless there. --RDBury (talk) 07:12, 26 April 2024 (UTC)

Thank you for teaching me so kindly. I'm going to re-read some of the references, keeping in mind what you've taught me. --SilverMatsu (talk) 16:02, 27 April 2024 (UTC)

Oblate spheroid

I thought of Googling orthographic projection ellipsoid and found these, did I interpret everything right?:

ν=a/(1-e²*(sinΦ)²)^1/2 (why not square root?) x=(ν+h)*cosΦ*cosλ y=(ν+h)*cosΦ*sinλ z=(ν*(1-e²)+h)*sinΦ This seems to be a simple spherical to Cartesian converter with latitudes (Φ) "massaged" so it's not slightly wrong (eccentricity²=0.00669437999014 so not much massaging). Then they convert that to topocentric Cartesian with a matrix I can't solve (now I know why galactic Cartesian's UVW!) but it seems like they also say surface points are U=ν*cosφ*sin(λ-λ_O) V=ν*(sinφ*cosφ_O-cosφ*sinφ_O*cos(λ-λ_O))+e²*(ν_O*sinφ_O-ν*sinφ)cosφ_O where _O means "of the topocentric origin". Did I get that right? If so then I can set an initial guess point at or about 0.25 circumference from the W-axis, use the formulae to find its U and V in "W-axis place"-centered coordinates and the test point is of course √(U²+V²) meters from the W-axis and the part of the ellipsoid with the most meters without being too far from the W-axis-test point plane is the limb of the Earth from infinite distance. The worst-case scenario for how spindly a pie the test points have to be in would be looking at ~the 45th parallel limb with the W-axis in the equator plane. The geocenter depth increases roughly quarter mile from 45.5N to 44.5N so ~56 meters poleward shortens the limb to W-axis line segment by 8 inches which is how much Earth curves in a mile. Azimuth accuracy needed increases "exponentially" with limb coordinate accuracy desired though so 10 miles accuracy would be a lot more than 10x easier than 1 mile. Sagittarian Milky Way (talk) 06:24, 26 April 2024 (UTC)

(Why not square root? Because the site may not have the facility to extend the radical bar over the whole expression.) —Tamfang (talk) 17:19, 7 May 2024 (UTC)

It's a pdf https://www.hydrometronics.com/downloads/Ellipsoidal%20Orthographic%20Projection.pdf since it has images couldn't they put a radical bar instead of 1/2th power? Anyway did I get it right, I never got to the part of school where they taught matrixes but maybe the stuff below is equivalent and not screwed up by my 100% truancy from late high school math conventions education. Sagittarian Milky Way (talk) 23:19, 7 May 2024 (UTC)

The text on these slides was obviously created using a word processing tool, and (apart from matters of taste) the choice of presentation may be affected by limitations of that tool. It all looks complicated, but actually the orthographic projection is as simple as it gets: map the 3D point $(x,y,z)$ to the 2D point $(x,y),$ "forgetting" one of the three spatial coordinates. For a sphere, one can use the transformation of spherical coordinates $(r,\theta ,\phi )$ to Cartesian coordinates $(x,y,z)$ (see the section Spherical coordinate system § Cartesian coordinates). A rotational ellipsoid requires an additional scaling of one of these coordinates. If the desired view of a 3D object is from an angle, rotate it in 3D space before projecting it. The general case is merely the chain of these by themselves standard, fairly simple transformations, of which spatial rotation is the most difficult. (It definitely helps to understand the notion of multiplying a 3×3 matrix and a 3D vector to get a new 3D vector; see Rotation matrix § In three dimensions.) A more natural perspective projection is only moderately more complex: one needs in general to also translate the object in 3D space, and the final mapping is given by $(x,y,z)\mapsto (x/z,y/z).$  --Lambiam 09:19, 8 May 2024 (UTC)

I long wondered what that was, then wondered how they could simplify 2-D lists, then wondered if it was multiplying everything by everything like a 2-dimensional version of first-outside-inside-last. When I read matrix (math) I'll probably slap forehead at how few seconds it takes to teach and how simple it is compared to how hard it is to understand the hieroglyphics completely without a dictionary or Rosetta Stone. Similar to when I treated the options, futures, future options and sports betting odds in the newspaper as puzzles for 5-6 years and though I gave up and Googled at age ~18 even ten or twenty more years might've not been enough despite each of these conventions being easily teachable in seconds (i.e. USA has its own odds convention for non-horse sports, why was one team always -105 or less (minus many thousands if they're very likely to win) the other team was +100 to +many thousands (always somewhat less positive than the other was negative) but sometimes it's -115 -105 or PK which obviously meant pick 'em/50% chance. Completely stumped me what that means, if it was consistently how much a win profits per $100 or how much to win $100 I would've figured it out but it was just the bet per $100 bet without the $ sign, minus="you bet this not him". Very anticlimactic, like Googling how magicians drop a cup being filled by a cup without it falling (ROT13: Pyrneguernq) Sagittarian Milky Way (talk) 16:28, 8 May 2024 (UTC)

April 27

"Distribution diagrams"

Distribution of (term node) sharing factor for a population of theorem proving runs

I'm trying to show the distribution characteristics of a numerical value in a (finite) population. To do so, I sort the values in ascending order, and then plot the feature values over the position of the value in the sorted sequence, as per the attached example. I'm probably not the first with that idea - is there a standard name for this kind of diagram? And/or is there a better way to visualise such data? --Stephan Schulz (talk) 13:36, 27 April 2024 (UTC)

If you switch axes (or turn your head sideways) this is the graph of a typical Cumulative distribution function. Perhaps it's better to call it a cumulative frequency instead of a distribution since you're plotting values observed and not the theoretical probability density, but the idea is the same. In particular, your graph resembles the second image shown in the article only turned sideways. The (usual) probability density is simply the derivative of the cumulative distribution function, so if you can estimate the derivative in your diagram that may give a better visual representation. The usual technique is to divide the range in to intervals, and then graph the number of occurrences in each interval. It seems to me that there might be a name in economics for the "sideways" version (enonomists seem to do a lot of things sideways), but I don't know what it would be. --RDBury (talk) 14:41, 27 April 2024 (UTC)

It is also customary, when plotting a cumulative distribution, to let the (now vertical) axis mark relative values in the range from 0 to 1 (or, equivalently and perhaps more commonly, from 0% to 100%) instead of an absolute ranking like from 1 to 7794 or whatever the sample size may be.  --Lambiam 15:06, 27 April 2024 (UTC)

See also Empirical distribution function and Quantile function. —Amble (talk) 00:13, 30 April 2024 (UTC)

May 3

Can Carmichael number be Lucas-Carmichael number?

Can Carmichael number be Lucas-Carmichael number?

Also, varying the signs, there are four different sequences for similar numbers:

1. squarefree composite numbers k such that p | k => p-1 | k-1
2. squarefree composite numbers k such that p | k => p-1 | k+1
3. squarefree composite numbers k such that p | k => p+1 | k-1
4. squarefree composite numbers k such that p | k => p+1 | k+1

the 1st sequence is Carmichael numbers, and the 4th sequence is Lucas-Carmichael numbers, but what are the 2nd sequence and the 3rd sequence? Are there any number in at least two of these four sequences? If so, are there any number in at least three of these four sequences? 61.224.150.139 (talk) 05:07, 3 May 2024 (UTC)

According to Lucas–Carmichael number, it is unknown whether there are any Lucas–Carmichael numbers that are also Carmichael numbers. GalacticShoe (talk) 05:52, 3 May 2024 (UTC)

2. The sequence is OEIS:A208728, and it starts $15,35,255,455,1295,2703,4355,6479,9215,\ldots$

3. The sequence is OEIS:A225711, and it starts $385,2737,6061,6721,17641,24769,25201,\ldots$

GalacticShoe (talk) 06:21, 3 May 2024 (UTC)

Do all numbers in any of these four sequences except 15 and 35 have at least three prime factors? 61.224.150.139 (talk) 06:41, 3 May 2024 (UTC)

Yes. You can show that:

1. If $p-1,q-1|pq-1$, then $p-1|q-1$ and $q-1|p-1$, implying $p-1=q-1\Rightarrow p=q$, which is disallowed.
2. If $p-1,q-1|pq+1$, then $p-1|q+1$ and $q-1|p+1$, implying either $p=q-2$ or $p=q+2$ (since they can't be equal.) The rest of this proof is left to the reader since I don't feel like writing it down, but based on the fact that $n|(n+2)^{2}\Leftrightarrow n|4$, it can be shown that $pq=15,35$ only.
3. If $p+1,q+1|pq-1$, then $p+1|q+1$ and $q+1|p+1$, implying $p+1=q+1\Rightarrow p=q$, which is disallowed.
4. If $p+1,q+1|pq+1$, then $p+1|q-1$ and $q+1|p-1$, implying $p\leq q-2$ and $p\geq q+2$, which is not possible.

GalacticShoe (talk) 07:52, 3 May 2024 (UTC)

Are all of these four sequences infinite? If so, do all of these four sequences contain infinitely many terms with exactly 3 prime factors? Also, do all of these four sequences contain infinitely many terms which are divisible by a given odd prime number? 2402:7500:943:D56F:909B:9877:85C8:AFAA (talk) 02:02, 5 May 2024 (UTC)

@GalacticShoe: 49.217.60.214 (talk) 05:02, 6 May 2024 (UTC)

In order for a sequence to be in both 1. and 2., this would require that all prime factors $p|k$ satisfy both $p-1|k-1,p-1|k+1\Rightarrow p-1|2\Rightarrow p=2,3$. The only squarefree composite number that is only composed of $2,3$ is $6$ which can easily be seen to not be in either sequence. Similarly, 3. and 4. would require all prime factors $p|k$ to satisfy both $p+1|k-1,p+1|k+1\Rightarrow p+1|2$ which does not hold for any primes $p$. Since 1. and 2. cannot coexist, nor can 3. and 4., this means that no number occupies three or more of the four sequences. GalacticShoe (talk) 06:25, 3 May 2024 (UTC)

Carmichael numbers are the numbers n such that $\lambda (n)$ divides n-1, where $\lambda (n)$ is the Carmichael lambda function (also called reduced totient function, since it is the reduced form of the Euler totient function) (sequence A002322 in the OEIS), and Lucas-Carmichael numbers should be the numbers n such that $\eta (n)$ divides n+1, and this $\eta (n)$ should be a reduced form of the Dedekind psi function, use the same reduce rule as the Carmichael lambda function to the Euler totient function (i.e. use the least common multiple in place of the multiplication for $\lambda (mn)$ and $\eta (mn)$ with coprime m, n), but I cannot even find this function 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, …) 2402:7500:943:D56F:909B:9877:85C8:AFAA (talk) 02:13, 5 May 2024 (UTC)

Factorial & primorial on wikipedia

I know the factorial notation n!. Recently I cam across 5# which I was unfamiliar with. Not knowing its name, (primorial), it proved hard to track down. I searched in wikipedia and Google for "n#" which seemed like the best bet. Both converted it to "n" and reported stuff about the 14th letter of the alphabet. I then thought maybe it is related to factorial, so I looked at wikipedia factorial (n! redirects to factorial on wikipedia, so that works if/when you don't know the term "factorial".)

So is there a way of making n# findable on wikipedia? If so how? -- SGBailey (talk) 21:21, 3 May 2024 (UTC)

Search for "#" and find Number sign#Mathematics. —Kusma (talk) 22:25, 3 May 2024 (UTC)

@SGBailey: "#" is WP:FORBIDDEN in page names. I'm actually impressed that a search on "#" gives Number sign as the only result. I examined redirects and saw the similar Unicode characters ﹟ and ＃. Thinking that it may help search, I have redirected N﹟ and N＃ to Primorial. The "go" feature of the search box ignores "#" in "n#" and goes directly to N, but if you force a real search on n# then the third or fourth (it varies) result for me is now "Primorial (redirect from N﹟)". PrimeHunter (talk) 19:57, 6 May 2024 (UTC)

May 5

Origin of notion that there are ב sub 2 many "curves"

(Sorry for awkward heading -- I couldn't get it to put the ב before the 2 because of some strange artifact of RTL rendering.)

I've seen in several places the claim that, as there are $\aleph _{0}$ natural numbers and ${\mathfrak {c}}=2^{\aleph _{0}}$ (sometimes improperly given as $\aleph _{1}$) real numbers, there are some greater number of "curves" (sometimes given as f or, again improperly, $\aleph _{2}$). Most recently I was reminded of it at our article on George Gamow's (generally excellent) book One Two Three... Infinity.

The usual complaint about these popularizations, a very valid one, is that they uncritically give these cardinalities as aleph numbers in a way that works only if the generalized continuum hypothesis holds. But there's another, quite serious, problem: The claim that there are more "curves" than real numbers is correct only if you have an extremely liberal notion of what constitutes a "curve".

One reasonable notion is that a "curve" is the image of the real line or the unit interval under a continuous function from the reals to Rⁿ (or similar space), but there are only $2^{\aleph _{0}}$ such functions, and therefore the same number of curves.

My best guess is that someone was taking "curve" to mean the graph of an arbitrary function. But these are not typically curves according to any obvious natural-language meaning; they're just scattered points in the plane.

So, question, what's my question? Does anyone know where this idea originated? Was it Gamow, some other popularizer, multiple sources? And what if anything should we do to clean up the text in our One Two Three... Infinity article? I'm thinking an explanatory footnote but ideally I'd want a source directly speaking to the misconception. --Trovatore (talk) 20:58, 5 May 2024 (UTC)

This Math Stack exchange entry is relevant, but it doesn't seem to cover what you're asking. One problem is that the statement is true by Wikipedia standards; you could cite the book. You would need a reliable source, such as a published article somewhere, to say it was wrong/vague/misleading in order to state that in our article. At the moment the article points out that you'd need the GCH to say what's in the book, but I guess that's supposed to be "common knowledge" (at least among mathies). — Preceding unsigned comment added by RDBury (talk • contribs) 23:49, 5 May 2024 (UTC)

This article aims to classify various subsets of the function space 𝐹(ℝ,ℝ) from a constructive-mathematics perspective. The Introduction states: "mathematicians have made numerous attempts to focus on special subsets of this vast vector space (e.g., all real-valued continuous functions [5])", where the cited text is:

Pugh, C.C. Real Mathematical Analysis, 1st ed.; Undergraduate texts in mathematics; Springer Science Business Media: New York, NY, USA, 2002; pp. 223–225.

The latter is available as a pdf here. The article itself denotes this subset as 𝐶(ℝ,ℝ) and concludes in Proposition 4 that 𝐶𝑎𝑟𝑑(𝐶(ℝ,ℝ)) = 𝑐. But this is of course outside the paradise that Cantor created for you.  --Lambiam 07:09, 6 May 2024 (UTC)

The reply given by a fellow Wikipedian to another Math Stack exchange question appears to imply that this also holds within the paradise.  --Lambiam 07:23, 6 May 2024 (UTC)

Cardinality of the continuum § Sets with cardinality of the continuum also lists, without citation, "the set of all continuous functions from $\mathbb {R}$ to $\mathbb {R}$".  --Lambiam 07:30, 6 May 2024 (UTC)

Discussed at Stack Exchange. Basically it's because a continuous function from $\mathbb {R}$ to $\mathbb {R}$ is uniquely determined by its values at rational points. AndrewWTaylor (talk) 16:11, 6 May 2024 (UTC)

I feel a bit of sympathy for him making those mistakes but he should have had a mathematician read through that chapter. NadVolum (talk) 16:40, 6 May 2024 (UTC)

Well, it's kind of the publisher's job to do fact checking. The statement was still in the 2012 Dover edition, so there have been multiple chances to fact check since the original 1947 publication. --RDBury (talk) 19:12, 6 May 2024 (UTC)

In general I'm skeptical of active attempts to use Wikipedia to correct readers' mathematical misconceptions — too much like righting great wrongs, and can easily become a POV magnet (like the old "What mathematics is not" section that once appeared in our mathematics article).

This one irks me, though, and tempts me to go back on that reasoning. I guess it's slightly personal, because I had internalized this bit about the cardinality of the set of curves, and (embarrassingly) didn't get it corrected till grad school. I had figured out for myself that there were only continuum-many analytic functions, because they're determined by the coefficients of the power series, but I conjectured that there were 2^𝔠 many C^∞ functions, and someone had to set me straight on that.

I think it's not just Gamow (whose book, I want to re-emphasize, is a big net positive). I've been trying to remember where else I might have seen it. I thumbed through Lilian Lieber's Infinity (which is a book that heavily influenced me) and didn't find it there. --Trovatore (talk) 20:00, 6 May 2024 (UTC)

Just a guess, but the article mentions "What is Mathematics?" by Richard Courant and R. Robbins as a source, and of all the sources it seems the most mathematical. It's a "Text to borrow" on Internet Archive so if you create an account you can view it for free. --RDBury (talk) 10:42, 7 May 2024 (UTC)

There is a parallel in the treatment of cardinal numbers between Gamow and Courant & Robbins up to the point where the latter write (p. 85): "A similar argument shows that the cardinal number of the points in a cube is no greater than the cardinal number of the segment." After that, they muse briefly on the fact that this is counterintuitive since the correspondence does not preserve dimension, but that this is possible because it is not continuous. That ends their treatment of cardinal numbers. Earlier they note (p. 84): "As a matter of fact, Cantor actually showed how to construct a whole sequence of infinite sets with greater and greater cardinal numbers." They even sketch the proof, but do not pursue the question of mathematical objects of higher cardinality than the continuum that are of interest by themselves.  --Lambiam 13:21, 7 May 2024 (UTC)

Digression: In fact it's challenging to come up with an object of larger cardinality that might naturally be considered by non-set-theorists. One possibility is βN, the Stone–Čech compactification of the natural numbers. I believe this is mentioned in an exercise in Folland's Real Analysis. --Trovatore (talk) 18:53, 7 May 2024 (UTC)

There's a relevant MathOverflow question about finding cardinalities beyond that of the continuum outside set theory. βN is given as an answer, but maybe the most elementary one offered is the set of all field automorphisms of C. But the answers do kind of make me agree with Gamow's surely intended point that it's difficult to find natural objects of size beyond 2^c, though not with his actual assertion. :) Double sharp (talk) 15:21, 8 May 2024 (UTC)

May 6

Find

Given x=3+2√2, find √x - 1/√x 171.79.74.205 (talk) 17:01, 6 May 2024 (UTC)

the contradictory part is that in the end, you get (√x-1/√x)^2 = 4, which will give you ±2; but √x which is √3+2√2 can be written as √(2-√1)^2 which is 2-√1 hence √x - 1/√x = -2 171.79.74.205 (talk) 17:10, 6 May 2024 (UTC)

"... can be written ... ". No, it can't. Not sure if this is an honest question or just trolling. --RDBury (talk) 18:43, 6 May 2024 (UTC)

I'm not sure what they that is about but can I suggest that 1/(a+√b) = (a-√b)/{(a+√b)(a-√b)} might help? NadVolum (talk) 20:33, 6 May 2024 (UTC)

Clarify, please. Do you mean:

• √x - (1/√x)
• (√x - 1)/√x
• √(x - 1)/√x
• something else...?

I'd suggest using LaTeX/MathJax code within <math>...</math> tags to format the expressions like ${\frac {{\sqrt {x}}-1}{\sqrt {x}}}$ etc. Please see WP:MATH for more info. --CiaPan (talk) 10:12, 7 May 2024 (UTC)

Sorry I misread the question and answered the wrong thing. But the square root of 3+2√2 is plus or minus 1+√2 and the original answer of ±2 is correct. NadVolum (talk) 11:07, 7 May 2024 (UTC)

√x is usually taken to mean the positive square root when x is positive. At least that's the notation used in Square root. That would make the answer 2. --RDBury (talk) 16:58, 7 May 2024 (UTC)

Only if you say the square root, and nobody has said that. It also doesn't matter whether the original √2 is positive or negative. NadVolum (talk) 17:07, 7 May 2024 (UTC)

The issue here is not a lack of the definite article, but the meaning of the symbol ${\sqrt {^{~}}}.$ Conventionally, when $a$ is a real number, ${\sqrt {a^{2}}}$ denotes the same as $|a|,$ so ${\sqrt {2}}$ is definitely positive.  --Lambiam 18:15, 7 May 2024 (UTC)

The answer is 2 if the meaning of "√x−1/√x" is (√x) − (1/√x). But note that the question uses "√3+2√2" with the meaning √3 + 2√2.  --Lambiam 18:25, 7 May 2024 (UTC)

Is this a homework problem? GalacticShoe (talk) 16:05, 7 May 2024 (UTC)

Possibly but they tried to check their solution. NadVolum (talk) 16:58, 7 May 2024 (UTC)