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April 18

Im stuck on this limit problem here

$\lim _{x\to 0}{\frac {\sin(3x)-\sin(x)}{x^{3}}}$

I have tried using L'hopital's rule, because of the $0/0$ then I got another result of the same thing, which doesn't seem right. NeUcLeIrDisaster3 (talk) 21:32, 18 April 2025 (UTC)

L'Hopital's rule very much can be iterated repeatedly, and that is likely the intended method. Though I'd double check that the numerator isn't $\sin(3x)-3\sin(x)$. Sesquilinear (talk) 23:09, 18 April 2025 (UTC)

Late response. Apologies. But, What's wrong with the numerator...if you don't mind me asking? NeUcLeIrDisaster3 (talk) 18:34, 27 April 2025 (UTC)

Using L'Hopital's rule gives me $${\frac {3\cos(3x)-\cos(x)}{3x^{2}}}$$ which is $2/0$, so you wouldn't be able to repeat. Sesquilinear (talk) 18:39, 27 April 2025 (UTC)

Okay. So this could be a typo then? NeUcLeIrDisaster3 (talk) 18:59, 27 April 2025 (UTC)

Well, that depends. What circumstances lead you to this problem? Sesquilinear (talk) 19:00, 27 April 2025 (UTC)

...nor should you need to. That limit simply diverges, no further analysis (pun not intended, but noticed ;-) needed. --Stephan Schulz (talk) 00:36, 28 April 2025 (UTC)

However, if this is an exercise meant to develop the student's skill in applying L'Hôpital's rule, it is rather likely that there is indeed a typo in the formula, as suggested by Sesquilinear. Compare these exercises: [1], [2].  ​‑‑Lambiam 09:51, 28 April 2025 (UTC)

$\sin x=x+x^{3}/6+...$ Tito Omburo (talk) 21:46, 27 April 2025 (UTC)

I think that's actually $\sinh x$. Your first plus should be a minus. Comes out the same for this problem, though. --Trovatore (talk) 23:39, 28 April 2025 (UTC)

So... ?  ​‑‑Lambiam 00:00, 28 April 2025 (UTC)

Well, it suggests a different, and often better, approach to the problem. The biggest problem with L'Hôpital's rule is that taking the derivative of an expression, other than in some special cases, is more likely to make the expression more complicated than simpler. So you get a formally correct transformation of the problem, but it doesn't really help you solve it.

Often better is to write out a few terms of the power series for the sub-expressions, see which terms matter and which ones don't, cancel some stuff out, and then take the limit. --Trovatore (talk) 23:45, 28 April 2025 (UTC)

April 24

Can a math PhD who never took a course in physics...

Solve a baseball bat velocity problem? You give them the mass of the bat and mass of the ball, and velocity of each, can they solve how far the baseball travels?

Also, in the math department, is there a course that covers quantum mechanics? Specifically stuff like Schrodinger's equations. Thanks. 24.136.10.82 (talk) 13:07, 24 April 2025 (UTC).

We have an article on Mathematical physics which I think covers what you're talking about. As for the bat and ball problem, you'd probably have to know the elasticity of the collision; this is the percentage of energy lost in the interaction. If completely inelastic then the ball just sticks to the bat and doesn't go anywhere, but if completely elastic then the ball will bounce off the bat with no loss in energy. Reality is always somewhere in between and depends on the materials the bat and ball are made of. --RDBury (talk) 16:12, 24 April 2025 (UTC)

To solve such baseball problems, you also have to apply Newton's laws of motion, which are taught in physics courses, but typically not in maths courses or maths textbooks – although there is a fair chance of students having encountered these laws in examples and exercises. A hypothetical mathematician completely unaware of these laws cannot solve such problems, regardless of their mathematical prowess.

From a pure mathematician's perspective, the only difference between mathematical physics and other fields of mathematics is the applicability of the mathematical models to physics problems, which is extrinsic to the mathematical aspects.  ​‑‑Lambiam 16:37, 24 April 2025 (UTC)

I'd say the basic laws of motion and Newton's law of universal gravity should both be covered in high school in most modern education systems. Elastic/inelastic collisions and air resistance less so. So it depends on how exact an answer you want. --Stephan Schulz (talk) 00:34, 28 April 2025 (UTC)

How to recover nonces from ECDSA signatures given the private key ?

Given :

    R: 1699b85f9fd4e3c6234bc0b3378a965a08ea4f76b5359998dec6123c20ff7b64
    S: 6db258553ff34e7928d877a93d219dfff683bdd6de8c54cbebafe028198285eb
    message hash: 5e39fb8e7f5ec05eab86c4f2618c5c96fb3c8c7ff38f37224084fffe50aaaeb0

I tried 0x6db258553ff34e7928d877a93d219dfff683bdd6de8c54cbebafe028198285eb*Mod(-(0x5e39fb8e7f5ec05eab86c4f2618c5c96fb3c8c7ff38f37224084fffe50aaaeb0+0x1c533b6bb7f0804e09960225e44877ac*0x1699b85f9fd4e3c6234bc0b3378a965a08ea4f76b5359998dec6123c20ff7b64),0xfffffffffffffffffffffffffffffffffffffffffffffffffffffffefffffc2f) which gives an incorrect result in Pari/ɢᴘ while the pubkey recovered from the signature match the $1c533b6bb7f0804e09960225e44877ac$ private key. 2A01:E0A:ACF:90B0:0:0:A03F:E788 (talk) 21:32, 24 April 2025 (UTC)

April 28

Must every symmetric/transitive relation be homogeneous, or it may also be heterogeneous?

In other words:

1. For given (disjoint) sets $X,Y,$ is it accepted to call a given relation R "symmetric", when: for all $x\in X,y\in Y,$ if $xRy$ then $yRx?$

2. For given (disjoint) sets $X,Y,Z,$ is it accepted to call a given relation R "transitive", when: for all $x\in X,y\in Y,z\in Z,$ if $xRy$ and $yRz$ then $xRz?$

However, if the term "symmetric/transitive" is not accepted for these heterogeneous relations, then do you have in mind a better name to describe them? HOTmag (talk) 10:06, 28 April 2025 (UTC)

1. Unless $R$ is a relation between $X\sqcup Y$ and $X\sqcup Y$, the consequent $yRx$ of item 1 does not make much sense. The statement "for all $x\in X,y\in Y,$ if $xRy$ then $yRx$" by itself normally already implies that $R$ is homogeneous.

2. This is more complicated. Here relation $R$ is apparently between $X\sqcup Y$ and $Y\sqcup Z,$ so it is not necessarily homogeneous. There is no higher mathematical authority ruling which abuses of language are condoned and which are proscribed. Personally, I would have no qualms declaring my non-homogeneous relation satisfying this condition to be transitive, but I can give no guarantee that this might not offend some lesser god. However, it may be wise to make the reader aware of the fact that the situation is not quite normal. A transitive homogeneous relation $R$ has the property that $R^{2}\subseteq R,$ which one can even use as the definition of transitivity, but for a heterogeneous relation this makes no sense.  ​‑‑Lambiam 14:54, 28 April 2025 (UTC)

Thank you. HOTmag (talk) 17:50, 28 April 2025 (UTC)

April 29

Density of Ulam numbers

Ulam numbers empirically seem to have a density of about 0.07. However, this paper in ArXiv says that they have zero density. I can't find that it has been published anywhere. What is the status of the density of Ulam numbers? Bubba73 ^{You talkin' to me?} 02:13, 29 April 2025 (UTC)

I guess it is open. Presumably, the paper was submitted to a journal. When the referees find holes in a purported proof, this is generally not made public, so we may simply not hear more about this. The paper on arXiv was originally submitted in 2020, but by now it has reached version 11, from 27 August 2023. The author identifies himself as "an ardent theory builder with very outlandish mathematical ideas drawn from intuition".^[3] Five years ago, a co-author of his on several papers^[4] published a paper "An Elementary Proof of the Twin Prime Conjecture",^[5] yet the consensus among number theorists appears to be this problem is also still open. The proof was published in a rather unknown journal. One would think the author submitted such an important result first to prestigious journals in number theory, so this strongly suggests it was rejected by these.  ​‑‑Lambiam 07:58, 29 April 2025 (UTC)

The Ulam number sequence in OEIS (OEIS:A002858) also doesn't seem to mention anything about an acceptance of the density 0 proof; rather, it just indicates that Stanisław Ulam himself believed the density to be 0, while empirical evidence suggests a density around 0.074. GalacticShoe (talk) 13:11, 29 April 2025 (UTC)

Based on the numbers up to 10¹², they appear to have a positive density. I got this data from Exploring the Beauty of Fascinating Numbers, by Shyam S. Gupta. The y-axis is density and the x-axis is the log₁₀ of the upper value.

10^4 to 10^12

10^6 to 10^12

10^8 to 10^12

Out to 10¹² it behaves as if the density is converging to about 0.074.... Bubba73 ^{You talkin' to me?} 01:34, 1 May 2025 (UTC)