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April 10

Kakeya Set

Hi. I have some questions about Kakeya Sets, just for fun.

1. Can a Kakeya set exist, given the additional restriction that it contain exactly one line segment for each angle? (Beyond the obvious answer of a circle)
2. Can such a set be made arbitrarily large? (my intuition was yes picturing a spiral shape, but I have a gut feeling that my intuition is wrong. )

Duomillia (talk) 00:32, 10 April 2025 (UTC)

I have an intuitive gut feeling that intuitions and gut feelings are the same thing. -- Jack of Oz ^{[pleasantries]} 00:57, 10 April 2025 (UTC)

My intuition tells me that we must not indulge in gut feelings.  ​‑‑Lambiam 11:39, 10 April 2025 (UTC)

My gut tells me not to eat ice cream after pizza. —Tamfang (talk) 18:29, 10 April 2025 (UTC)

Isn't the closure of the deltoid shown as the first illustration in our article Kakeya set an example meeting your additional restriction? Also, isn't every superset of a Kakeya set, up to $\mathbb {R} ^{2},$ also a Kakeya set?  ​‑‑Lambiam 11:47, 10 April 2025 (UTC)

To clarify, for 1. when you say "exactly one line segment for each angle", do you mean line segments centered at the origin? The example of the circle you gave contains multiple line segments for any particular orientation centered at any given point within the circle. GalacticShoe (talk) 17:02, 10 April 2025 (UTC)

If you think of the line segment as being oriented, if it makes an angle $\alpha$ with the horizontal (oriented from $-\infty$ to $+\infty$), turning it around by half a turn changes the angle to $\alpha +\pi$ (modulo $2\pi$). So when making a full turn in a disk of diameter $1$ it attains each angle $0\leq \alpha <2\pi$ precisely once.  ​‑‑Lambiam 17:26, 10 April 2025 (UTC)

Ah I see where I went off the rails, I failed to notice the "unit" part of "unit line segment" in the definition of a Kakeya set, in which case yeah the unit circle would clearly work. I imagine the Reuleaux triangle, or any Reuleaux polygon, would be another example. GalacticShoe (talk) 17:35, 10 April 2025 (UTC)

Reduced Dedekind psi function?

The Euler's totient function is:

$\varphi (n)={\begin{cases}1&{\text{if }}n=1\\(p-1)p^{r-1}&{\text{if }}n=p^{r}{\text{ with }}p{\text{ prime}}\\\varphi (a)\varphi (b)&{\text{if }}n=ab{\text{ with }}gcd(a,b)=1\\\end{cases}}$

and we have the “reduced” Euler's totient function: (sequence A011773 in the OEIS)

$\lambda (n)={\begin{cases}1&{\text{if }}n=1\\(p-1)p^{r-1}&{\text{if }}n=p^{r}{\text{ with }}p{\text{ prime}}\\lcm(\lambda (a),\lambda (b))&{\text{if }}n=ab{\text{ with }}gcd(a,b)=1\\\end{cases}}$

Also, the Dedekind psi function is:

$\psi (n)={\begin{cases}1&{\text{if }}n=1\\(p+1)p^{r-1}&{\text{if }}n=p^{r}{\text{ with }}p{\text{ prime}}\\\psi (a)\psi (b)&{\text{if }}n=ab{\text{ with }}gcd(a,b)=1\\\end{cases}}$

and we have the “reduced” Dedekind psi function:

$\gamma (n)={\begin{cases}1&{\text{if }}n=1\\(p+1)p^{r-1}&{\text{if }}n=p^{r}{\text{ with }}p{\text{ prime}}\\lcm(\gamma (a),\gamma (b))&{\text{if }}n=ab{\text{ with }}gcd(a,b)=1\\\end{cases}}$

the values of the “reduced” Dedekind psi function $\gamma (1)$ to $\gamma (24)$ are 1, 3, 4, 6, 6, 12, 8, 12, 12, 6, 12, 12, 14, 24, 12, 24, 18, 12, 20, 6, 8, 12, 24, 12, but this sequence does not in OEIS, thus does this function exist in number theory? 220.132.216.52 (talk) 02:54, 10 April 2025 (UTC)

If the sequence is not in the OEIS that's usually a good indication that it's not well known. If it were it would be much more likely to be found there than here. --RDBury (talk) 13:53, 15 April 2025 (UTC)

April 14

Unit usage

1. Do Canadians ever measure room space in square metres?
2. Do Canadians use metric units to measure size of things? Are licence plates measured in millimetres there?
3. Is it so that even in French-speaking Canada, most people give their height in feet/inches and their weight in pounds?
4. Does United Kingdom use kilometre and km/h in any official purposes?--40bus (talk) 20:17, 14 April 2025 (UTC)

   These questions are not about mathematics. Try the Miscellaneous section.  ​‑‑Lambiam 22:03, 14 April 2025 (UTC)

We have an article, Metrication in Canada, which I assume would answer most of these questions. Otherwise it's more a culture question than a math question. --RDBury (talk) 08:32, 15 April 2025 (UTC)

April 16

Bayes’ theorem with multiple evidences of different weights

How is Bayes’ theorem calculated when multiple pieces of evidence that support or rebut a hypothesis with different strength amounts have to be considered? Here is an example scenario.

There are two jars, each filled with 20 balls. The left jar has 18 green balls and two orange balls. The right jar has five green balls and 15 orange balls. Each ball weighs about 70 grams regardless of color. Each jar sits on top of its own electronic scale. You close your eyes and are given one of the balls at random. The ball is green, but the right jar’s scale reading is 70 grams lighter than what it previously was. How would this be computed? Primal Groudon (talk) 21:38, 16 April 2025 (UTC)

We use the following notations for various events:

$L$: the ball was drawn from the left jar.

$G$: the ball is green.

$W$: the right jar is lighter than before.

(These each have complementary events, which if they need to be denoted can be done by using an overbar; for example, ${\overline {G}}$ would mean: the ball is orange. Then $P(G\cap {\overline {G}})=0$ and $P(G\cup {\overline {G}})=1.$)

In the set up, we know by prior real-world knowledge that jars do not get spontaneously lighter, so $P(L\cap W)=0.$

If we try to apply Bayes' theorem we do not directly get anywhere. Ii is easier to go back to basics, the definition of conditional probability:

$P(A|B)={\frac {P(A\cap B)}{P(B)}},{\text{ provided that }}P(B)>0.$

The role of $A$ – the event whose likelihood is to be determined – is taken by $L,$ while $B$ – the observed event – is $G\cap W.$ Then we get:

$P(L|G\cap W)={\frac {P(L\cap G\cap W)}{P(G\cap W)}}.$

We know that $P(L\cap W)=0.$ Thus, a fortiory, $P(L\cap G\cap W)=0,$ and so $P(L|G\cap W)=0.$

The problem in applying Bayes' theorem is that it uses the conditional probability with the events swapped, $P(G\cap W|L),$ which is not directly available. It can likewise be determined by applying the basic definition of conditional probability; however, this route is an unnecessary detour.  ​‑‑Lambiam 22:51, 16 April 2025 (UTC)

What’s the minimal size of a nonce leakage so that the private can be recovered from a single signature in practice ?

There’re a lot of papers on how to recover a private key from a nonce leakage in a ᴇᴄᴅꜱᴀ signature. But the less bits are known the more signatures are required.

Now if I don’t know anything about private key, how much higher order or lower order bits leakage are required at minimum in order to recover a private key from a single signature ? I’m interested in secp256k1. 2A01:E0A:ACF:90B0:0:0:A03F:E788 (talk) 22:37, 16 April 2025 (UTC)

According to the abstract of this paper, the answer is: 12 bits.  ​‑‑Lambiam 23:09, 16 April 2025 (UTC)

The question is how to do it in pratice ? The paper seems only theoritical and do not seems to speak on how to implement it. This might means for actually doing it that the number of bits is larger. 37.171.242.50 (talk) 10:17, 17 April 2025 (UTC)

If you have the capability of mounting side-channel attacks to give you 12 bits, surely you are also able to use their method of attack, which they qualify as being "very practical", to obtain the coveted secret key.  ​‑‑Lambiam 13:56, 17 April 2025 (UTC)

I just temporarirly uploaded it to https://jumpshare.com/s/236VsVoUccTfoWhSv9go. It seems to require several signatures and not just 1. 78.246.6.147 (talk) 15:08, 17 April 2025 (UTC)

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)

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)

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)