Timeline for Cartesian dissimilarity of a function $\ f:A^3\to A^3\ $ and its inverse
Current License: CC BY-SA 4.0
31 events
| when toggle format | what | by | license | comment | |
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| May 21, 2020 at 3:04 | review | Close votes | |||
| May 22, 2020 at 3:02 | |||||
| May 14, 2020 at 22:05 | history | edited | Wlod AA | CC BY-SA 4.0 |
TeX typo
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| May 14, 2020 at 18:36 | comment | added | Wlod AA | I've accepted @AlekseiKulikov answer. Let's also remember the YCor's contributions. I hope that this thread is only the beginning of the topic of the Cartesian asymmetric of pairs $\ f\ g\ $ of functions inverse one to another. One possible development direction is about the algebra induced by the situation (sub-directions: additional assumptions). Another, let's go beyond dimension $3$. | |
| May 14, 2020 at 18:27 | vote | accept | Wlod AA | ||
| May 13, 2020 at 20:38 | comment | added | Wlod AA | @YCor, of course, the question belongs to combinatorics, it's most natural; thank you, YCor. (Somehow, I missed the combinatorics tag). | |
| May 13, 2020 at 20:23 | comment | added | YCor | OP: "(This question [...] belongs to [...] Foundations of Mathematics. (Please, someone reattach the related tag to my question)." I added "co.combinatorics" (to which it primarily belongs), removed "lo.logic" and later Andres Caicedo removed "set-theory". If there is a motivation from foundations, feel free to put back one of these tags or both (and erase the parenthesis, maybe even put the motivation at the beginning). As far as I'm concerned I won't try to revert (but please keep co.combinatorics since it's interesting for its own sake as combinatorial question). | |
| May 13, 2020 at 19:26 | history | edited | Wlod AA | CC BY-SA 4.0 |
PS
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| May 13, 2020 at 17:38 | comment | added | YCor | The problem of existence is solved (by Alexei Kulikov) as there are solutions for all $n\ge 3$. Still counting solutions is natural. Actually, it sounds even more natural to count $u_n$, the cardinal of the set $U_n$ of permutations of $[n]^3$ of the form $(x,y)\mapsto (a(y,z),b(z,x),c(x,y))$, and $v_n$, the cardinal of the subset $V_n$ of those for which no coordinate of the inverse is constant w.r.t some coordinate. For instance $U_n$ is stable under pre and post-composition by permutations of coordinates (which is an action of $Sym(n)^6$ on $Sym([n]^3)$). | |
| May 13, 2020 at 17:19 | comment | added | Wlod AA | @YCor, thank you for the pro-conjecture support and for your computations. #### "I'll have a look for 𝑛=4." -- where is John von Neumann when we need him? | |
| May 13, 2020 at 15:20 | answer | added | Aleksei Kulikov | timeline score: 4 | |
| May 13, 2020 at 15:11 | comment | added | YCor | With computer (Sagemath) if not mistaken, I checked that there is no solution for $n=2$. Just by brute force checking for all $4096$ triples of functions $u,v,w,2^2\to 2$, whether $(x,y,z)\mapsto (u(y,z),v(z,x),w(x,y))$ is injective and the inverse function satisfies that its coordinates depends on all arguments. I'll have a look for $n=4$. | |
| May 13, 2020 at 14:26 | comment | added | YCor | @AlexM. this "conjecture ban" is ridiculous in this case. It concerns "famous conjectures" and variants thereof. In a question like this, calling it "conjecture" is like saying "I strongly believe that the answer to my question is yes", and I see no point in your personal initiative to discourage this. | |
| May 13, 2020 at 13:33 | history | edited | Andrés E. Caicedo |
edited tags
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| May 13, 2020 at 10:46 | comment | added | Alex M. | @WlodAA: It is very reasonable, in fact, if you think about it for a minute. MO is a place that attempts to produce, and then store, definite answers. A conjecture, on the other hand, might not have answers, by its very nature. A conjecture might generate an interesting discussion, but no definite answer. Therefore, it would be more appropriate for a forum, which MO is not. Furthermore, there are plenty of wannabe researchers who would flood MO with conjectures if there were allowed (think of how easy it is to produce them in additive number theory). This is why they are not accepted. | |
| May 13, 2020 at 10:17 | comment | added | Wlod AA | (I still can't believe it that an original conjecture can be against MO rules, how ridiculous!) | |
| May 13, 2020 at 10:12 | comment | added | Wlod AA | @AlexM., I don't care. I am into mathematics. That's why somehow I am still here but just barely. Perhaps I'll leave MO after this incident for good. | |
| May 13, 2020 at 9:00 | comment | added | Alex M. | @WlodAA: So your question is the conjecture? If so, then please notice that conjectures are considered off-topic on MO (even if they are of research level), so your question will likely get closed (even though it might be mathematically interesting). | |
| May 13, 2020 at 8:20 | history | edited | Wlod AA | CC BY-SA 4.0 |
emphasis, bold font **odd**.
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| May 13, 2020 at 8:02 | history | edited | Wlod AA | CC BY-SA 4.0 |
fixed a typo
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| May 13, 2020 at 8:00 | comment | added | Wlod AA | @YCor, I gave the odd case examples. The rest is unclear. Earlier I was a bit sloppy about the even case when there is an odd divisor > 1. | |
| May 13, 2020 at 7:57 | comment | added | Wlod AA | @kodlu, yes, as YCor said; I've added the definition anyway. | |
| May 13, 2020 at 7:52 | history | edited | Wlod AA | CC BY-SA 4.0 |
less sloppy :)
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| May 13, 2020 at 7:16 | history | edited | Wlod AA | CC BY-SA 4.0 |
diagonal product
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| May 13, 2020 at 6:47 | comment | added | YCor | Could you give the examples you have in mind when $|A|$ has an odd prime divisor? | |
| May 13, 2020 at 6:46 | comment | added | YCor | I assume $(f_1\triangle f_2\triangle f_3)(x_1,x_2,x_3)$ just means $(f_1(x_1,x_2,x_3),f_2(x_1,x_2,x_3),f_3(x_1,x_2,x_3))$. | |
| May 13, 2020 at 5:43 | comment | added | kodlu | can you define the operation $\Delta$? | |
| May 13, 2020 at 4:33 | review | Close votes | |||
| May 13, 2020 at 22:02 | |||||
| May 13, 2020 at 4:23 | history | edited | YCor |
edited tags
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| May 13, 2020 at 4:01 | history | edited | Wlod AA | CC BY-SA 4.0 |
typo
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| May 12, 2020 at 23:59 | history | edited | Wlod AA | CC BY-SA 4.0 |
TeX typo in the title
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| May 12, 2020 at 23:51 | history | asked | Wlod AA | CC BY-SA 4.0 |