Thread for Non-CA Academic Questions

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Re: Thread for Non-CA Academic Questions

Post by Macbi » March 8th, 2020, 4:12 am

V_k are the von Neumann hierarchy. They're a recursively defined family of sets with one for each ordinal. The first is V_0 which is just {}. Then each time you go up one you take the powerset. So V_1 is {{}}, V_2 is {{{}},{}}, V_3 is {{{{}},{}},{{{}}},{{}},{}}, and so on. To define V_k for a limit ordinal k you just take the union of V_j for all j < k. The hierarchy gets bigger at each stage; l ≤ k implies V_l ⊆ V_k. And it eventually covers everything; every set is contained in some V_k (and hence in all higher V_k).

A model of ZFC is a pair (S,ε), where S is a set and ε is a relation on S, such that every axiom of ZFC is true when applied to (S,ε) with ε in place of ∈. There's a theorem which says that a theory is consistent if and only if it has a model. So in the case of ZFC we cannot prove that it has a model, since we cannot prove that it's consistent. But if we assume a large cardinal axiom then we can prove it has a model, which is generally how large cardinal axioms imply the consistency of smaller large cardinal axioms. For example if k is inaccessible then (V_k,∈) is a model (this kind of model, where ε is actually taken to be ∈ restricted to a smaller set, is called a transitive model).

To see why (V_k,∈) is a model when k is inaccessible, consider each axiom of ZFC. Aside from Foundation and Extensionality, which obviously hold in any transitive model, each axiom tells us how to construct a new set from some old ones. So the way a model could fail to obey them would be if it contained some sets, but didn't contain the set that could be constructed from them. But since k is a limit ordinal, every set in V_k is contained in V_l for some l < k. One shows that the sets constructed from V_l have to lie in V_p, where p is constructed from l using powersets and unions. But by definition of 'inaccessible' the cardinals less than k are closed under powersets and unions, so p < k and hence V_p ⊆ V_k.

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Re: Thread for Non-CA Academic Questions

Post by Moosey » March 13th, 2020, 12:19 pm

is the triangle-wave version of the infinite harmonic series finite for an uncountable set of xs? Or perhaps all real xs?
compare it to the harmonic series here

For reference, this is my definition
f(x) = sum from 1 to inf of |xn - round(xn)|/n

also for fun I will provide a detail of a "tooth" of this function:
looks like it could actually be a tooth
looks like it could actually be a tooth
Screen Shot 2020-03-13 at 12.21.56 PM.png (52.58 KiB) Viewed 3070 times
It appears to exhibit some self-similarity
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Re: Thread for Non-CA Academic Questions

Post by A for awesome » March 13th, 2020, 1:34 pm

Moosey wrote:
March 13th, 2020, 12:19 pm
is the triangle-wave version of the infinite harmonic series finite for an uncountable set of xs? Or perhaps all real xs?
compare it to the harmonic series here

For reference, this is my definition
f(x) = sum from 1 to inf of |xn - round(xn)|/n

also for fun I will provide a detail of a "tooth" of this function:
Screen Shot 2020-03-13 at 12.21.56 PM.png

It appears to exhibit some self-similarity
It should only be finite for integer x — for non-integer values, |xn - round(xn)| will be nonzero for a set of values of n with nonzero asymptotic density on Z, and for rational non-integers will have a nonzero "average" value in the limit as n —> infinity and for irrational numbers will have an "average" value of 1/2 in the same limit. This means that the harmonic series should still diverge for these values (unless I'm missing something by not being formal enough). I suspect that the function given by f(x)/g(x) as given in the Desmos link is another example of one that's continuous everywhere but differentiable nowhere (because there appear to be cusps at odd-denominator rational values, which I believe are dense on the reals), but am not sure how to prove it. (Or it could end up being discontinuous in the limit potentially, I guess.)
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Re: Thread for Non-CA Academic Questions

Post by Moosey » March 25th, 2020, 7:43 pm

Can anyone explain Buchholz's OCF? I understand Madore's and have an idea of how for many OCFs psi_a(n) is supposed to work.
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Re: Thread for Non-CA Academic Questions

Post by EvinZL » April 8th, 2020, 8:57 pm

Moosey wrote:
August 19th, 2019, 7:24 pm
I, on The random posts thread, wrote:e = Σ(infinity)(x=0)(1/x!)
What's Σ(infinity)(x=1)(1/2^^x)? It looks to be about 0.8125...

EDIT: fun fact: Π(n)(x=1)(1+1/x) = n+1
What's Π(infinity)(x=1)(1+1/(2^x))?
And Π(infinity)(x=1)(1+1/(x!))? that seems to be ~3.68215...
Those are my questions.
Additionally, what is Σ(infinity)(x=0)(sin((x/x+1)π))?
sin((x/x+1)π)~pi/x
Therefore it diverges
In fact its value is z(1)/1!-z(3)/3!+z(5)/5!-... by using the expansion where z is the Riemann zeta function
Π(infinity)(x=1)(1+1/(2^x)) is likely not expressible in closed form, because let f(z)=Π(infinity)(x=1)(1+z^x) so your question asks for f(1/2). When f(z) is expanded in powers of z the result is 1+Σp*(n)z^n where p*(n) is the number of partitions of n into unequal parts, and partitions are notably hard to handle.
Also, Σ(infinity)(x=1)(1/2^^x) is transcendental by Liouville's theorem

Code: Select all

x = 59, y = 12, rule = B2i3-kq4j/S2-i3
.2A7.2A35.2A7.2A$3A7.3A15.3A15.3A7.3A$.2A7.2A15.A3.A15.2A7.2A$26.A5.A
$26.A5.A$26.3A.3A2$26.3A.3A$26.A5.A$26.A5.A$27.A3.A$28.3A!

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Re: Thread for Non-CA Academic Questions

Post by Moosey » August 20th, 2020, 1:37 pm

Does there exist a number n such that in gijswijt's sequence, the sequence a(1) through a(n) is the same as the repetition of a(1) through a(n/4) four times? If so, what is n?
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Re: Thread for Non-CA Academic Questions

Post by wzkchem5 » August 22nd, 2020, 6:46 am

EvinZL wrote:
April 8th, 2020, 8:57 pm
Moosey wrote:
August 19th, 2019, 7:24 pm
I, on The random posts thread, wrote:e = Σ(infinity)(x=0)(1/x!)
What's Σ(infinity)(x=1)(1/2^^x)? It looks to be about 0.8125...

EDIT: fun fact: Π(n)(x=1)(1+1/x) = n+1
What's Π(infinity)(x=1)(1+1/(2^x))?
And Π(infinity)(x=1)(1+1/(x!))? that seems to be ~3.68215...
Those are my questions.
Additionally, what is Σ(infinity)(x=0)(sin((x/x+1)π))?
sin((x/x+1)π)~pi/x
Therefore it diverges
In fact its value is z(1)/1!-z(3)/3!+z(5)/5!-... by using the expansion where z is the Riemann zeta function
Π(infinity)(x=1)(1+1/(2^x)) is likely not expressible in closed form, because let f(z)=Π(infinity)(x=1)(1+z^x) so your question asks for f(1/2). When f(z) is expanded in powers of z the result is 1+Σp*(n)z^n where p*(n) is the number of partitions of n into unequal parts, and partitions are notably hard to handle.
Also, Σ(infinity)(x=1)(1/2^^x) is transcendental by Liouville's theorem
An interesting problem that I solved when I was in high school (and I was very proud of it at that time):
What's Σ(infinity)(x=0)(1/(x^4+1))?
Hint: first calculate Σ(infinity)(x=0)(1/(x^2+i))
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Re: Thread for Non-CA Academic Questions

Post by Moosey » August 22nd, 2020, 7:49 pm

A fun problem I posted on the discord
take any tree. any two nodes sharing the same parent may be deleted, if they have the same branches above them, and the entire tree may have a copy of itself with a single leaf node removed attached to any of its nodes. Can every tree be reduced to a single node?
Some challenges:
(1) reduce (((())())) and some other trees (any will do)
(2) prove or disprove that every tree can be reduced to a single node
(3) optional-- create a googological function f based on this and find the first few values of f(n).
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Re: Thread for Non-CA Academic Questions

Post by Moosey » August 23rd, 2020, 10:57 am

another problem I posted on the discord:
trapped knight generalization:
call the n-knight the trapped knight which can return to a square n times after it has been stepped on. It prefers the squares it has stepped on the least, even if they have a larger value than another accessible square that has been stepped on more times. (the 0-knight is a conventional trapped knight, for instance.)
define knight(n) = the time before the n-knight is trapped.
(1) how fast does knight(n) grow?
(2) do all knights halt; that is, is knight(n) defined for all n?
(3) what are the first few values of knight(n)
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Re: Thread for Non-CA Academic Questions

Post by Schiaparelliorbust » December 9th, 2020, 3:17 am

I just noticed this thread (I hope I'm not necroposting) and I've had this question for a while: How do adaptations form? I understand how some more straightforward ones can form (larger/smaller body (parts), change of shape (to some extent), change of chemicals, change in function + exaptation etc.) but more complex ones have always stumped me. How do things like Helicoprion's tooth whorl or this snake's spider-like tail form? In evolution, complex things like eyes can evolve because every step of the way towards a fully fledged eye offers an advantage to the organism. First simple detection of the presence or absence of light, then perhaps you can detect more brightness levels (pitch dark, dark, dim, bright, very bright), then you can detect direction, then color, then shape and some more refinements until you get to an eye. This has happened at least three times independently (at least the latter stages) in chordates (vertebrates), molluscs, and arthropods. Something like wheels though would probably never evolve because anything other than a fully working wheels with axles and everything would offer any advantage. I find the adaptations that I listed at the start are more like wheels than eyes. It seems like the evolutionary pathway is too elaborate and the evolutionary pressure simply isn't enough.
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Re: Thread for Non-CA Academic Questions

Post by toroidalet » December 11th, 2020, 9:42 pm

The fossil fandom wiki states that the two main hypotheses for the use of Helicoprion's tooth whorl were either to feed on ammonites or that it would fling them out into a school of fish to catch them. In the first case, it seems plausible that specimens with ever-longer and rounder jaws could get more ammonite, so that over time it gradually evolved a spiral-shaped jaw that it could stick into an ammonite. In the second case, it could be argued that specimens with longer and more flexible jaws caught more fish and that curling up the jaw helped conceal it so the fish wouldn't run away as soon. (evolutionary arguments of this form are ridiculously easy to make and almost impossible to disprove)
Unfortunately (unless you like swimming out in the ocean), Helicoprion is extinct, and we can never know what its spiral jaw was used for and therefore how it evolved. Still, there is almost always an evolutionary argument (or maybe you could argue something based on sexual selection if a feature has no benefit whatsoever).
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Re: Thread for Non-CA Academic Questions

Post by Schiaparelliorbust » December 12th, 2020, 7:04 am

toroidalet wrote:
December 11th, 2020, 9:42 pm
The fossil fandom wiki states that the two main hypotheses for the use of Helicoprion's tooth whorl were either to feed on ammonites or that it would fling them out into a school of fish to catch them. In the first case, it seems plausible that specimens with ever-longer and rounder jaws could get more ammonite, so that over time it gradually evolved a spiral-shaped jaw that it could stick into an ammonite. In the second case, it could be argued that specimens with longer and more flexible jaws caught more fish and that curling up the jaw helped conceal it so the fish wouldn't run away as soon. (evolutionary arguments of this form are ridiculously easy to make and almost impossible to disprove)
Unfortunately (unless you like swimming out in the ocean), Helicoprion is extinct, and we can never know what its spiral jaw was used for and therefore how it evolved. Still, there is almost always an evolutionary argument (or maybe you could argue something based on sexual selection if a feature has no benefit whatsoever).
I guess that makes sense. I still sometimes have a hard time imagining how things like this could ever happen, but you have to take into account the vast timescales over which this stuff happened. I thought of Helicoprion's tooth whorl as being a distinct trait that either existed or didn't, but your explanation shows how it could have gradually evolved. Certain adaptations still make me wonder, especially behavioral ones, but they all must have some explanation, even if it's not directly based on survival/genetics (sexual selection, genetic drift, social change etc).
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Re: Thread for Non-CA Academic Questions

Post by muzik » May 6th, 2021, 1:13 pm

Since the Fibonacci sequence (which approximates the golden ratio) can be found within Pascal's triangle, it is also possible to find the sequences that approximate the silver ratio (Pell, ...) and higher metallic means within it or similar arithmetic devices?

Likewise for the Tribonacci sequence and higher order versions of it.

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Re: Thread for Non-CA Academic Questions

Post by ColorfulGabrielsp138 » May 8th, 2021, 10:38 pm

Code: Select all

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Code: Select all

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Code: Select all

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Code: Select all

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Code: Select all

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D!
Are there any other examples?
EDIT: Tried some soup searches, but no luck

Code: Select all

x = 21, y = 21, rule = LifeColorful
11.E$10.3E$10.E.2E$13.E4$2.2B$.2B$2B$.2B15.2D$19.2D$18.2D$17.2D4$7.C$
7.2C.C$8.3C$9.C!
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Re: Thread for Non-CA Academic Questions

Post by GUYTU6J » May 9th, 2021, 3:50 am

ColorfulGabrielsp138 wrote:
May 8th, 2021, 10:38 pm
[a long list of RLEs showing fractions that I think needs merging into one RLE, or better, rendering with LaTex and making an image]
Are there any other examples?
These are called Anomalous Cancellation.
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Midst broken fountains, mouldering walls.)
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Re: Thread for Non-CA Academic Questions

Post by Caenbe » June 10th, 2021, 12:39 pm

muzik wrote:
May 6th, 2021, 1:13 pm
Since the Fibonacci sequence (which approximates the golden ratio) can be found within Pascal's triangle, it is also possible to find the sequences that approximate the silver ratio (Pell, ...) and higher metallic means within it or similar arithmetic devices?

Likewise for the Tribonacci sequence and higher order versions of it.
Yes, to a certain extent. I remember discovering that you can get a recurrence relation of the form a_n = a_(n-j) + a_(n-k), for any 0<j≤k with j and k coprime, by summing the entries along a line of slope depending on j and k. (So, none of the things you mentioned, but you can get the plastic ratio.) Unfortunately, I didn't bother to write down a proof, but it shouldn't be hard to reconstruct.
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