Thread for Non-CA Academic Questions
Re: Thread for Non-CA Academic Questions
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.
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.
Re: Thread for Non-CA Academic Questions
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: It appears to exhibit some self-similarity
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: It appears to exhibit some self-similarity
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Re: Thread for Non-CA Academic Questions
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.)Moosey wrote: ↑March 13th, 2020, 12:19 pmis 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
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Re: Thread for Non-CA Academic Questions
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
sin((x/x+1)π)~pi/xMoosey wrote: ↑August 19th, 2019, 7:24 pmThose are my questions.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...
Additionally, what is Σ(infinity)(x=0)(sin((x/x+1)π))?
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
Re: Thread for Non-CA Academic Questions
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
An interesting problem that I solved when I was in high school (and I was very proud of it at that time):EvinZL wrote: ↑April 8th, 2020, 8:57 pmsin((x/x+1)π)~pi/xMoosey wrote: ↑August 19th, 2019, 7:24 pmThose are my questions.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...
Additionally, what is Σ(infinity)(x=0)(sin((x/x+1)π))?
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
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
A fun problem I posted on the discord
(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).
Some challenges: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?
(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
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
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
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).
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
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).toroidalet wrote: ↑December 11th, 2020, 9:42 pmThe 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
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.
Likewise for the Tribonacci sequence and higher order versions of it.
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Re: Thread for Non-CA Academic Questions
Code: Select all
x = 18, y = 13, rule = LifeHistory
2.AD3A9.A$2.A.E11.A$2.A.AEA9.A$2.A.A.E9.A$2.A.3AD8.A$9.5A$.7A7.3A$9.5A
$D3A.A.A7.A.A$.E3.A.A7.A.A$.AEA.3A7.3A$.A.E3.A9.A$.3AD2.A9.A!
Code: Select all
x = 18, y = 13, rule = LifeHistory
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9.5A$D3A.3A7.3A$.E3.A9.A$.AEA.3A7.3A$.A.E3.A9.A$.3AD3A7.3A!
Code: Select all
x = 18, y = 13, rule = LifeHistory
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Code: Select all
x = 18, y = 13, rule = LifeHistory
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EA.3A7.3A7.3A$3.E.A.A7.A.A7.A$.3AD3A7.3A7.3A!
Code: Select all
x = 18, y = 13, rule = LifeHistory
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$13.5A$.11A7.3A$13.5A$D3A.3AD3A7.3A$.D.A3.A.E11.A$.AEA.3A.AEA7.3A$3.E
.A3.A.E7.A$.3AD3A.3AD6.3A!
Code: Select all
x = 18, y = 13, rule = LifeHistory
4.AD3A13.A$4.A.D.A13.A$4.A.AEA13.A$4.A3.E13.A$4.A.3AD12.A$13.5A$.11A7.
7A$13.5A$D3A.3A.3A7.3A.3A$.D.A3.A.A11.A.A$.AEA.3A.3A7.3A.3A$3.E.A5.A7.
A5.A$.3AD3A.3A7.3A.3A!
Code: Select all
x = 18, y = 13, rule = LifeHistory
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A$.11A7.3A$13.5A.A.A$2.D3AD3A9.3A$3.E.A.E$3.AEA.AEA$3.A.E.A.E$3.3AD3A
D!
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
These are called Anomalous Cancellation.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?
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Re: Thread for Non-CA Academic Questions
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.muzik wrote: ↑May 6th, 2021, 1:13 pmSince 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
That’s needlessly complex; suppose to get contradiction that the sequence is bounded, then because it’s also monotonous it tends to a limit, which must be a fixed point of x+1/x. But it has no fixed point, so we have a contradiction.toroidalet wrote: ↑April 26th, 2019, 12:02 amI believe that for k=0 (xn+1=xn(+1/xn)) it diverges. There's an inductive proof which goes something like this:(Obviously if x0 is negative it will decrease infinitely.)Provided the sequence reaches n, it will reach n+1, since each time it increases by at least 1/(n+1), so after n+1 applications of this the new number will be ≥n+1.
I'm not sure how to approach k≥1; perhaps the behaviour depends on the seed values.
\sum_{n=1}^\infty H_n/n^2 = \zeta(3)
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Re: Thread for Non-CA Academic Questions
This is rather complicated, and I don't think I should put this anywhere else. Given any method of tiling, is there a limit to finite aperiodicity? An infinite aperiodic pattern in a periodic grid has the capability to support patterns that do not contain an infinite number of patterns. A 1-cell thick infinite line could be constructed that follows an aperiodic binary pattern, so there is no limit, but change the tiling method to a Penrose tiling, and now a problem occurs. This is a tiling that must take up 2 dimensions and must contain every pattern that can exist. Say we have two different infinite Penrose tilings. Any finite pattern in 1 can be found in the other, regardless of the pattern. While the section of the pattern may be the only overlapping section with that offset. Larger and larger patterns will still fit, so increasing the pattern infinitely to an infinite pattern size would result in that pattern fitting the Penrose tiling, and actually being an exact replica. This pattern again is the overlapping section of the tiling, and this means our two different Penrose tilings are actually the same exact tiling, but with transformations. Is there a specific factor that allows for multiple infinite aperiodic patterns, or is it just based off of how you are tiling the plane?
Code: Select all
x = 36, y = 28, rule = TripleLife
17.G$17.3G$20.G$19.2G11$9.EF$8.FG.GD$8.DGAGF$10.DGD5$2.2G$3.G30.2G$3G
25.2G5.G$G27.G.G.3G$21.2G7.G.G$21.2G7.2G!
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Re: Thread for Non-CA Academic Questions
I'm not sure what you're trying to say, but
is incorrect. Consider a tiling with perfect mirror symmetry (or rotational symmetry, it doesn't really matter). You can make a tiling that provably is not the same tiling by using every choice to disrupt an axis of reflection. This different tiling does in fact contain arbitrarily large mirror-symmetric regions, but none of them can expand infinitely. At every size there are matching regions, but eventually every such pair eventually differs.Larger and larger patterns will still fit, so increasing the pattern infinitely to an infinite pattern size would result in that pattern fitting the Penrose tiling, and actually being an exact replica.
It depends on the tiles. I believe the Robinson tiles can only tile the plane a few different ways.Is there a specific factor that allows for multiple infinite aperiodic patterns, or is it just based off of how you are tiling the plane?
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Re: Thread for Non-CA Academic Questions
Can someone provide an RLE for the graph of Tupper's self-referential formula at (±2^16, ±2^16) (say) around the origin?
How large would the plot range be if it covers all of the 16×16, 8×32, ..., 1×256 random soups? (No, not that I want to feed said pattern to apgsearch and complete soup searching at once; this is probably a terrible idea.)
How large would the plot range be if it covers all of the 16×16, 8×32, ..., 1×256 random soups? (No, not that I want to feed said pattern to apgsearch and complete soup searching at once; this is probably a terrible idea.)
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Re: Thread for Non-CA Academic Questions
What does the "include studio" mean in this code?
I remember seeing this several times before, such as
I remember seeing this several times before, such as
Code: Select all
#include <cstdio>
#include <cstring>
int main(){
char s1[500]="cent quatorze ";
char s2[500]="mille cinq ";
strcat(s2,s1); // Swap the two strings.
strcat(s1,s2); // Swap back.
puts(s1);
return 0;
}
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Re: Thread for Non-CA Academic Questions
"cstdio" stands for "C standard input and output". It does exactly what it says and allows you to input and output things.
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Re: Thread for Non-CA Academic Questions
Then what should the code print?
It doesn't look like the code that I've studied previously.
Does it just swap the strings and back as expected?
It doesn't look like the code that I've studied previously.
Does it just swap the strings and back as expected?
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