/sci/ - Science & Math

e.g. [math]1=0[/math] does the job, assuming Explosion and the Peano axiom that no successor equal 0.

>>12492926
Like how the value at [math]k=0[/math] of the Fourier transform of [math]f[/math] is its integral.

Here's some thing fun for you making use of your insight:

A: Think of an arbotrary oder polynomial [math]p[/math] with coefficients in the positive integers.
B: Done.
A: What's [math]p(1)[/math]?
B: *tells*
A: What's [math]p(p(1))[/math]?
B: *tells*
A: [math]p[/math] is ...

Semirelated is the Gödel beta-function.

>>10904185
You're just looking at functions that generate a finite monoid and ask for the order of the group element

I think you're just asking for monoid representation theory of finite groups where the mutliplication is given by function concatentation.

For the invertible functions (groups), see the list of options here
https://en.wikipedia.org/wiki/List_of_small_groups
In particular, for cyclic grups, in C, for f(x):=x*exp(2pi/n), you get f^(n)(x)=x.

Your example are probably finite subgroups of
https://en.wikipedia.org/wiki/Modular_group

Made a video elborating on
https://en.wikipedia.org/wiki/Hereditarily_finite_set

https://youtu.be/-kLNlzbNRQQ

>>9646196
You know that Cantor was working on periodic function when he derailed math forever

>... Heine proposed that Cantor solve an open problem that had eluded Peter Gustav Lejeune Dirichlet, Rudolf Lipschitz, Bernhard Riemann, and Heine himself: the uniqueness of the representation of a function by trigonometric series. Cantor solved this difficult problem in 1869. It was while working on this problem that he discovered transfinite ordinals, which occurred as indices n in the nth derived set Sn of a set S of zeros of a trigonometric series. Given a trigonometric series f(x) with S as its set of zeros, Cantor had discovered a procedure that produced another trigonometric series
>...and then he noticed that Sω would also have to have a set of limit points Sω+1, and so on. He had examples that went on forever, and so here was a naturally occurring infinite sequence of infinite numbers ω, ω + 1, ω + 2, ...[37]
>Between 1870 and 1872, Cantor published more papers on trigonometric series, and also a paper defining irrational numbers as convergent sequences of rational numbers. Dedekind, whom Cantor befriended in 1872, cited this paper later that year, in the paper where he first set out his celebrated definition of real numbers by Dedekind cuts. While extending the notion of number by means of his revolutionary concept of infinite cardinality, Cantor was paradoxically opposed to theories of infinitesimals of his contemporaries Otto Stolz and Paul du Bois-Reymond, describing them as both "an abomination" and "a cholera bacillus of mathematics".[38] Cantor also published an erroneous "proof" of the inconsistency of infinitesimals.[39]

>>9646249
Ah, looks very QFT motivated. How or why does the Weierstrass elliptic function enter the frame?

>>9520791
Hey man, long time no see.

For better or worse, I never really stopped shitposting on /sci/. During that time when I was around the corner form you in Germany, I also got into HoTT a little bit and last year eventually started doing youtube videos on dependent type theory, although it diverged into a smart contract programming series and now I'm waving back into math. Those things can be combined though, e.g. in
https://youtu.be/T9z3YQkKCLo
And I never stopped following Schreiber at the nLab who does physics in categories that are models for HoTT etc.
https://youtu.be/_O41kh0z_UM
I remember teaching you hom-functors with paint drawings :D Good you know more than me now.
in any case, one may move from one institution and faculty to the other, but I consider the distrinction between physics, math and computer science merely a social seperation of academics, and not a line you could evend draw throught the subject themselves. Best be looking at what all of those have to offer and do your own thing in the middle.

I was thinking about doing a light Curry-Howard shilling clip on the weekend..

Do we know anything about the distribution of biases?

E.g. if irl the biases are all just 10% chance of getting heads, then whatever general stategy we come up with here, if will be worse than "after every flip, choose another coin". And, further, while this would be the best stategry for the cruel 10% case, it would in actuality still be futile.

So for the sake of the argument, let me consider the case where the biases are randomly choosen, evenly, between 0 and 1, for each coin.
Then you'd probably want to make a Bayesian update of your estimation of the bias of the coin at head. You'd eventually stick with one where you find a bias >0.5, but how high is good enough will depend on T.

1st flip:
So let's say you flip a coin and it's tails. There is no reason to believe the coin has bias >0.5, so why not change coin. And then you're back to flip 1.
On the other hand say you flip a coin and it's heads. Then there's so far no reason to believe the coins bias is below 0.5, so chaning to another coin would be a bad move.
2nd flip:
Now you flip again. If heads comes up, you'd natually stick with it.
On the other hand, if tails comes up, you're at 50/50. Since we assume random distribtuion of biases over all coins, there's no reason to change coin either.
3nd flip:
If tails comes up, you're 1:2 down and flipping will probably be best.
Now if heads comes up, you're 2:1 up and stick with it.

Okay so now you keep playing that game and beyond 3 flips with a coin, you get a better and better estimate of the estimate simply form your flipping (frequentist interpretation).
But so e.g. say you find your coin is most likely a 67% heads bias coin. I don't know, but whether going on and finding an even better coin will depend on how many flips are left (and so how big T is). Because if T is very large, then trying to find e.g. a 80% heads bias coin is not unrealistic and then, after you found it, the flips beyond will make you mad gains.
There's what I meant above.

>>9396370
>programming intuitions
what exactly did you mean by this?

>>9396655
Might be worth pointing out that you see the exponential 2^n=m there because that sum up to m (harmonc numbers H_m) asymptotically are, roughly
1/2 + log(m)

>>9398851
I agree with >>9398871 and think Grothendieck was more driven by being better than others than by "how pure math is" as a get-go.

I have a strong disagreement with the guy in the Mathoverflow post at the end, where he says that to study philsophy and reason about ethics theories, you must know about simple mathematical theories such as linear algebra. Mathematical, formalized logics don't have a monopoly on reasoning - in fact I think the core formalization that exists since Frege is good and the most effective one to get to a point where we can build iPods, but there are many good non-formalizable notions of reasoning. You can use logic and math to work out staticas of some data you gather, but logic won't help you with a good timely (!) theory about ethics any more than it will help you when you have a fight with your girlfriend. Knowing
(A => B) => (not B => not A)
helps me even beyond math, but that doesn't mean it's necessary to reason as a human. I wouldn't go into this fuzzy talk if the guy in the MathOverflow post not literally cry about death and the reason to live.

>>9398369
I think it's a cool subject because it gives procedural meaning to plain propositions. You see some sum formula where terms cancel each other and you may algebraically prove it, but you can alsouse inclusion-exclusion and see what the cancelations actually amount to. Or in some cases you device a counting procedure where you pocket items in this and that way (double counting). This is refreshing in the same way that geometric "proofs" of the pytagorean theorem (or similar things) give meaning to equalities.
That being said, combinatorics can also be ugly in the way that number theory is - ad hoc proves left and right.

>>9396859
:^)

>>9370077
This should do
https://discordapp.com/invite/R8v48YA
They just switched from Slack to Discord a month ago.

Also, I'm not a progrmamer. But for example, last month they finished a smart competition where the top 10 people won $20,000 and only 17 people even handed in a script. Mind you, as of today, created 5 billion dollar total (and even this year, 1 NEO went from 15c to $50), so they have the money.

>>9370098
>How long have you been developing/learning?
On NEO, 3 months.
>What kind of background do you have?
I'm a theoretical physicist.
>What is your end-game? What are your goals within the crypto community?
Right now, I want to develope a decentralized voting platform. If you're interested, tell me your name on Discord and I write you.

I got into it because I like Python and the Ethereum market cap is already 20 times that of NEO. All those crytpos will double in worth within a few months, so it doesn't really matter where you put your money in.