/sci/ - Science & Math

>>12528882
That cola-wine or what it is, is bound to fall over.

Also, what is the poster refering to as made for fun - the world? Don't women too?
Or is it a Man as in humanity? But then the women seem to enjoy themselves.

>>12465872
Well, to compute determinants of the form
det(A+B)
where the determinants of A and B are simpler (in my case, they are unitary matrices or derivatives of them).
I'm starting to think the formula is a relative of Jacobi's formula, which would suggest it works for higher dimensions too..

>>12283998
Given that C embeds into GL(2,R) and bigger direct sums of such, we'll quickly find infinitely many solutions.

For one, your condition on matrices implies
det(X)=exp(trace(A))

Your requirement is however much more strict than
>X∈G and X∈g simultaneously

>>11590389
I looked a bit into the topic of this paper - actually computing zeta function zeros.
I was aware that it's a theorem that the Riemann hypothesis was [math] \Pi_0 [/math] in the arithmetic hierarchy, i.e. there's a natural number problem such that to find a counter example you can brute force it discretely. But I always found it dubious how one would verify an actual zero on the critical strip and I'm still a bit nebulous about Riemanns computation thereof in the 1850's. But it was through that paper brought to my attention that
[math] \pi^{-s / 2} \Gamma(s / 2) \zeta(s) [/math]
is real for [math] s= tfrac{1}{2} + t [/math].
(Here's a small task for u: prove this!)
It means that a sign change long t actually guarantees you got the bounds of a zero.
Sidenote, it looks like Turings last paper is on finding zeros and hell this is an ugly looking subject
https://en.wikipedia.org/wiki/Turing%27s_method

Now I tinkered around with my own approaches, in particular via the Eta function representation convergence for Re(s)>0 and

[math] \zeta(x+i\,y) = \dfrac{2^{i\,y}}{2^{i\,y}-2^{1-x}}\cdot\sum_{n=1}^\infty \dfrac{(-1)^{n-1}}{n^x} \left[ \cos(y\, \log(n))-i\sin(y\, \log(n))\right] [/math]

I do some introductory discussion here
https://youtu.be/Fl3XgPpvSNI

>>11290146
>most constructivists deny its validity
Doubtful. Cantors proof is constructive, see below

>>11290151
>But if he were to find such a bijection, that would indicate that Cantor's argument is flawed
No, he'd have shown a contradition of the system used.
If [math]E[/math] is the claim that there's a bijection, he'd have shown
[math] E \land \neg E [/math]
which is not a problem with Cantor's arugment but with the logic/the theory.

>Constructivists aren't mathematicians.
Every pure intuitionistic proof is also, without changing of words used, a classical proof, so no issue can come from that side.
E.g.
https://youtu.be/0Sl0bcXyTAo
https://youtu.be/rHsuesTdFLM

Cantors proof establishes the non-existence of a bijection.
What constructive mathematics can have an issue with is deciding disjunctive statements
such as LEM [math] E \lor \neg E [/math]
or trichotomy relations such as [math] (a<b) \lor (a=b) \lor (a>b) [/math]
e.g. the total ordering of larger ordinals or, indeed, cardinals.

Not Cantors theorem breaks with classical provability, but e.g.
https://en.wikipedia.org/wiki/Schr%C3%B6der%E2%80%93Bernstein_theorem
telling you when two cardinals relate as a=b or a<b.

As a consequence, there's constructive models where N and P(N) are not naively ordered like |N| < |P(N)|, but e.g. P(N) may be subcountable

https://en.wikipedia.org/wiki/Subcountability

>>11289624
In what sense? Anyway, the answer is most certainly no.
Here's a few pointers:

[math] \omega^\omega [/math], or [math] {\omega_0}^{\omega_0} [/math], is a countable ordinal in the smaller to medium range.
You can describe the order type it represents in Peano arithmetic.
It represents, roughly, the collection of all arbitrary sized but finite lists whos entries are natural numbers.
As a countable ordinal, it's cardinality is that of the natural numbers, [math] | {\omega_0}^{\omega_0} | = |\omega_0| = | {\mathbb N} | [/math].
Consider the description of countable ordinal types beyond infinity here:
https://youtu.be/EAAC9dCV9_k

[math] { \mathbb R } [/math] is of size of the continuum,
[math] |{ \mathbb R }| = | P({\mathbb N}) | =| 2^{ | {\mathbb N} | } | = | | \{0, 1\}|^{ | {\mathbb N} | } | = | | \{0, 1\}|^{ | \omega_0 | } | [/math]
and [math] | P({\mathbb N}) | \ge | \omega_1 | [/math] where [math] \omega_1 [/math] is set holding all countable ordinals, i.e. the smallest uncountable ordinal. This one is uncountable, so long long past all the easily compresensible ordinals.

To make the comparison sharp, [math] | P({\mathbb N}) | = | \omega_1 | [/math], is to claim the continuum hypothesis.
The size of [math] { \mathbb R } [/math] w.r.t. to ordinals is independent of ZFC, there's no good answer to it.
So your question was doomed from the start...

>It seems though like that definition might be a special case of some stuff like topos or sheaves or something.
What is "that definition" in this sentence now? Sheaf-categories fulfill those properties, so the latter is the more general variant, for what it's worth.
I just wanted to point you to it, in case you're interested. I was a bit scared of looking at your formulas.
>Has analysis, algebra, topology etc. just already "filled" the space of possible mathematical definitions?
"No", but I think it's a matter of (Kolmogorov) complexity.

>>11280330

[math] (P\land (P\to Q))\to Q [/math]

is proven by

[math] \lambda \langle p, f\rangle.\, f(p) [/math]

In words, given a proof of [math] P\land (P\to Q) [/math], use modus ponens to conclude [math] Q [/math].

The principle of non-contradition is the special case of the above with Q a false proposition, i.e.

[math](P\land (P\to F))\to F [/math]

which you may write as

[math] \neg(P\land \neg P) [/math]

----

This in turn happens to be a classical equivalent of LEM,

[math] \neg P \lor P [/math]

So going from here, and weaken it to

[math] \neg P \lor \neg \neg P [/math]

and then use distributivity,

[math] (\neg R\lor \neg S) \to \neg (R\land S) [/math],

then you also get

[math] \neg(P\land \neg P) [/math]

>>5654537
>DHoTT
>Omega-groupoide
>free theorems
didn't get that.

>There's people that want...
I don't see any of these programs getting more users than Haskell or Lisp and I feel these aren't even used enough to call it worth is.
I don't *really* get the purpose of the veryfy-math-only programs. If nobody else is interested in it, I don't see why they are. Okay, I don't know maybe that's no good point of view, but do you see something which you'd call acutal progress in that field?

18 is impressive, what else do you do?
(mathematically or otherwise)

PS: Go find yourself and actualy problem to solve. Otherwise you get nowhere.

http://www.ll.mit.edu/news/DRACO.html

>>5426779
>.99.. could only be described as 3 * 1/3, but 1/3 is not an infinite string of threes behind a decimal. Its a finite string depending on how many times we choose to engage in the fractional number operation which certainly will yield another three.
This is an odd thing to say, and I think I'm having a bit of trouble understanding it. Are you saying that if I only bother to do 1/3 to four decimal places, it would be equal not to 0.33... but just 0.3333? That would be incorrect-a number is not made different by how far we go to calculate it.

Not finals, but I've got my A-levels cumming up between now and late june which I'm studying my ass off for. Fucking complex numbers, how do they work?

>inb4 underage b&
>inb4 LOLDONGS babbys first complex numbers

What would you do if you studied something, got your degree in it, started working and realize you fucking hate it?

I am kind of in that situation and I do not know what to do..
Perhaps it's noteworthy to mention that I live in Germany where degrees are free, no tuition costs (in most States).