/sci/ - Science & Math

>>10901162
[math] {d^n \over dx^n} f(g(x))
=\sum \frac{n!}{m_1!\,m_2!\,\cdots\,m_n!}\cdot
f^{(m_1+\cdots+m_n)}(g(x))\cdot
\prod_{j=1}^n\left(\frac{g^{(j)}(x)}{j!}\right)^{m_j} [/math]

>>10904088
Is it clear that all linear maps of a field extension (say in Q(2^(1/3))) can be represented by field element multiplication in the algebra?

The hardware implementation makes us of a Boolean algebra and on the bitlevel you may think of it as induced by e.g. the NOR connective, i.e the "are they both off" check, i.e. in terms of polynomials over Z_2 represented as
NOR(bit1, bit2) = 1 + bit1 + bit2 + bit1 * bit2
You can read all about the implication of the corresponding circuit gate solving the "are they both off" e.g. at
https://en.wikipedia.org/wiki/NOR_gate

And then sure, if you fix bit2 to being off
NOR(bit1, 0) = 1 + bit2 =: NOT(bit1)
so that
OR(bit1, bit2) = NOT(NOR(bit1, bit2)) = NOR(NOR(bit1, bit2), 0)
and then you may read the graph/circuit
OR(NOT(bit1), bit2) = NOR(NOR(NOR(bit1, 0), bit2), 0)
i.e.
1 + bit1 + bit1 * bit2
as the material implication.

If you want to take this to more practical level, you'll need something like an additon from it
https://en.wikipedia.org/wiki/Adder_(electronics)

and on the procedural programming level, you also get abstraction of those 0's and 1's (those might be a byte in size or more, i.e. holding 7 unnecessary bits for each boolean, for historical reasons) and programmers are free to compose them with program level conditionals too - which will in general be not, or and and.

That said, you'll find even the most exotic logics being translated to type systems and functioning as type level building blocks for e.g.
https://en.wikipedia.org/wiki/Curry%E2%80%93Howard_correspondence
The intuitionistic case is the default one here - and arises from the closeness of type former to the realizability interpretation of logical proofs.

But you also get linear logic implementations nowadays, which don't have weakening, and thus can or should be seen as relevant logics.

I left academia and do some light crypto trading.
But I still wage in the field of Augmented Reality, for the experience.

Let
a=[a_k, a_{k-1},...a_1]
b=[b_m, b_{m-1},...b_1]
be two lists of length k and m, respectively, where each number is either 0 or 1. Let them represent binary numbers, e.g.
[1,1,0,1] = 2^3+2^2+0*2^1+2^0 = 8 + 4 + 1 = 13.
and let + denote the addition of numbers.

Now let
c = a + b
What are the components c_j in terms of the components of a and b, algebraically.
The issue is, if you code it up (which I did, in Python), that you have a lot of if-statements, in particular the stuff related to the carry over bits. So it's easy e.g. for c_1 but hard for later c_j.
A recursive formula would be fine too, but I'd like it purely algebraically, not with stuff like if-clauses and codepathways.

Thanks.

I wrote my thesis (computational chemistry) and successfully defended it. Now I must edit some last typos and print it before I get the document. I had a Russian advisor who was pretty tough to work with some times, but all in all it went well and since I worked at an agency, I didn't have to tutor, which I really apprechiated.
Now I got an interesting well paying job in a pretty much unrelated mathy engineering field.
When I started I also got into crypto and now have enougn money to sustain myself 10+ years without work, so I now really just go to work out of interest.

http://www.math.harvard.edu/~lurie/

Spectral Algebraic Geometry: 2252 pages.
>Roughly 66% done
Higher Topos Theory: 949 pages.

What is he up to?
Has anyone ever read his work?

Do you guys like the sums of divisor function (sigma1)?
What's interesting about it to you?

https://en.wikipedia.org/wiki/Divisor_function
https://wikimedia.org/api/rest_v1/media/math/render/svg/e84ff27cd7cd28822f3e3ee0235f86d1859ff0b7

>>9309622
to quote myself in >>9305415
>I just always assumed it's some guy who wrote his own tools to do this stuff - essentially the pieces that the Wolfram team didn't care for.

The person just took an interest in stuff nobody else does. I'm sure 95% of Erdos meme theorems can be formalized and the techniques be automated too, but it's uninteresting. Go into a library at a math university and pull out a book from the 70's that's not been touched for 20 years, and you'll find many mathematical theories that have been forgotten. And I mean studies of objects like groups without units and stuff like this. Or the whole potential theory spiel. Or first-order theories in logic and thorough investigations of their models, where the axioms are essentially random and without motivation. 97% of math discovered isn't used anymore. If Wolfram doesn't have a normal form and closed solution for it, doesn't mean the algorithm to do it doesn't exist.

>>9305183
because Riemann integration "is solved" to the extent that it can, and in what is now known as computer science. Essentially
https://en.wikipedia.org/wiki/Risch_algorithm
I just always assumed it's some guy who wrote his own tools to do this stuff - essentially the pieces that the Wolfram team didn't care for.

What has Lurie been up to in the last 2 years?

http://www.math.harvard.edu/~lurie/

First .PDF, Spectral Algebraic Geometry: 2252 pages.
>Roughly 66% done
Second .PDF, Higher Topos Theory: 949 pages.
Seriously, what is this stuff?