/sci/ - Science & Math

>>12425425
Yes, sure.

His whole "colored complex numbers" is just the 2 dim rep of the (non-Hamiltonian!) [math]\left(\frac{{\mathbb F}}{-1, 1}\right)[/math]-quaternion algebra.
https://en.wikipedia.org/wiki/Quaternion_algebra

Obviously there's nothing new under the sun about small matrices.
His citing habits are not extensive.

>>12224266
>>12224290
So does nobody have any input?

>>12229891
In what context do you even have to prove this?

What you want is "function extensionality" and apriori it's not a feature of each of Martin Löf's type theories. In HoTT iirc it follows.

Of course they main/famous axiom to prove equality of two types is univalence, saying that A==B -> (A=B) where by == I mean type equivalence, which is when you get find functons (equivalences) putting A and B as images of another

>>11003901
Not sure what you're getting at but I should probably recompile it again. Also has quite a physishits bias when I look at it today.

I just digged out the source and saw that I trimmed the following

Microlocal Analysis
Deformation
Quantum Groups
Fourier Analysis
Topological Groups
Representation Theory
Special Functions
Module Theory
PDEs
Lie Groups
Harmonic Analysis
Wavelet Analysis
Potential Theory
Integrable Systems
Turbulence
Chaos Theory
Dynamical Systems
Metric Spaces
Stability Theory
Optimal Control Theory
Ergodic Theory
Calculus Of Variations
Optimization
Bifurcation Theory
Measure Theory
Stochastic
Fuzzy measures
Bayesian Inference
Statistics
Martingale Theory
Regression Analysis
Probability Theory
Estimation Theory
Game Theory

Here's the full list, I'm happy for comments (best in that file right there)

https://gist.github.com/Nikolaj-K/502bf35e1fb818df85f33de47668a793

>>10926070
I'm pondering about doing a review of this year old unfinished manuscript on CFZ set theory, which has a model in a popular modern type theory
https://www1.maths.leeds.ac.uk/~rathjen/book.pdf

In that case it will end up here
https://www.youtube.com/c/NikolajKuntner

One of my all time favorite SE answered is one of two from the philosphy page that asked about the history of things, see the answer here
https://philosophy.stackexchange.com/questions/2617/how-did-first-order-logic-come-to-be-the-dominant-formal-logic

For logic, model theory and sets I really like those lecture notes by simpson, in particular the introduction
http://www.personal.psu.edu/t20/notes/

If you want a digestible read on how sets emerge from cats, I recommend the first 150 pages or so of Goldblatts Topoi, introducing categories from scatch
https://www.amazon.com/Topoi-Categorial-Analysis-Logic-Mathematics/dp/0486450260

There's also a paper by Adwodey
>From Sets to Types to Categories to Sets
where he shows how the movement between them could/would work.
https://www.andrew.cmu.edu/user/awodey/preprints/stcsFinal.pdf

One of my favorite math history books is
>Modern Algebra and the Rise of Mathematical Structures
that covers logic and algebra and what it meant to people and how they thought about it from Galois to Grothendieck, I really recommend it
https://www.amazon.com/Modern-Algebra-Rise-Mathematical-Structures/dp/3764370025
Has a lot of pages dealing with Hilbert as well

>>10890933
>>10891001
Here's a list of undecidable tasks:
https://en.wikipedia.org/wiki/List_of_undecidable_problems

And those are problems that aren't dependent of any particular set theory axioms.
Statements indpendent of standard set theory ZFC, which is a notion not as severe as undecidablility, are listed here:
https://en.wikipedia.org/wiki/List_of_statements_independent_of_ZFC
E.g. ZFC set theory doesn't let you prove
>If Y is a bigger set than X, then Y also has more subsets than X
(Counterexamples exist for permissible sets of size between N and R)

>>9848795
>(1)
is that a question?

>(2)
https://mathoverflow.net/questions/182412/why-do-roots-of-polynomials-tend-to-have-absolute-value-close-to-1

>>9848801
If z ranges over the sample zeros, and n is fixed, then

[math] u_k =min(| z- e^{i k/n} |) [/math]

for k from 1 to n/2 are n/2 of those roots (each close to e^{i k/n}) and so the sum over the section lengths [math] |u_k-u_{k-1}| [/math] will go to pi.

>But what I've found is that while professors often provide lip service to math as a pursuit of transcendent truth, their actual motivation is entertainment.

What's bad about entertainment?

The sad part, the real redpill, is that a major reason is the hope for fame.
(That's true for all academics.)

>As if the goal of mathematics was to fill up your free time, not some higher understanding.
I don't quite see the difference.
But to a large extent, I'm a formalist and I don't believe in any strong sense of
>truth

Men choose rules and play games.
(And if you consider ALL possible game rule, you're moreover rewarded that some mirror natural processes, and thus there are some applications.)

>>8800573
I actually tried to brige the two results, but I didn't get anything nice.

You have

[math] 1+2+3+4+... m = \lim_{z\to 1} \sum_{k=0}^m k \, z^k [/math]
but the limit m to infinity doesn't exist.

Now introduce a function smoothening

[math] \langle f \rangle (k) := \int_k^{k+1} f(t)\, dt [/math]

If you consider

[math] \sum_{k=0}^m \left( k \, z^k - (1-q) \langle k \, z^k \rangle \right) [/math]

you get an expression which for z to 1 and then q to 1 goes to
[math] m(m+1)/2[/math]
as it should
and if you take the same expression and go with m to infinity and with q to 0 and THEN with z to 1, you get
[math] -1/12[/math]

I found the regularized sum has a closed expression, but it it's not worth looking at much, as it looks as follows:

[math] \dfrac{(1-q) (z-1)^2 \left(z^{m+1}-1\right)+z \log (z) \left(z^m \left((m+1) (q-1)(z-1)^2+(m (z-1)-1) \log(z)\right)+\log (z)\right)}{(z-1)^2 \log ^2(z)} [/math]

>>8686556
I guess you're right. If I may reword what you say, the sad aspect is that notable progress in pure maths appears to happen on a scale longer than one humans lifetime.

On the one hand, it's to do with our brains, e.g.
https://en.wikipedia.org/wiki/Numerical_cognition
E.g. there are experiments suggesting humans/monkeys can count roughly to 7 in an instant, but not really more. I.e. when you see a picture with 1.000.000 dots for a second, a human has no capability of telling if he saw 100.000 or 1.000.000 or 10.000.000. Those are all just "many" and he can't grasp the difference, really. If, however, you're shown 2 points, or 4 points, for a second, you know exactly you saw 2 or 4. The quantity where it becomes fuzzy is about 7. If you see 12 points, you can't really tell if it was 11 or 12 or 13.
You can teach addition in first class at school, but it might possibly be impossible to teach Lie group theory and functional integrals there, just because human brains aren't capable and not made for it.
Even if we have e.g. attempts like Homotopy Type Theory where rewrite higher homotopy theory, usually very hard to learn, in a way that takes some of it's complicated aspects as starting point (thus making it more accessible), I don't think all relevant mathematical theories and idea even have a concise rewrite in that sense.
In fact I'd say that complexity theory is telling us about the inherent difference in how simple some notion can be expressed.

On the other hand, even if we were twice as smart, I think the mathematical universe is basically endless and in this supersmart parallel world, the same would have happened: The "lower hanging fruits" were put together in a few hundret years (they'd just get further than us), and the new generation - even if much smarter than use - would find themselves with super hard problems they'd not be able to solvequickly.

>>8686590
>physics is always just missionary
kek'd

>>8686556
I guess you're right. If I may reword what you say, the sad aspect is that notable progress in pure maths appears to happen on a scale longer than one humans lifetime.

On the one hand, it's to do with our brains, e.g.
https://en.wikipedia.org/wiki/Numerical_cognition
E.g. there are experiments suggesting humans/monkeys can count roughly to 7 in an instant, but not really more. I.e. when you see a picture with 1.000.000 dots for a second, a human has no capability of telling if he saw 100.000 or 1.000.000 or 10.000.000. Those are all just "many" and he can't grasp the difference, really. If, however, you're shown 2 points, or 4 points, for a second, you know exactly you saw 2 or 4. The quantity where it becomes fuzzy is about 7. If you see 12 points, you can't really tell if it was 11 or 12 or 13.
You can teach addition in first class at school, but it might possibly be impossible to teach Lie group theory and functional integrals there, just because human brains aren't capable and not made for it.
Even if we have e.g. attempts like Homotopy Type Theory where rewrite higher homotopy theory, usually very hard to learn, in a way that takes some of it's complicated aspects as starting point (thus making it more accessible), I don't think all relevant mathematical theories and idea even have a concise rewrite in that sense.
In fact I'd say that complexity theory is telling us about the inherent difference in how simple some notion can be expressed.

On the other hand, even if we were twice as smart, I think the mathematical universe is basically endless and in this supersmart parallel world, the same would have happened: The "lower hanging fruits" were put together in a few hundret years (they'd just get further than us), and the new generation - even if much smarter than use - would find themselves with super hard problems they'd not be able to solvequickly.

>>8583912
Study any engineering field while knowing learn C/C++/Java or even Python and do an internship of 1 month or a 20 hours job each year or each second year. Then you will always find a job, independent of why you studied.
If you study computer science and do code monkey stuff, you'll also always find a job and it'll be enough to live off.
To actually get to a high pay position, you'd be a person liking to work towards a managing job, which means eventually getting off the duty work you do at the start. The can work at a car company with aerospace engineering (think engines) just as with a computer science job at a tech company, or electrical engineering at a semiconductor firm. I can't give you the details which of those works better, and I doubt anybody can because times change.
In any case, going into a degree because of money is fucking retarded imho

>>8421068
Let me throw in some buzzwords and you'll see where you can pic something up.

Numbers like the charge and the spin characterize different fields (functions on space of space time) out of a certain class of fields that are relevant for physics.

Spin always relates to certain symmetry groups and their so called representation. Do you know what a representation is? E.g. the rotations in the plane can be represented by a 2x2 matrix (SO(2)) with an angle parameter t, or as exp(+it), or exp(-it), and it's all rotations but written down differently. Spin in 2 complex dimensions, it's tied to the rotation group in 3 real dimension (SO(3)). The charge is somewhat like the difference between + and - in the exponent there.
With isospin and fields, the notions of spin and (electrical) charge are brought together.

All the relevant fields are tied to some spin representation (i.e. have a spin).
Supersymmetry is so to speak an abstract kind of rotation where different representations mix.

>why would all the bosons have spin in the first place
Almost by definition of what "boson" means.

PS what do you do in CS exactly? Join the Robotics book reading
Good night.

I'm a theoretical physicist who seems to be ending up in statistics coding now, like is often the case.
So if you cut Engineering, sure - I'm in.

>>8253049
You get on my nerves, bro. No, touching isn't a particularly interesting example. My point is (as I emphasized) that all words describing things are like touch - they don't actually work.

>>8253876
I'm a physicist

I'd love it if there were a clean answer to this, but afaik the forget functor concept is nowhere formalized, really.
You can do it in terms of sets via equivalence relations, but this is cumbersome and ugly as night.

=== Derivation rules ===

Let <span class="math"> A [/spoiler] , <span class="math"> B [/spoiler] and <span class="math"> C [/spoiler] be some claims and consider the last sentence.

We might take the following argument to be a proof of this statement:

If we have established that <span class="math"> A \land B [/spoiler] holds, then in particular <span class="math"> B [/spoiler] holds. If moreover we have established that <span class="math"> B \Rightarrow C [/spoiler] holds, then, as <span class="math"> B [/spoiler] is the condition <span class="math"> B \Rightarrow C [/spoiler] , we can put the two together to establish that <span class="math"> C [/spoiler] holds.

We introduce notation to capture this sentence, which is briefly discussed before Formally, Derivations can be considered trees and the proof presented as

<span class="math"> { \frac{{ \frac{(A \land B) \ : \ \mathrm{True} } {B \ : \ \mathrm{True} } } ( \land E_2) \ \ \ \ \ (B \Rightarrow C) \ : \ \mathrm{True} } {C \ : \ \mathrm{True} } } ( \Rightarrow E) [/spoiler]

intermezzo:
http://strawpoll.me/5724986

As derivations only go from true propositions to other true ones, we may drop the " <span class="math"> : \mathrm{True} [/spoiler] " and write

<span class="math"> { \frac{{ \frac{(A \land B)} {B} } ( \land E_2) \ \ \ \ \ (B \Rightarrow C)} {C} } ( \Rightarrow E) [/spoiler]

In this proof, the following derivation rules have been adopted and used used (rules also written without " <span class="math"> : \mathrm{True} [/spoiler] "):

* <span class="math"> \land [/spoiler] - Elimination rule:

<span class="math"> { \frac{X \land Y} {Y} } ( \land E_1) [/spoiler]

resp.

<span class="math"> { \frac{X \land Y} {X} } ( \land E_2) [/spoiler]

* <span class="math"> \Rightarrow [/spoiler] -Elimination rule:

<span class="math"> { \frac{X \ \ \ \ \ X \Rightarrow Y} {Y} } ( \Rightarrow E) [/spoiler]

A logical or formal mathematical theory is specified by specifying the logical language, the derivation rules and a collections of axioms. Such an axiom system would be <span class="math"> (A \land B) \ : \ \mathrm{True} [/spoiler] and <span class="math"> B \ : \ \mathrm{True} [/spoiler] , from which we could follow <span class="math"> C \ : \ \mathrm{True} [/spoiler] .

We have that

<span class="math">\frac{1}{1-x}=1+x+x^2+x^3+...[/spoiler]

for x in the interval (-1,1).

What is true is that for x not 1, we have

<span class="math">\frac{1}{1-x}-1+\frac{1}{1-1/x}=0[/spoiler]

However, x to 1/x maps (-1,1) to R\[-1,1] and so we cannot expand both terms at the same time to give the result in OPs pic. Hence in that sense we cannot make much sense of it.
Naively it's plain wrong because for positive x it's summing up infinite positive terms and we'd expect a positive number at best.

What we can do is the following:

Let
<span class="math">r(t):=a+b·(1+e^{i\pi t})[/spoiler]
be the arc path in the complex plane with
<span class="math">r(0):=a+2b[/spoiler]
and
<span class="math">r(1):=a[/spoiler]
some complex numbers.
Choose an |a|<1 and an |b|>1.
Then outside the disc, for |z|>1, we have

<span class="math">h(z):=\sum_{n=0}^\infty \frac {1} {z^n} = \frac {1} {1-1/z}[/spoiler]

with

<span class="math">h'(z) = -\frac {1} {(1-z)^2}[/spoiler]

Now

<span class="math">h(r(0)) + \int_0^T h'(r(t))·dr(t)= 1-\frac {1} {(1-a)-b·(1+e^{i\pi T})} [/spoiler]

Here I've started with the well defined value at <span class="math">h(a+2b)[/spoiler] and computed the value of the analytic continuation, by summing up all the changes of the function along the line r(t).
As I don't pass the problematic point z=1 (I enter the disc from another side), the change is always finite and I can integrate it up.

And indeed, when it reaches the point a at T=1, it becomes the inverse of
<span class="math">\sum_{n=0}^\infty a^n=\frac {1} {1-a}-1[/spoiler]
and so the result (this interpretation of the equation) is zero.

The q-Pochhammer symbol is a bigger part of the Kac book - but it's really only a few pages long, the "book" I mean. I think it's written by a student.

I actually came across q-analogs when learning about q-deformations of SU(2) in the context of quantum field theories on non-commutative spacetimes.
http://en.wikipedia.org/wiki/Noncommutative_quantum_field_theory

I did my Masters in that general area, although I sicked to the differential geometric aspects. The classical ones, if you will.

Then I read about q-analogs via wikipedia.org/wiki/Tsallis_entropy

The Joy of Cats state/category stuff examples sound different from what you described, but do you know the category of context stuff - that sounds more like it. One approach to it is found in http://bookzz.org/book/874776/085691

I don't know why you ask to ask if you should recommend something, that's literally one line. Not that I don't have enough to read and no time at all :)

>for me, it's been equalizers of internal and external commutative operators on certain varieties
Why?