/sci/ - Science & Math

First for Operator Algebras and Connes' Embedding Problem

>>15898763
Hi anons I've come here to shit on the thread. Basically I met no professors during undergrad but did well (top of the class) in some courses. Now I'm applying to graduate programs (not a PhD, most people go through a Master's first). Is this a good email to send to a professor?
>>>/adv/30276795
>>>/adv/

>>15898859
>I met no professors during undergrad but did well (top of the class) in some courses.
Big oof.
That said I would remove the second and third paragraphs and write about why you're interested in the program and put in a short line about how although you don't know each other well you'd be grateful for the letter instead.

I used to travel around Berkeley on my bike with my shotgun and blow out the windows of houses that had this sign
cops never caught me

>>15898763
Homotope these balls nigga

I will use LEM and there's nothing you can do about it.

What the fuck is the asymptotic behavior of a Fourier transform supposed to tell me? I can't seem to find any resources on this shit.

Omega is not infinit decimal.

>[math]x^{(-)}[/math] is exponentiation
>[math]x_{(-)}[/math] is evaluation of an indexing function
Is there any context in which these two operations can be meaningfully said to be duals of each other?
I suspect that Lie theory might provide such a connection, but I haven't gotten around to studying it yet.

>>15899574
posted in the wrong thread. pic rel

>>15899579
>>15899614

>>15898763
Where can I find past questions of the math subject
GRE to use for practice?

>>15900528
Indian?

>>15900708
>>15900528
No, I'm American

>>15900762
cope

>>15898556

Please help me, employed anon.... again these are all my courses, and a masters program I'm thinking of doing on the right column. How do I grow up to be like you.

putnam was fun this year.
I (probably) didn't get a 0 so I am happy because I expected to get a 0. I only solved A1 and B1 but I had a lot of fun doing it. Did anyone else take it? What did you think?

>>15901114
First time taking it, I’m exhausted. I think i did okay hopefully i’ll place, How’d u solve B1? I got A1,A2, A4, but i couldn’t get any full solutions for the B part. I got close on B5, and I’ll get partial credit for B6, but i had no clue for B1-B4.

How much bigger is the first Mahlo Cardinal compared to the first Inaccessible?

>>15901146
For b1 screwed around with a few of the mxn grids until i noticed a pattern. I noticed that under all legal moves and configurations, there was always a path of empty spaces from the bottom left corner to the top right corner. I argued (without a good proof) that such a path would exist in all legal configurations. So the problem became how many ways can you permute "up" and "right" movements with (m-1) "up"s and (n-1) "right"s. I got (m+n-2)!/((m-1)!*(n-1)!). I probably wont get full points because my solution was nonrigorous but I think the amswer is right and the solution kinda makes sense. After handing it in the professor proctoring it said he got the same answer. Let me know if this makes sense. How did you do the other A problems?

>>15901114
I got 20pts if I’m lucky, realistically 10? At least 1. I didn’t study, so I think for someone who cares more ~30pts is quite reasonably attainable. It must have been a little easier this year.

>>15901226
I didnt study either, i just watch a lot of youtube videos. There was a class in my school for preparing for it but i didnt attend because i had work. Which problems did you solve? I think A1 was kinda free

>>15901232
Yeah A1 didn’t meet my expectations. This is my first time but it did seem too easy from what I hear. I did B1 as well but after reading this thread I may have misread the question. I thought B1 and B2 both seemed very manageable. I worked on A3 rather than A2, and I think A2 I might have had a chance with if I tried. It seemed doable in retrospect

>>15898763
What's a good book on learning how to count. Today I got filtered trying to count the ways to arrange red and green balls.

Also, what's a good book for learning how to drive definite forms of sums. Today I got filtered by trying to do the sum of the first n square numbers.

>t. Putnam A1 and B1

>>15901187
How did you find that perm? I got (m+n-2) choose min(m,n) but it was a wild guess. I am DOGASS at counting. I found a proof that for all unique legal configurations of coins there exists a unique path:

First I noticed that if there's no path an illegal move has been made, whose contrapositive means that if only legal moves then there's always a path.

Then I noticed that if two configurations have the same path then they're the same configuration
Proof:
Suppose they're different configurations and they have the same path, that means at least one coin is in a different position, which means at least one open spot doesn't match. for the same path to exist and this new spot to appear, either a coin has to be moved off the grid or on top of another coin, both of which is illegal. So either the latter configuration is illegal or it is the same configuration

Then if there's two different paths to one unique configuration then the maximal overlap would still require an additional open space, meaning, again, either a coin is not on the grid, or two are on top of each other, again this is illegal.

So it's an equivalent problem and the number of ways to move from one corner to the other (moving orthogonally) has the same result. Which corresponds to the number of ways you can arrange (m-1) up and (n-1) right moves.

Then I got stuck trying to count that.

>>15901290
I just realized if I had written min(n-1,m-1) I would have been correct but only technically, and the rest of my proof is good (in my opinion) and had high potential for better than 1 point (I got A1 by a guess too lmao)

>>15901259
>ways to arrange red and green balls.

Anon. That's just basic combinations and permutations. You don't need a book for that.

>>15901187
A2 was nice. You solve n=2, then you prove by induction. Assume true for n then divide p_n+2 by p_n and you get the base case again.

A4 is fairly simple if you remember the cords of an icosahedron. The vertices of an icosahedron are at 0,+-1,+-phi. Thus to solve the answer you just wanna show that any number in the reals can be approximated by n*phi + m. This proof is similar to irrational steps around a circle.

>>15901290
I am not in a good state to explain it right now but i used the formula for permutations with repetition. I have it memorized because of my job. I cant really type it here but if you look up "formula for permutations with repetitions" you should find it and understand it. I only had to permute two distinct moves ("up" and "right")with repetition.
Tomorrow when i am more able i will try doing A2 and A4 based on what you said, thanks for the info anon

>>15901499
I got filtered by an easy problem. I want a book that builds. I want to learn to count, not just solve that one problem.

>"Applied analysis"
>it's just calculus
>"applied statistical modelling"
>it's just regular stats
>"modelling and computation"
>it's just numerical methods, not a single line of code
>"fields and vector analysis"
>it's just regular vector calculus
Is it just europe or do all unis pick the worst and most misleading names possible for their courses?

>>15902124
In Europe its customary to hype the hell out of course names. In the US, a standard course in analysis is called something like "introduction to real analysis". In Europe the same course would be called "hypertheoretical analysis and measure theory" or something.
Will never get over Germans taking "analysis I" year 1 of their their CS or engineering degree, and having the stupidity and ego to believe its the same course as real analysis for math majors in the USA.

>>15902359
>Will never get over Germans taking "analysis I" year 1 of their their CS or engineering degree, and having the stupidity and ego to believe its the same course as real analysis for math majors in the USA.
Those classes are more advanced than the real analysis classes for math majors in the USA. Where did you get the idea that they don't include measure theory?

>>15902478
Lol no they are not dude. There’s a reason why US universities are the best in the world. Yeah they cost you an arm and a leg, but pretending like they aren’t just superior is silly

>>15902478
>>15902922

I got all As in my math classes (inc. both real analysis, topology, algebra, advanced linear algebra), but Cs in my government, philosophy, and speech classes. giving me a GPA of about 3.0. Is it over for me?

Does anyone here have experience with automatic theorem provers?
I am trying to prove in Isabelle that appending [ 1 ] with itself n times yields me a list such that any element of it is 1. I've been trying all day and it's driving me crazy.
Here's what I have so far.
[/code]
theory Scratch
imports Main
begin

fun repeat_append :: "nat 'a list 'a list"
where
"repeat_append (0::nat) (l::'a list) = l"
| "repeat_append (n::nat) (l::'a list) = repeat_append ( (n::nat) - (1::nat) ) ( append l [ l ! 0 ] ) "

theorem "(n::nat) < (m::nat) - 1 ⟹ nth (repeat_append m [ (1::nat) ]) (n::nat) = (1::nat)"
[/code]

Does anyone here have experience with automatic theorem provers?
I am trying to prove in Isabelle that appending [ 1 ] with itself n times yields me a list such that any element of it is 1. I've been trying all day and it's driving me crazy.
Here's what I have so far.

theory Scratch
imports Main
begin

fun repeat_append :: "nat 'a list 'a list"
where
"repeat_append (0::nat) (l::'a list) = l"
| "repeat_append (n::nat) (l::'a list) = repeat_append ( (n::nat) - (1::nat) ) ( append l [ l ! 0 ] ) "

theorem "(n::nat) < (m::nat) - 1 ⟹ nth (repeat_append m [ (1::nat) ]) (n::nat) = (1::nat)"


Does anyone here have experience with automatic theorem provers?
I am trying to prove in Isabelle that appending [ 1 ] with itself n times yields me a list such that any element of it is 1. I've been trying all day and it's driving me crazy.
Here's what I have so far.

theory Scratch
imports Main
begin

fun repeat_append :: "nat 'a list 'a list"
where
"repeat_append (0::nat) (l::'a list) = l"
| "repeat_append (n::nat) (l::'a list) = repeat_append ( (n::nat) - (1::nat) ) ( append l [ l ! 0 ] ) "

theorem "(n::nat) < (m::nat) - 1 ⟹ nth (repeat_append m [ (1::nat) ]) (n::nat) = (1::nat)"

>>15903458
I've never met a theorem prover whose notation I liked enough to make me want to use it.
If (as it seems) your objective is to acquire an understanding of automated reasoning, rather than analyzing the formal properties of some particular program, then you'd learn far more from writing your own theorem prover (designed to formalize your own modes of reasoning, e.g. constructive/classical/type-theoretic) rather than deciphering the (highly complicated) semantics of existing ATP programming languages, for the same reason that as a math student, you'd learn far more from writing your own proofs rather than reading existing proofs from a textbook.

>>15903134
>but Cs in my government, philosophy, and speech classes
bro it's literally all just rote memorization and bullshiting your way through

is this still accurate?

>>15898851
>Connes' Embedding Problem
do you understand the proof in https://arxiv.org/abs/2001.04383

>>15904156
>women studies
>random signals and noises, naive lie theory
giggled

>>15904156
>all of the above is standard education up to high school (europe)

Not a joke, Euros actually say this.

Is much of modern mathematics a cargo cult science? i.e. imitating the behavior of real mathematics but having no significance.

>>15904240
What does this even mean? Most modern matematics is government agencies working on secret projects, compartmentalizing certain portions of their work, and making mathslaves at universities solve parts of those problems.

>>15904143
Yeah, I just didn't have the will to put up with the busywork. So I did the usual ignore-homeworks-if-possible, panic-study-for-1-hour-before-exams etc.

>>15904251
>Most modern matematics is government agencies working on secret projects
Do you get your math news from pol or something?

Previous anon asking for help again.
Since Isabelle is too complicated and I don't know how it works, I thought I'd try another theorem prover called Prover9 which is simpler and it's based in FOL.
Now I am trying to prove that a function defined as f(x) = f(x-1), f(0) = 1 implies f(3) = 1.
But the prover cannot find a proof.

Assumptions:
N1 = Suc(N0).
N2 = Suc(N1).
N3 = Suc(N2).

(dec(x) = y) <-> dec(suc(y)) = y.

(-x=N0) -> f(x) = f(dec(x)).
x=N0 -> f(x) = N1.

Goals:
f(N3) = N1.

Am I missing some assumption?

How do I prove that if [math] H [/math] is a normal subgroup of [math] G [/math] such that there is a homomorphism [math] \pi : G \to H [/math] satisfying [math] \forall h \in H \; \pi (h) = h [/math], then [math] G [/math] is isomorphic to [math] H \times G / H [/math]?

>>15904654
Turns out it wasn't being able to prove (-x=N0) because I haven't axiomatized any number not to be equal to any other.

>>15901834
Anon how did A2 and A4 go for you?

I've been working on B5 and deduced that [math]\pi^2 = (1, m) (2,2m\mod{n})(3,3m\mod{n})\ldots(n, nm\mod{n})[/math] and so we can use properties of square permutations to deduce some bounds. Firstly [math](n, nm \mod{n}) = (n)[/math] and similarly we have for any [math]k \perp n[/math] we have a nontrivial transposition because if it was trivial we would have [math]mk\equiv k \pmod n \Rightarrow m\equiv 1 \pmod n[/math] which is a contradiction.

We also have all square permutations are even and we have at least [math]\phi(n)[/math] non-trivial permutations but I have no idea how to proceed from here

>>15904676
Consider the map [math]\phi : G\rightarrow H\times G/H[/math] given by [math]\phi(g)=(\pi(g),gH)[/math]
Verify this is an isomorphism

Hate the feeling of having too many classes at once. University format is so ass. You get in the zone in a topic, start diving in deep, find something tremendously interesting, and then its time to move on to another subtopic or focus on another course altogether. In a semester or two you're already too rusty to pick up at that point without another month of dedicated review. This stops in grad school right? No more getting pulled in all directions?

>>15904755
This is what I have so far. At first it seemed promising but it looks like this automatic theorem prover has more or less the same limitations as the more complex proof assistants. I am trying to prove that 3 ≤ 2 is false but it's been running for a while and hasn't been able to either prove it or detect that it isn't provable.

Assumptions:

N1 = Suc ( N0 ) .
N2 = Suc ( N1 ) .
N3 = Suc ( N2 ) .

- (N1 = N2).
- (N1 = N3).
- (N2 = N3).

% Definition of Dec
all x all y ( ( Dec ( x ) = y ) <-> Suc(y) = x ).
all x all y ( x = y <-> y = x).

% Definition of repeat_append
all n all x ((n=N0) -> (repeat_append(n,x) = Nil)).
all n all x ((-(n=N0)) -> repeat_append(n,x) = Cons(x, repeat_append(Dec(n),x))).

% Definition of take_n
all x all n all y all l all r ((n = N0) -> (((x = Cons(l, r)) & y = l) -> (take_n (x, n) = y))).
all n all x all y all l all r ((-(n=N0)) -> ((x = Cons(l, r) & y = take_n(r,Dec(n))) -> (take_n (x, n) = y))).

% Definition of dec_m_by_n
%all n all x all y all r ((n = N0) -> (dec_m_by_n(m, n) = m)).
%all n all x all y all r (-(n=N0)) -> (((dec_m_by_n(m, n) = y) <-> (y = dec_m_by_n(Dec(m), Dec(n))))).

%Definition of Incr_m_by_n
all m all n ((n = N0) -> Incr_m_by_n(m, n) = m).
all m all n ((n != N0) -> Incr_m_by_n(m, n) = Incr_m_by_n(Suc(m), Dec(n))).

% Inequality
all a all b ((exists n (Incr_m_by_n(a, n) = b)) -> (le(a,b))).
all a all b ((exists n (Incr_m_by_n(b, n) = a)) -> (ge(a,b))).
all a all b ((exists n (Incr_m_by_n(a, Suc(n)) = b)) -> (lt(a,b))).
all a all b ((exists n (Incr_m_by_n(b, Suc(n)) = a)) -> (gt(a,b))).

%causes -le(N1,N0). to be true, why?
%all a all b (gt(a,b) -> (a != b)).

Goals:
%all n all m ( (lt(n,m)) -> (take_n(repeat_append(m,N1), n) = N1)).
-le(N3,N2).

>>15904824
not surjective

>>15904236
all memes aside, euro highschools have different levels of maths with the most advanced being on the level of precalc with maybe tidbits of single variable calculus.
You are required to take the advanced maths in hs to be able to take most math courses in uni, this is why most euro unis don't have "precalc" or "trigo" or any other very rudimentary math courses

>>15904824
>>15905809
Nevermind. Thanks.

>>15905858
The Abitur in Germany which is the high school graduation exam everyone there needs to take does include differential and integral calculus among other things.

>>15905948
according to wikipedia, math is optional on the abitur
>differential and integral calculus
those are big words for basic integration and differentiation.

kind of weird that you guys call stats "stochastic"

>>15905858
>You are required to take the advanced maths in hs to be able to take most math courses in uni, this is why most euro unis don't have "precalc" or "trigo" or any other very rudimentary math courses
This is good, and your highschools are 100% better than American ones in every way. While some US highschools (guessing like 30%) prepare students for university with proper PreCalculus courses and fewer still have calculus I (analysis I) as an offering, most don't. Most US high school teachers are absolute morons actually.

US colleges need to offer remedial math courses to compensate for the terrible k-12 education most freshmen have. It isn't that these freshmen are all incapable, they were just never given the chance.

>>15906041
>math is optional
Germany is split into 16 Bunderländer. All of them have slight differences in their educational systems. In many of them math is required.

>>15905858
"Advanced math" taught by high school teachers. I'm sure that's a good idea.

>>15906054
as everything else in the united states, it completely depends on where youre from, wealth, or in exceptional circumstances (thomas Jefferson, stuyvesant) purely ability. In my high school i took single var calc when i was 15, and multi, complex, and lin algebra when I was 16 and 17. It was good preparation for Uni, since my teachers were actually professors. Some kids took single var calc when they were freshmen; there was a kid in my class who was 12 (bizarre situation).

>>15906339
Whoa. Do you have any more demographic information? I took precalculus my junior year, and there wasn't anything left to take my senior year so I just got told to go home at noon everyday. It was essentially a waste of an entire academic year, and I graduated at 16 before going to my local SLAC which had very little in terms of math curriculum.

>>15906425
nyc stuyvesant high school crazy stuff. Public high school 3600 kids with half the kids impoverished

>>15902922
American universities attract the best talent, that's for certain. So in that sense, the very best American universities will have the best concentration of talent in the world (such as MIT, Caltech), and in that sense they will be better. But in terms of what is taught or the quality of instruction in isolation, I have little reason to think it would be better. but of course the advantage of having such concentrated talent can't be overstated imo, as it also forces/encourages you to push yourself and meet extremely high achieving people
Please don't mind the file attached (it's a translation of Hitler Youth handbooks from 1938), this is for linking to another thread.

>>15907229
Good bedtime stories for the kids. Thanks.

>Given a catenary [math]y = a \cosh{ \frac{x}{a} }[/math] with symmetry about the y-axis, find in terms of [math]a[/math], [math]b[/math] and [math]x[/math], where [math]a[/math] and [math]b[/math] are constants, the formula of the parabola for which the area bounded between the two curves over the interval [math][-b, b][/math] is at its minimum.
How does one go about solving this?

>>15905866
Hope it's not too opaque how I came up with that. It's a very natural map, given the information

>>15908331
I also came up with it, but I was initially not able to prove it's isomorphic.

>last day of class
>professor stop me at the door
>"You need to go to graduate school. I've noticed that your solutions to the homework problems always stand out, and often you have a better solution than me or the book."
>t-thanks you too
BASED?

>>15898763
I feel like that last line was made by the lizard people to stop us from getting anywhere meaningful with math too quickly

>>15908217
You can try the Ansatz [math]y = c + d x + e x^2[/math] and then set the partial derivatives of
[eqn] \int_{-b}^b \left| c + dx + ex^2 - a \cosh \left( \frac{x}{a} \right) \right| dx[/eqn]
with respect to c, d and e equal to zero. This gives you three equations that you can solve for c, d and e.

>an infinite set may be equivalent to one or more of its proper subsets
This is not math, this is insanity.

Not sure how to ask this in a non-crass way, but what is with the stick-up-the-ass people have over real analysis? If you already took calculus, its pretty easy and intuitive? This course was overhyped to hell and back all over the internet and its a fucking joke. You aren't learning anything new, and its hard to be forced to as you can quite literally memorize everything if that is the way you "learn" things. Algebra was way worse and gave the sensation memed on about with real analysis.

>>15908842
maybe you just have a special type of autism, because in many cases these exact reactions are switched (as you have undoubtedly noticed)
analysis introduces a bunch of concepts that are initially unintuitive (e.g. continuity of the Dirichlet function versus the Thomae function, or Stieltjes integrals), and if you have a poor lecturer it's not hard to see why there would be confusion
algebra, in contrast, is pretty straightforward, and a lot of basic groups/fields/etc. have obvious real-world examples that make them far less arcane to the average student

>>15908842
do you eat your corn in rows like a typewriter or columns/spirals?

>>15908875
not him but I dislike biting into corn that doesn't have corn surrounding it so I hop randomly

>>15908342
Congratz. He wants to write you a recommendstion letter.

>>15909012
that’s some impressive autism right there
https://bentilly.blogspot.com/2010/08/analysis-vs-algebra-predicts-eating.html?m=1

>>15909061
>take funny meme
>turn it into pedo tranime

>>15909061
>>15909121
10 in base 3

>>15909129
Fuck, should be 11

>>15908870
I bet your "algebra" course said nothing about group actions, sylow groups, solvable groups, field of fractions, symmetric polynomials, etc.

>>15909188
As much as you would like me to say it finished at Gaussian elimination, it had all of the above, unfortunately
in fact, Sylow's theorems alone got an entire week dedicated largely to proving them

>>15909129
>>15909174
>10 in base 3 has the same value as 3 in base 10
>11 in base 3 does not have the same value as 3 in base 11
What makes this hold for 10 but not 11?

>>15909232
So what are the obvious real-world examples of short exact sequences?

>>15909249
I didn't say anything about short exact sequences having real-world examples, and I could ask you the same question about whatever arbitrary, bizarre, "haha gotcha" function they introduce as the most extreme counterexample possible in analysis.
Be more literate.

>>15909065
>All of the analysts were eating in spirals, and the algebraists in rows.
TIL how I fucking ate corn as a child predicted my interest in analysis
what in the fuck

yes i am joking you autists but it's a very funny blog post thanks anon

>>15909065
Reading things like this and posts on math reddit, makes me realize most people into math are quite stupid.

Brainlet here.

Why if x/y = a then y/x = 1/a ?

>>15909549
10/5 = 2, then 5/10 = 1/2

>>15909536
Everyone is varying degrees of stupid including you.

>king of topology
>failure at algebra
Should I be scared of algebraic topology? I'm taking it in summer

>>15910298
>>king of topology
>>failure at algebra
Why? How does this even happen?

Hello I'm a community college student about to enter my final semester before transferring for a math degree. Before I do, I really want to have some kind of project under my belt. I've taken Calc I-III, an intro to proofs class, and intro to ODEs so far. I know this isn't much but is there anything interesting I can do to really push myself? My basic idea is to get some really difficult problem (feasible for a Sophomore) and then take an honest crack at it while learning new math in the process of trying to understand it, but I don't know what to do. There's so much math to do every which way that I'm lost.

How do I prove or disprove the following?
[eqn]\lim_{x \to{ \infty}} \left( \frac{a_1^x + a_2^x + … + a_n^x}{n} \right)^{ \frac{1}{x}} = \max (a_1, a_2, …, a_n) \\
\lim_{x \to{- \infty}} \left( \frac{a_1^x + a_2^x + … + a_n^x}{n} \right)^{ \frac{1}{x}} = \min (a_1, a_2, …, a_n) \\
\lim_{x \to{0}} \left( \frac{a_1^x + a_2^x + … + a_n^x}{n} \right)^{ \frac{1}{x}} = \sqrt[n]{a_1a_2…a_n}[/eqn]

Is PA a social contract?

>>15911112
The fact that so many civilizations "created" the exact same basic arithmetic separately without being in contact shows that there is something more fundamental to 1+1=2 than "it's just a consequence of arbitrary axioms"

>>15911167
Prehistoric civilizations develop arithmetic for one of two reasons:
>If they recognize the right to property ownership, they'll develop an accounting system to take stock of their wealth
>If they track the seasons (or worship the sky gods), they'll develop a calendar system to count the passing of days
I guess you could argue that capitalism and astronomy are "fundamental" in a sense, but that isn't how the word is normally understood in mathematics.

>>15910951
For the first one note that [math]\ \lim_{x\rightarrow \infty} n^{1/x}=1[/math]. So we get [math]\ \lim_{x\rightarrow \infty}(a_{1}^{x}+...+ a_{n}^{x})^{1/x}= \lim_{x\rightarrow \infty}exp(ln(a_{1}^{x}+...+ a_{n}^{x})^{1/x})=\lim_{x\rightarrow \infty}exp(\frac{ln(a_{1}^{x}+...+ a_{n}^{x})}{x})[/math].
If there is a greatest element [math]\ a_{k}[/math], then [math]\ \frac{a_{i}}{a_{k}}<1[/math] for all [math]\ i\neq k[/math] and we get [math]\ \lim_{x\rightarrow \infty}exp(\frac{ln(a_{k}^{x})+ln(\frac{\sum_{i\neq k}a_{i}^{x}}{a_{k}^{x}}+ 1)}{x})[/math]. And so your greatest element pops out.
I'm fairly certain the others follow something similar.

>>15909562
>10/5 = 2
?

>>15909549
Because if you multiply them together you get 1

>>15898763
>sit in on category theory seminar
>mfw it's all HoTT

When will they learn

>>15909549
first equivalence -> x = y*a
second equivalence -> y*a = x

>>15910550
algebra is too magical for me
I never understood anything about group actions

https://math.stackexchange.com/questions/236212/prove-that-simultaneously-diagonalizable-matrices-commute

Does anyone recognize what book this exercise is from?

>>15910892
Maybe ask an instructor who knows you better. If it’s learning you’re after, it’s never too early to open up a standard textbook in analysis or topology and just do (almost) every exercise in the book in order, you’ll probably learn the fastest that way.

I wish we had programs like this in my country. This is newtype level stuff.

https://youtu.be/s6OmqXCsYt8?t=314

>>15909234
Every base is base 10.

>>15912316
me with my last brain cell trying to remember why multiplication works

What is the data limit for what can be stated in math?

Shouldnt that simplest statement lie at the bottom of all things?

>>15912983
Nuh uh

Do we have an equivalent of Sylow theorems for ring?

How do I cope with the fact that what I've been trying to prove for months is most likely unprovable? Like how the fuck do I move on? I keep coming back to it and it's so much wasted time and effort.

>>15913593
You didn't obtain partial results? What about showing that a big conjecture implies it? Where are your potential counterexamples? Proofs of similar results under different assumptions?

What the fuck have you been doing?

arrays and for loops will get you further into math then 1000 textbooks about linear algebra or calculus

>>15911759
Alright, thanks. I still can't figure out the last one though. I think I might need something else.

>>15899072
S := {x in {0, 1} | x=1 or 'continuum hypothesis holds'}

has no least element.
Proof me wrong, classitards.

>>15914230
Use L'Hospital's rule. The proofs are all on wikipedia.
https://en.wikipedia.org/wiki/Generalized_mean

>>15899072
The subset

[math]S := \{n in \{0,1\} \mid n=1 \lor (\text{continuum hypothesis holds}) }[/math]

having a least element can't be rejected.
But no least element provably exists.

Prove me wrong, classitards.

[math]S := \{n in \{0,1\} \mid n=1\lor(\text{continuum hypothesis holds})}[/math]

[math]S := \{n in \{0,1\} \mid n=1\lor(\mathrm{continuum\ hypothesis\ holds})}[/math]

Suc(Dec(x)) = x
gt(Incr_m_by_n(x,Suc(y)),x)
Knowing only those axioms in first order logic, can it be proven that
gt(Incr_m_by_n(x,y),x)

I'm trying to determine if the automated theorem prover I am using is buggy or I am missing something.

>>15879419

>>another day since graduation
>>another day where I haven't used cohomology once
>Remember to spend your time in university learning useful things, not sucking your professor's dick.

It is self-evident that every element of the second cohomology group H(C_{2^n}, C^n) is bijective to the set of "arithmetic" binary encodings of the elements of C_{2^n}, where I don't feel like defining "arithmetic" because I'd have to refer to my dissertation again and that brings up bad memories.

For example, the "canonical" binary encoding of C4 maps 0 -> 00, 1 -> 01, 2 -> 10, 3 -> 11. Under this encoding, the "carry" function consists of the well-known cocycle z(a,b) = a&b, combined with the coboundary of 0.

An alternate arithmetic encoding would be 0 -> 11, 1 -> 10, 2 -> 01, 3 -> 00. Here, the "carry" function consists of the same coycle, z(a,b) = a&b, combined with the coboundary of 1.

Give me a constructive method to determine this encoding -> coboundary map, and I swear on my mother's life (and on her future grave too, why not), I will get to a job making $500k / year (provided you have a PhD or are in your last year), and together we will build GPUs that are about 25% smaller and 20% more power-efficient than anything that can be build today.

I swear to god, you fucking nerds don't understand how lucky you are to have sat through a full PhD study of topology. You guys are actually so defeatist and self-pitying that you can't see the practical applications of what you've learned when it comes to computing, which just so happens to be the only industry that actually scales into the future. You could be making bank if you actually applied that topology PhD instead of crying about it.

>>15914548
I love you.

>>15914548

Correction. I only need a constructive method to determine that encoding -> coboundary map over C_{2^n} for finite n. And not even n that large, 32 would be sufficient, because that's about how big things get in practice.

An acceptable, weaker, result would be a constructive method to determine the coboundary of H(C_2^{n-2} X C_2, C_2^{n-3} X C_2 X C_2) associated with the "canonical" binary encoding, for 2<n<32. That won't let you make literally all GPUs 25% smaller, but it'll make a 5% dent or so. You want the general result if possible because it uniquely lets you circumvent the automated tools we use today, and those automated tools are absolute garbage.

>>15914550
I love you more.

>>15914549
>>15914552

I am not even memeing here, I am equally jealous of you nerds and frustrated that you don't want to apply the theory that you've learned. Your whole field is sitting on a gold mine and you're all too depressed to realize it.

Please fucking prove me wrong, work your fancy math on this, and I will get you a job offer next month and a copy of Pierre Grillet's The Cohomology of Commutative Semigroups for you to use in GPU design.

If what I'm asking for is an intractable problem, well fuck me, I never had formal training in math, guess we'll just have to wait a decade for some young guy to work on this before we can have power-efficient hardware. But based on my 3 months of trying to self-teach algebraic cohomology, this seems like something that could be figured out.

>>15914555
I agree. Marry me.

>>15914560

a/s/l?

>>15914561
20F USA

>>15914563

I should've just asked for which US state, that actually matters, unlike a/s. I'll be in CA, OH, DC, OK, TX this month, if you're in any of those 5 states let's grab dinner.

>>15914565
I'm down for dinner. TX.

>>15914567

Austin or College Station, which of the universities are you at? Let's say, Friday the 22nd? I'll have to ask my friends for some restaurant recs: I only really remember Barley Swine, which was really good, but it's brunch not dinner.

>>15914569
I'm in Houston. I know a few good places here. My treat since you're visiting.

>>15914572

Let's me look, yeah, Houston works. Either the 21st or 22nd works pretty w/ my travel plans.

Do you wanna swap contacts now, or just set a time and place and recognize each other there?

>>15914572
>>15914576

If you do want to swap contacts now and you're worried about man-in-the-middle interception, I've got identifying information in the form of a half-finished never-submitted journal paper that's obviously written by the same person posting in this thread.

>>15914576
Let's swap contacts, get to know each other a bit and meet up on the 22nd. Send me your info - LinkedIn if you have the balls, or reddit/discord.

>>15914578
thanks, anon, while I'd love to read more of their thoughts, don't want to get it without permission.

>>15914580

No LinkedIn.
Discord: tde__ as username (2 trailing underscores) or 313067998430756864 as the unique ID

>>15914584

This was me as well, btw, just talking in the 3rd person for whatever reason. 4chan wasn't designed for 1:1 convos.

>professor's slides on real analysis overview refer to nonstandard analysis as "non-std anal"

>>15914496
So I've converted it to TPTP format and checked it against many provers and they say it's true so it must be true.
But I don't understand why.
This is the theorem in TPTP format:

fof(f1,axiom, ! [X] : suc(dec(X)) = X).

fof(f2,axiom, ! [X] : ! [Y] : gt(incr_m_by_n(X,suc(Y)),X)).

fof(f3,conjecture, ! [X] : ! [Y] : gt(incr_m_by_n(X,Y),X)).

What's the trendiest research topic in mathematics that's isn't AI/ML/datascience, etc?

>>15914496
What is the type of x and y? Are Suc, Dec, incr_m_by_n, and/or gt getting replaced with something unexpected by overload resolution? Suc(Dec(x)) = x is trivially false in Peano arithmetic for example, set x to 0. So the conclusion (along with anything else) would become provable by ex falso quodlibet.

>>15914746
First order logic is untyped.
But yes, it's because I used that instead of dec(suc(x)).
I think it was making the replacement suc(Y) by suc(dec(x)) and from that getting incr_m_by_n((X,X),X).
What threw me off is that in the proof it said it was using "paramodulation" which seemed more complicated than just variable substitution.

>>15909121
>funny meme
Anon, I...

How can two metric spaces be homeomorphic but not "uniformly" homeomorphic?

>>15914936
Because there are spaces for which there exists a homeomorphisms but there does not exist a uniform homeomorphism. Uniform continuity is a stronger property than regular continuity.

>>15914975
I'm asking what kind of properties are preserved by unif. hom. but not hom.

I'm thinking "metric" properties like completeness. But then, what properties are preserved by isometries but not unif. hom?

>>15914321
I'm fucking retarded, I forgot Wikipedia existed. Thanks again.
>I have to take both the left hand side limit and the right hand side limit separately
No wonder I couldn't figure it out.

>>15914593
Will anon get raped? Maybe just mugged? Actually given a job for knowing basic cohomology that many first-year grad students know? Taking all bets!

>>15914936
I believe (0,1) and R are not uniformly homeomorphic, ie no homeomorphism between them is uniform.

Very nice. MathSorcerer bros, your response?
Smell status: exquisite

>>15915372
Put me on for tree fiddy

I have a bachelors in physics, am 30 years old, and work as a code monkey for a school. I want to get my masters, and maybe even PhD, in mathematics since I get free tuition for working here. I realized in undergrad that I didn't really like physics, just the math in it. Despite this I didn't switch because I thought it was too late. I know the most I could hope for now is some low level associate professor or something like a community college but I truly believe I would be happier doing that than my job now.

To anyone who is actually in the field, do those jobs at least pay enough to sustain yourself? I am married and my wife works too. Obviously I don't really expect to make what I make now, but I am hopeful it would still be decent.

>what does majoring in math look like.

I saw this meme problem on twitter and im too tired to actually solve it. Im just wondering if theres a better way than trying to do an integral, ilI feel like its going to give me more trouble than it should. I know its fairly basic calculus but i struggled a lot more than i shouldve back then and that was years ago.

>>15898763
Hello I am the retard who was wondering why is Jacobson's Basic Algebra considered a graduate level book. I now know. It is because except for the absolute foundational definitions, it leaves almost every major result in undergraduate algebra courses either as an exercise or just skips it completely assuming you already know it. FUCK. Now I have to start all over again with a more basic algebra book.

>>15916926
https://www.youtube.com/watch?v=nuJHLqfheUA
For the record, the exact value is 12 arcsin 1/√5 + 2π - 8.

>>15899103
The asymptotic behavior of a Fourier transform provides insights into how the transform behaves as the frequency variable becomes very large or very small. In other words, it describes the behavior of the Fourier transform in the limit as the frequency goes to infinity or zero.

>>15917427
Thanks. I dont know if it wouldve accord to me to use geometry like that, especially since i wouldve liked a more exact answer, i dont think that was a problem with the method though, he couldve kept more exact values but it was good to see an alternative approach.
If i think i can do something with calculus i try to force it even if there could be another way.

Reject axioms, embrace self-evident truths. What do you say?

Is numbers a sin? Only 1.

>>15918359
every complex number is a sin you nit

>>15917788
The video didn't have an exact answer because it tried to resolve the arcsin too early instead of right at the end. You'll get the exact answer if you leave the arcsin to the end.

>>15902124
>>"fields and vector analysis"
>>it's just regular vector calculus
tf were you expecting, brainlet post

For [math]x \in \mathh{R}[/math] do we get [math](-1)^x \in \mathbb{R}[/math]? For rational numbers I think the answer should be every fraction reduced in lowest terms with an odd denominator, but what about irrational numbers?

>>15919486
Of course not
[eqn](-1)^x = \cos(\pi x) + i \sin(\pi x)[/eqn]
It's only real for integer values of x.

>>15919533
But [math]x = -1[/math] is the only real solution to [math]x^3 = -1[/math], therefore [math](-1)^{1/3} = -1[/math]

>>15919576
That's not how you define exponentiation.
https://mathworld.wolfram.com/ComplexExponentiation.html

3=1
Fin

hey /mg/ I haven't been here in a while -- has anyone seen yukari poster recently?

those yukari edits were peak /sci/

Why don't you use [math] \text{Con} \mathrm \TeX \text t [/math]?

>>15908791
How many even numbers are there?

>>15898763
What's the fastest algorithm for finding a Grobner basis over rational numbers?

>>15914548
Link for further reading?

>>15918254
Solipsism is a wonderful head trip.