[math]/\mathfrak{mg}/[/math]

Joy of analysis edition
Previously >>16683009

Is there a generic notion of “local property” in topology? Some properties like local connectedness require you to check every open neighborhood, while others like local compactness only require existence of a single neighborhood with that property. I don’t want to autistically remember whether something like local Hausdorffness is one or the other.

what happened to anitranny
hope life is good whatever happened

>>16712268
Last I heard she was in the /mg/ discord. But I'm not there so I don't know if that still holds.

>>16712226 (OP)
Nice kot

How can I find the determinant of the following square matrix?
[eqn] A =
\frac{1}{3}
\begin{bmatrix}
1 & 1 & & & & & & 1 \\
1 & 1 & 1 & & & & & \\
& 1 & 1 & \ddots & & & & \\
& & 1& \ddots& & & & \\
& & & \ddots & & 1 & & \\
& & & & & 1 & 1 & \\
& & & & & 1 & 1 & 1 \\
1 & & & & & & 1 & 1 \\
\end{bmatrix}
[/eqn]
More specifically, I just want to know whether [math] | \det A | \leq 1 [/math] or not.

>>16712267
I think the differences between these concepts are important, even though the word "local" is used for all of them.

Local connectedness checks every open nbhd because it's checking for whether the space is connected "no matter how much you zoom in."

Whereas local compactness only needs a single neighborhood, since it only needs things to be compact "if you zoom in enough."

>>16712666
Maybe calculate some smaller versions of that and try to find a pattern

>>16712669
I understand that, but how do I possibly formalize this. Say I invent some new global property needed for my work. How do I know whether to localize it using all open neighborhoods or just a single neighborhood?

>>16712666
Laplace expansion?

Since [math]\begin{vmatrix}1 & 1 \\ 1 & 1 \end{vmatrix}= 0[/math] then doesn't that mean the determinant of every sub matrix and hence A itself also have to be zero?

>>16712708
The 8×8 case equals -1

>>16712666
https://en.wikipedia.org/wiki/Circulant_matrix?useskin=vector#Determinant

>>16712713
No it doesn't.

>>16712666
The blank parts are 0?

Seems people interpret them to be 1, in which case detA would be 0 from having two equal columns

Springer 50% flash sale with coupon FLSH50.
What are /mg/bros buying?
Im buying Switzer, Marker, and looking for something else.

>>16712716
Thanks, using the Circulant matrix determinant formula, I think we can say the determinant of the matrix A in >>16712666 is 0 if n is a multiple of 3.

>>16712708
However, for the cases n=4,5, the determinant doesn't be 0, according to WolframAlpha:

https://www.wolframalpha.com/input?i=determinant+%5B%5B1%2C1%2C0%2C1%5D%2C%5B1%2C1%2C1%2C0%5D%2C%5B0%2C1%2C1%2C1%5D%2C%5B1%2C0%2C1%2C1%5D%5D

https://www.wolframalpha.com/input?i=determinant+%5B%5B1%2C1%2C0%2C0%2C1%5D%2C%5B1%2C1%2C1%2C0%2C0%5D%2C%5B0%2C1%2C1%2C1%2C0%5D%2C%5B0%2C0%2C1%2C1%2C1%5D%2C%5B1%2C0%2C0%2C1%2C1%5D%5D

I think the determinant should be nonzero for every n not a multiple of 3. I think we can show this with the Circulant matrix determinant formula, but I need to review how complex roots of unity work.

>>16712267
I think the more or less standard definition of locally X is "every neighbourhood of every point contains an X neighbourhood", i.e. such neighbourhoods form what's called a "local base" (at every point); the neighbourhoods are usually not required to be open, unless the text is one of those that uses the "neighbourhood"="open set" convention. Concerning the two definitions you mention
1) The two definitions of local connectedness (every neighbourhood contains connected neighbourhood vs contains open connected neighbourhood) are actually equivalent.
2) Local compactness is a bit problematic (there are at least ~4 major conventions that are not always interchangeable), but for Hausdorff spaces they are ALL equivalent, so you might as well remember the definition I mention above.

I’m flying through the Algebra chapter but I can’t even prove the first Elementary Calculus problem. What does that mean? Am I autistic?

I'll just post this just in case someone solves it

>>16713160
Hey, I remember getting distracted by this for a couple days a couple years ago. You can get an exact solution for up to k = 5 points and an interval for k > 5, by breaking all the possible shapes down into their component subshapes. For k = 4, use the result of https://doi.org/10.2307/1426176 and permute the order of the points. For k > 4, use https://doi.org/10.1214/16-BJPS315

>>16712771
Thanks for the heads up

>>16712413
after quitting shitposting to become a teacher?

Could someone link the discord?

https://www.wolframalpha.com/input?i=Flatten%5B%7B%7Cx%E2%80%93y%7C%2B%7Cx%2By%7C%3D24%2C%7Cx%E2%80%93y%7C%2B%7C%281%E2%80%932%2F%E2%88%9A3%29*x%2By%7C%3D24%E2%80%938%E2%88%9A3%2CTable%5B%28x%E2%80%933*%283%E2%80%93%E2%88%9A3%29k%29%5E2%2B%28y%E2%80%933*%28%E2%88%9A3%E2%80%931%29k%29%5E2%3D%286%E2%88%9A3%E2%80%936%29%5E2%2C%7Bk%2C%E2%80%932%2C2%2C2%7D%5D%2CTable%5B%28x%2B4k%E2%88%9A3%29%5E2%2B%28y%E2%80%934k%E2%88%9A3%29%5E2%3D%2812%E2%80%934%E2%88%9A3%29%5E2%2C%7Bk%2C%E2%80%931%2C1%2C2%7D%5D%7D%5D

um.. what happened to probability statistics general? Also, how hard is it to get into a top AI/DS/ML program if I'm from tier 3 uni with a BS in stats?
>inb4 /scg/
let this pass just this once lads

Had to turn down my dream scholarship because of my mother's huge debt. It's the hardest thing I've ever done in my life.
Have any of you guys turned down a scholarship and got it back by applying another year?
I explined to them the situation but they said is imposible to "freeze" my application.
Bros, I need some cope right now.

>>16714866
your mother's huge debt doesn't affect you in any way
just accept it and go

>>16714914
She cant pay it by her own. Lawyers have come to my house to sue her for not paying.

>>16712232
TOP FUCKING KEK

You can do it anons! Keep working and thinking hard it'll pay off

>>16714866
>>16715302
literally not your problem

>>16714866
Math is for the rich. How many times do I need to tell you this?

>>16713157
Show me the solution for the hardest Algebra chapter’s problem. And show me your attempt for the 1st problem in Elementary Calculus chapter.

Half the fucking obstruction to understanding the Yoneda lemma is how every single reference doesn’t bother stating it in terms of functor natural isomorphisms. Like do you really, absolutely, need a c in picrel when omitting it would make it crystal clear that this is a natural isomorphism of functors rather than an isomorphism of objects?

>>16715918
For me, natural isomorphisms are best understood when written with the object specified explicitly as "c", "x", etc, but this probably depends on the experiences and background of the person reading it

>>16715918
Isn't what you're asking for just the yoneda embedding?

>>16716184
It conceals the fact that they’re natural transformations by making them morphisms. It’s like currying in a sense. Can be useful in some contexts, but obfuscating in others. And when I people talk of natural isomorphisms, I want to see the actual natural transformations stated as opposed to their components. Makes the arrow autism crystal clear.
>>16716191
I’m not asking, just complaining.

bump

>17 year old Hannah Cairo refutes a 40 year old conjecture
arxiv.org/abs/2502.06137

>>16717709
I wonder if her parents are professional mathematicians nepobabying her into academia...

>>16717709
cute

>>16716274
>I want to see the actual natural transformations stated as opposed to their components
Do you know what a natural transformation is? It's nothing but its components.

>>16717763
It's like saying "a vector is just its basis components". Natural transformations in their "component-free" form are better to think about because they're arrows between functors. So a natural isomorphism isn't just any isomorphism; it's a much more powerful statement about isomorphism of functors. When you make these componentwise statements, that picture is lost and I have to wonder what fucking functor you're talking about.

Consider these two expressions
[math]\mathrm{Hom}(F(A),B) \cong \mathrm{Hom}(A,G(B))[/math]
[math]\mathrm{Hom}(F(-),-) \cong \mathrm{Hom}(-,G(-))[/math]
The first is a bijection of sets. Could be any bijection you want. The second is an isomorphism of functors. An adjunction between F and G isn't just any random bijection of hom sets as in the first example. It has to be a natural isomorphism of the second example, so that there is a canonical bijective map specified in the first.

>>16717759
Man face

>>16717709
>>16717756
>>16717759
>her
It's a tranny

>Modern Quantum Mechanics is at odds with Relativity
>Einstein said "god doesn't play dice"
>Biggest opposition is Bell Inequalities
I am having problems accepting probability ruling quantum mechanics.
How can probability determine the function of particles? How does that make sense?

>>16717808
do you have a source for that or are you just memeing

>>16717756
the subject is not arcane enough for a teenager to be suspect

>>16717808
your mom is a tranny

I'd like to minimize the function [math] f : S^{n-1} \rightarrow \mathbb{R} [/math] given by [math] f(u) = \sum_{i=1}^{N} ( ||v_i||^2 - (v_i \cdot u)^2 ) [/math], where:
[math] S^{n-1} [/math] is the origin-centered unit sphere in [math] \mathbb{R}^n [/math],
[math] v_1,\ldots,v_N [/math] is a collection of N points in [math] \mathbb{R}^n [/math],
and we assume [math] v_1,\ldots,v_N [/math] are linearly independent in [math] \mathbb{R}^n [/math].

However, I'm running into some difficulties, and I was wondering if anyone here could help.

Using Lagrange multipliers, I found the possible critical points of f on [math] S^{n-1} [/math] are at points [math] u \in S^{n-1} [/math] such that u is an eigenvector of the matrix [math] V := \sum_{i=1}^{N} v_i v_i^T [/math].

However, when I looked at the 2nd derivative of f, I found its Hessian matrix is [math] D_2 f = (-2)V [/math]. The problem is that this is negative-definite (since V is positive-definite), which would imply f can't have any local minima, even on [math] S^{n-1} [/math]. But this obviously doesn't make sense, since f must attain a local minimum on [math] S^{n-1} [/math].

Can anyone see where my mistake is?

>>16717851
>do you have a source for that or are you just memeing
I'm not memeing and my source is my own two eyes. Look at """her""" body, face and hairline for more than a few seconds and get back to me.

Besides, you didn't think there were actual women in mathematics, did you? It's trannies all the way down.

>>16717869
We're in a mathematics general and you're thinking about dicks.

>>16717851
Do you seriously believe a teen can just get into abstract harmonic analysis on her own? It’s tiger parenting, 100%. Especially considering she’s starting her PhD at 17. You need a lot of strings pulled in the modern academia to get into college at such a young age and a teenage can’t do that on their own.

>>16717881
>her
>she

>>16717782
Those canonical bijective maps are the components of the natural isomorphism dude. Have fun learning category theory.

>>16717890
I know they’re components. Jesus Christ. When you just indicate an isomorphism of objects, there’s no canonical isomorphism implied. When you indicate a natural isomorphism of functors, it induces a canonical isomorphism on the underlying objects. It’s literally notation abuse to go “objects A and B are naturally isomorphic” when a natural isomorphism is about functors between categories, not objects inside categories.

>>16717759
Hmm...

>>16717881
i think she can solve a problem in abstract harmonic analysis on her own

the analytical concepts here are accessible for her age group. more recent theory requires psychological maturity.

as she states in her paper, one only needs to study the theory of fourier restrictions.

>>16717868
Typo, it should say [math] v_1,\ldots,v_N [/math] are a spanning set for [math] \mathbb{R}^n [/math], instead of linearly independent

Also I figured out the issue, basically the Hessian of f restricted to [math] S^{n-1} [/math] isn't the same as taking the Hessian of the unrestricted f and restricting to the tangent space of [math] S^{n-1} [/math]

>>16717881
Tiger parenting is less plausible.
As predictable /sci/ is full of jealous midwits who cannot accept the fact that some people are born with a big block v8 for a brain.
Enjoy your 1.3L inline 4 brain, faggot.

>>16717951
Didn't know someone with a big brain can just stumble upon a niche area of math in their early teens without external input and parental access to journal articles. Impressive, very nice.

>>16717925
See >>16717956. Her reference list includes journal articles and the timespan is from 1970 to 2024. Someone in their early teens does not have access to journal articles (they're not free, you know) and spends their time doing this kind of research without their parents' oversight. It's clearly the classic case of Jewish parents trying to game their child into a tenured position by shadow writing for her.

>>16717956
So now the mathematical achievement has to be done with no external input. Shifting the goalposts there buddy. And this is not a niche area of math, harmonic analysis is one of the biggest analysis areas after PDEs. Also, most journal articles are trivial to find on the internet, I'm sure she could have figured it out with her intellect.

You are clueless if you think a very special 17 year old kid with some guidance is not capable of solving this problem. Read this article https://english.elpais.com/science-tech/2025-07-01/a-17-year-old-teen-refutes-a-mathematical-conjecture-proposed-40-years-ago.html.
Seems like she would have mentioned her parents being mathematicians or helping her somehow if that was the case.

Most importantly to refute your moronic point, this quote: "The program director recognized Cairo’s extraordinary mathematical talent — another goal of this type of activity is to identify people with a special talent for mathematics and foster their interest and ability — and invited her to teach in future editions. And she did. Now, at her new university, the University of Maryland, where she’ll begin her Ph.D. this fall, she hopes to start her own group."

You can't fake this shit, it is too complex to memorise stuff and be able to adequately teach it purely off of memorisation if her parents truly "ghostwrote" stuff.

She most likely is a math prodigy and no amount of seething on your part will change that.

>>16717981
>So now the mathematical achievement has to be done with no external input.
You are the one who claimed that she did it on her own. Which makes your statement
>Shifting the goalposts there buddy.
extremely ironic.
>most journal articles are trivial to find on the internet, I'm sure she could have figured it out with her intellect.
Most journal behind a paywall. Bitch, please.
>if you think a very special 17 year old kid with some guidance is not capable of solving this problem.
Oh, I don't think that at all. The key word here is "with some guidance". You know, it is expected of graduate students to work independently, without guidance. But hey, if she's just a teenager whose parents handhold her through it, then it's different. She's just too smart to figure it out on her own.
>Seems like she would have mentioned her parents being mathematicians or helping her somehow if that was the case.
Seems to be you're an extremely naive goy if you think people would just admit to shadow writing articles for their kids to push them through academia.
>The program director recognized Cairo’s extraordinary mathematical talent
Oh, wow, a dude said something. I'm sure it has nothing to do with her parents pulling strings.
>You can't fake this shit
I didn't say it's fake. I said it's the classic case of young children getting bereft of their childhood because their neurotic Jewish parents want to live vicariously through their kids and gain recognition.
>She most likely is a math prodigy and no amount of seething on your part will change that.
You are most likely a naive idiot who hasn't had experience with academia. This stuff happens all the time. It's a consequence of "holistic judgement" bullshit in the US, which leads some parents to do essentially what Khan from KOTH did. The sheer naivete of yours is laughable. I suggest you read up on this prodigy and how his Jewish parents ruined his life
https://en.wikipedia.org/wiki/William_James_Sidis

>>16718021
>You are the one who claimed that she did it on her own.
I'm not, I'm a different poster than the guy you were arguing with. Just wanted to jump in because you sounded unnecessarily salty and conspiratorial over the fact that somebody potentially did something excellent.

>Most journal behind a paywall. Bitch, please
There are websites you know... Yes, not all papers are on those websites but if you are a motivated kid, which she is, you can email people for access or visit libraries. It is not such a barrier you think it is.

>Oh, I don't think that at all. The key word here is "with some guidance". You know, it is expected of graduate students to work independently, without guidance. But hey, if she's just a teenager whose parents handhold her through it, then it's different. She's just too smart to figure it out on her own.
What point are you making exactly? And just randomly assuming her parents wrote her paper and then shaping your entire view of the situation around that.

>Oh, wow, a dude said something. I'm sure it has nothing to do with her parents pulling strings.
See my direct point above.

>I didn't say it's fake. I said it's the classic case of young children getting bereft of their childhood because their neurotic Jewish parents want to live vicariously through their kids and gain recognition.
Why make that assumption and state it so definitively? By all accounts she seems happy to pursue her passion. If you would have stated your opinion from a position of concern and acknowledgement that this happens to some child prodigies I would have agreed because that's reasonable.
All you've done is make yourself seem jealous and bitter.

>>16718021
>You are most likely a naive idiot who hasn't had experience with academia.
I'm naive for not instantly assuming something you don't know about somebody you know nothing about apart from her name and that she might have proved something interesting.

>This stuff happens all the time. It's a consequence of "holistic judgement" bullshit in the US, which leads some parents to do essentially what Khan from KOTH did.
I'm sure it happens, I just disagree with instantly concluding, like you did, that this is what happened to Hannah Cairo. Don't know what Khan from KOTH is.

>The sheer naivete of yours is laughable. I suggest you read up on this prodigy and how his Jewish parents ruined his life
https://en.wikipedia.org/wiki/William_James_Sidis
I will not read up on that because by your summary it wouldn't prove anything. Something happened to some people so it must have happened to this kid? And it wouldn't be 4chan if you wouldn't keep invoking religion or race somehow ayy?

comparing myself to others mathwise is making me feel insecure about taking calc 2 at my cc... and since i was a retard before this is my 2nd time taking it...

>>16718070
stop comparing yourself to everyone more accomplished than you. who knows, maybe the people who you're comparing yourself to have been doing advanced math since they were young or have a natural talent for it. they might have been preparing for a stem academic career since they could first read and write.
you are on your own path with your own set of struggles and hardships. it gets easier every day, but you have to do it every day. stay disciplined and focused and get help when you need it. don't be afraid to go back and review fundamentals. i bombed my calc exams until i realized i sucked at trig and algebra and had to go back and relearn it properly by self studying.

>>16718121
thanks. honestly it's more of an ego problem since im mainly comparing against my cousins who are the same age as me and have already passed the course. and the fact that they thought i already passed the course through taking the ap calc bc exam. it just feels embarrassing having to explain that i couldn't get a 4 or 5 on that exam, but it is what it is.

>>16718055
>you sounded unnecessarily salty and conspiratorial over the fact that somebody potentially did something excellent
For every Gauss out there, there are 100 kids who are pushed to the limit by their grifter parents at all costs for the sake of fame. It's extremely common among Jews, Indians, and the Chinese. It's the unfortunate truth. Here's a video of a guy going over one of these "prodigies" who claims to be some megapolyglot.
https://www.youtube.com/watch?v=ppdCYE212W4
>you can email people for access or visit libraries.
What people? Her parents' cronies? What libraries? University libraries that are off limits to non-students? Give me a break. This idea that "every kid out there can just access journal articles and do professional research at a young age without their parents' direct involvement" is just bullshit. No matter how smart you are, that's not the necessary resource here. It's your parents' connections.

>>16717851
>https://www.youtube.com/watch?v=3ZeH_8sTyKA

Complex Variable exam tomorrow, wish me luck.

>>16719023
Good luck to you!
May your success be conformally mapped and analytically continued...
https://www.youtube.com/watch?v=g6LmapJ6k70

>>16712226 (OP)
Hey, I want to make a favour for a friend but I'm not well-versed in the mathematical sciences. Can you recommend some textbooks or articles for chaos theory, game theory, and pure mathematics? My friend just got admitted to a Bachelor of Sciences and they are studying Pure Mathematics; I want to get them some gifts (studies, student resources, learning materials) to celebrate. What resources have you guys used and what is best for a fairly advanced student who hasn't been in the milieu of university just yet? Thanks so much in advance. Sorry for being lazy and asking but my discipline is in the Arts.

>>16719169
Just looking at my stack to my left, I can see a couple
of books that might be suitable:
Nonlinear Dynamics and Chaos by Strogatz
Understanding Analysis by Abbott
Introduction to Linear Algebra by Gillett

Besides the texts and more they'll read in the courses,
I'll also throw in an iPad for multiple uses, quality paper
and stationery for writing, and a coffeehouse to chill and
ponder. All the mathheads need some respite now and then.

>>16719177
Thanks! I found two of these and will send it to them. I think an iPad will be good but I might find a secondhand model.

>>16717709
>>16717759
cameltoe in pictures should be required for proof it is a natal female
if you cannot show cameltoe, i will assume your achievements were done by a man

>>16719237
She's 17. Back to europe faggot

>>16719256
>SHE'S ONLY 17 YEARS 364 DAYS 23 HOURS 59 MINUTES 59 SECONDS AND 999 MILLISECONDS YOU SICK YUROPOOR

>>16719237
Not necessary, simply listen to the first few seconds of the provided media: >>16718310

>>16719261
It's not 4chan.co.uk, limey.

I don't know how they treat your kind there, but I have a hunch it's at the bottom of the Thames.

>>16719299
Why are Americans so fucking retarded that they assume every European is a Brit?

>>16719256
>cameltoe is literally the same as rape
back to reddιt
>>16719272
cameltoe is super necessary

>>16718310
>This like.... generally like....the overall things like....we like to see like ....in restriction theory.
Why do zoomers talk like that?

>>16719422
Not just zoomers. I attended seminar talks where 50-something boomers talked like that. This has started in the 70s when beatniks thought it’s hip and cool to talk like what trash because their silent gen parents are too square or something.

Is being born probability or predetermined?

>>16719299
>have a hunch
whatever quasimodo

>>16719349
Because Brits are the only ones in Europe that are even worth acknowledging (as awful as they are). The rest of you are barely even worth keeping as vassal states. If the rich in America didn't like vacationing in Europe and oggling the historic buildings in your cities, you'd have long been turned into one long strip mall with an inferior version of the McDonald's you already have. (the erdbeerkuchen at the Austrian McCafe has no business being as good as it is, I'll give you that).

>>16719956
>all that cope implicitly saying you’re retarded and don’t know basic geography

Subtopic:

What are some absolutely esoteric math concepts you still can't grasp even after months of toiling them?

>>16712666

Not a hw thread.

>>16717868

Not a hw thread.

>>16720179

NY FUCKING PA, people. Don't ask us for help, we're just people who go to church, like math, and occasionally post racist/sexist memes.

[eqn]/\mathfrak{mg}/\not = /\mathrm{hw}/[/eqn]

>>16712666
A nice observation is that this represents an operation which takes the average of "consecutive" basis vectors taken three at a time. Viewed that way, the determinant must be bounded by 1 because this map couldn't possibly increase volume.

>>16720182

This isn't a hw thread. NYPA applies here.

>>16712226 (OP)

Analysis, huh? Makes me want to barf.

I'm a group theorist by trade. I'll pass.

>>16720185
>I play legos for a living
based

>>16720186

I...er...uh...what?

Did I stutter?

>>16720187

How the fuck do you equate group theory to Legos?

My current area of research is on the (2,5) Burnside group finiteness problem, and it's absolutely nothing like playing with Legos. Get lost.

>>16720187
>picrel
bruh what the fuck is that

>>16720193
how the fuck did this get on arxiv? the site is truly dead, ts is quite literally middle school math with a group theory spin

>>16712666
not >>16720182, but their idea is pretty smart, so imma just expand on it.
Your matrix M transforms the n-dim cube of edge length 1 to some n-dim prism, whose volume equals |det M|.

[math] \tfrac{\Vert e_1 + e_2 + e_3 \Vert}{3} \leq \tfrac{\Vert e_1 \Vert + \Vert e_2 \Vert + \Vert e_3 \Vert}{3} = 1 [/math]

will be true for any set of 3 diff basis vectors. So M transforms each cube edge on the origin to a prism edge on the origin, each of which being the average sum of some 3 cube edges will have magnitude <= 1. The max volume of a prism given all the edges on the origin is just the product of all their lengths, which at max is 1, so the volume |det M| must be <= 1

Seems a lot more "fun" than using that Circulant Matrix equation, though the latter gave you more info.

>>16720214
>their idea is pretty smart
thanks anon

>>16720187
>Bastardas
KEK

>>16720214
The same argument shows that if the maximum column-sum of a square matrix M is <=1, then |det M| <= 1, or

1-Norm [math] \Vert M \Vert_1 \leq 1 \ \Rightarrow\ \vert \det M \vert \leq 1 [/math]

>>16720187
>Gemma Bastardas

>>16720214

Stop shitting up the general, this is NOT A FUCKING HOMEWORK THREAD.

>>16720239

Stop adding to this, okay?

Just tell them what it is. This is [math]\underline{\mathrm{not}}[/math] a homework thread, and they should take it somewhere else. Period.

>have vague idea of how to define concepts such as "mathematics" and "number"
>complete degree
>now have absolutely no idea of how to define "mathematics" or "number"

>>16720259
>he finally realized that what’s a number depends on which category you’re working in

My degree needs so much math
How do i make myself like math more so i can tolerate it

>>16720503
What's not to like about math? It's literally all so fun. Name a specific thing you don't understand or don't like. I'm 99% sure it's just because you're confused about it.

>>16720223
Ok last thing. The furthest this method goes as far as I could tell is that the side length is at most the max length of the column vectors, which in this case is [math] \sqrt{3}/3 [/math]. Had to think a little to convince myself this was smaller that the max |eigenvalue| of M, so I can't do any better with this method.

[math] | \det M | \leq \max_j \Vert M_{:j} \Vert_2^n = \tfrac{\sqrt{3}}{3}^n [/math]

>>16720574

Not a HW GENERAL.

>>16720558
Fumbling one thing in a long problem then getting the whole thing wrong

Student or teacher
Opinions on pearson

>>16712819
The determinant is 0 if [math]3|n[/math], 1 if [math]3\nmid n[/math]and [math]n[/math] is odd, or -1 if [math]3\nmid n[/math] and [math]n[/math] is even.

>>16720627
if n = 2 the determinant is 0, not -1, wym?

>>16720631
for matrices larger than 3x3, obviously.

Not. A. Hw. Thread.

Stop ruminating over this one anon's homework problem. This is a thread to discuss math, not a free service to do homework people.

>>16720180

This. This right here.

>>16720214

>>16720216

Samefag.

>>16720180
>>16720659
>>16720720

>>16720627
>>16720633
Makes no sense for the determinant not to converge to zero as n goes to infinity, as there's zero chance that the matrix is preserving volume. Did you forget that that 1/3 applies to all n rows/columns of the matrix. Looks like you only applied it once.

>>16720770
Very true; the base matrix oscillates between 0, -3, and 3, and I thoughtlessly multiplied that by 1/3

>>16720770

>>16720846

WHY ARE WE STILL ON THIS?! This isn't a hw thread. You know what, forget it.

>>16720901
fuck off retard

>>16721048

Yes, because letting people shit up their thread with homework instead of going to >>>/wsr/ is totally based...

>>16721127

>their

This*

>>16720574

>>16721184
I prefer [math] \tfrac{1}{3^{n/2}} [/math]. Matches up with the actual answer [math] \tfrac{1}{3^{n-1}} [/math] from >>16720846

>>16721184
I prefer [math] \displaystyle \frac{1}{3^{n/2}} [/math] . Matches up with the actual answer [math] \displaystyle \frac{1}{3^{n=1}},\ 0,\ \pm \frac{1}{3^{n-1}} [/math] from >>16720846

Is there a good geometric interpretation or intuition as to what the coordinate ring of an algebraic set is? Or maybe some motivation. Will it make more sense when I start studying schemes?

Any UK anons know if I'm screwed or not? Had reasonable grades before (good not great, was very relaxed). Third year finals come around and I just got mad at petty personal stuff and crashed out. Flunked final year exams and now missed my masters offer. Oops. Will they likely just want my money? Or should I make new plans?

>>16721380
it's the ring of nice (regular) functions on the algebraic set (with values in the field). More precisely, we have a canonical isomorphism between the coordinate ring of an algebraic set X and the set (ring) of morphisms of varieties from X to the affine line. This is also the preferred geometric interpretation in the context of schemes.

>>16721986
Ok I think I get it — so taking the quotient by the ideal associated to the algebraic set is like removing functions which would misbehave on that set, is it something like that?
I’m reading Hatcher and he defined the coordinate ring but did not explain it (or maybe I missed some detail). I do see how it holds good information because of integral domains and fields as quotients of polynomial rings, but it seemed at first more like an ad hoc diagnostic tool than a natural construction.

>>16722098
I see I should have just waited for chapter 3 he goes into it. It seems the ideal doesn’t really misbehave, it’s just the kernel of the homomorphism

I would like to present my paint diagrams for cochain complexes for my notes, and hope you enjoy them as much as I do

>>16722311
the last triangle is the wrong color, i realize now

I'm going back to school for applied math and can only take two of these electives. What should I take?

>>16722583

4393, 4336.

>>16722583
Depends entirely on what your knowledge level is and what you want to do or are interested in.

>>16712666
treat it as an adjacency matrix
ignoring the 1/3, and letting n>2
det = [x^n] (1-x^2)/(1-x+x^2) + (2x^3)/(1+x)

Suppose we have n complex vectors paired as the columns of an upper diagonal matrix whose diagonal is real, and we wish to transform these vectors via some unitary matrix U to make all n vectors real. Starting off with just n=2 vectors, the first column of U needs to be real to keep the first vector real, and if I did the math right, I think U should look like

[math] UM = \left[ \begin{matrix}
\cos\theta & -e^{i\psi}\sin\theta \\
\sin\theta & e^{i\psi}\cos\theta \\
\end{matrix} \right]
\left[ \begin{matrix}
R_{11} & R_{12}e^{i\phi} \\
0 & R_{22}
\end{matrix} \right] [/math]

But if this is true, then the only way to make the second vector real is if [math] \tfrac{R_{12}^2}{R_{22}^2}\sin^2\phi = \sin^2\psi [/math], so if the diagonal of vector #2 is larger than the magnitude of the other lengths. I'd venture to guess that for n=3, something similar will need to happen like [math] R_{33}^2 \geq R_{23}^2 + R_{13}^2 [/math].

It's expected that you can't turn all [math] \mathbb{R}^{n\times n} [/math] into [math] \mathbb{C}^{n\times n} [/math] or backwards, since they got different dimensions. Im pretty sure you can turn all [math] \mathbb{C}^{n\times n} [/math] into the upper diagonal matrix M with a real diagonal with householders or wtv. But what is this requirement that the diagonal has to dominate in some way? Seems sort of random

>>16712226 (OP)
I HAV ALWAYs (sorry for caps) sucked at maths and want to teach myself maths from the ground up. i've tried a couple times over the years with khanacademy's thing which is nice as an intro but i tend to flake away from it--not unique, i do this with all sorts of things. i have read and downloaded recommended textbooks, that sort of thing but from the very basic beginnings, where do i start?
i went through public school and all that, i was taught that 1+1=2 but i don't ever remember learning WHY and i definitely never made it to any course teaching a proof
tldr what are some texts/exercises to start from for a "how to learn maths for fucking retards" please and thank you

What's the intuition behind connections in Riemannian geometry?

I did fuckall over the summer and the fall semester is coming in fast for my ass. What chapters of Thomas' Calculus are pertinent to a (general college) Calc 2 curriculum? I believe it starts at chapter 8 with integration by parts.

I'd read through the entirety of the Calc 1 portion and ease my way into the relevant Calc 2 portion, but for the sake of time I think it'd be best to just go back and reference anything that I don't understand whenever appropriate. Is that a good approach?

>>16723954
You can think of them as being similar to matrix operations, but for nearby tangent planes. They are like linear functions but they "bend" in some sense around the manifold.

For example, if q and p are distinct points on a n-sphere, the tangent planes are defined by the mapping:
[math]
T_p S^{n-1} = \left{\underline{v}\in \mathbb{R}^n; \underline{v} = (I_n - \underline{p}\underline{p}^T) \underline{u}, \forall \underline{u} \in \mathbb{R}^n\right\}
[/math]
And
[math]
T_q S^{n-1} = \left \{ \underline{v} \in \mathbb{R}^n ; \underline{v} = (I_n - \underline{q} \underline{q}^T) \underline{u}, \forall \underline{u} \in \mathbb{R}^{n}\right \}
[/math]

By definition, these two tangent spaces are rank n-1 sub-spaces in [math]\mathbb{R}^n[/math], and a connection in this case would be a mapping:
[math]
H: T_p S^{n-1} \to T_q S{n-1}
[/math]
which, would take the form of an n x n matrix of functions on the components of the tangent plane at p.

>>16723942
Most aren't really taught TRUE whys, more like gists or real work examples to convince you of the truth, rather than a real proof. If you want real build up, look for a Real Analysis textbook. Tao has a simple af one that I rec. But honestly, most normal people don't care bout this. If you want to be good at algebra, just do algebra problems; if you want to be good at high school calculus, do those practice problems.

>>16723954
Conservative potentials if you’re a physicist. EM potential and gravitational potential.

Bros, I need help and advice.
Basically, I don't have the slightest idea what to do with my career. I just know that I like studying mathematics, which is usually considered pure mathematics (no programming or numerical methods). However, I was accepted into this program:
https://www.universite-paris-saclay.fr/en/education/master/mathematics-and-applications/m1-mathematiques-et-interactions-site-evry

The first year has programming and such, but in the second year, I'm supposed to be able to take this program:
https://www.universite-paris-saclay.fr/en/education/master/mathematics-and-applications/m2-analysis-modeling-simulation#programme, which is supposed to be more theoretical.
Do you think it's a good program for someone like me, who doesn't like programming at all? I know it's a very good university, but honestly, I don't really care about that; I just want to study math.
I don't know if I really like the program, but I'm afraid that if I reject it, I won't be accepted into another program and I'll lose the opportunity to study math.

>>16724591
I don't know anything about that program or that school, but I have faith that you'll be able to learn the modeling and simulation stuff as you go.

Mathematicians generally make pretty crappy software engineers, but I've never met a working mathematician that couldn't do what they needed to do in R/Python/Matlab/Mathematica. Unless you go for a topic that's on the edge of "applied" topics anyways (e.g., PDEs, analysis, probability theory, or the cryptographic side of algebra), you probably won't need much more than basic for-loops, some basic data management practices, and a bit of elementary numerical analysis to help approximate integrals or derivatives as needed.

Anons were you always good at math? Do you think you can get better at math? I revisited some of my old textbooks and realized I’d forgotten basically everything I learned in undergrad. Nothing stuck. Am I just stupid?

>>16724672
write notes down. Everyone forgets stuff so they go back to their notes or textbooks

>>16724672
>I revisited some of my old textbooks and realized I’d forgotten basically everything I learned in undergrad
Obviously, If you don't review the concepts you've have learned, in semi-regular intervals, then you tend to forget those concepts eventually. I can't recall many concepts I've learned in biology or geography or history in school; not because there's something wrong with my learning capabilities, but because I haven't made the effort to review, since I usually never use those concepts in my studies or work.
>Am I just stupid?
Just like any skill, practice make perfect; as long as you make a put effort into disciplining yourself in whatever study plan you decide to follow, you'll make progress at a rate you find comfortable.

Also, if you aren't doing so already, I recommend using spaced repetition software to make reviewing more efficient.

>>16724684
I learned about it via chess 27 years later than I should’ve. Any recommendations? This is the first semester I’ll be actively trying to implement it. Glad I’m not the only one who’s thought of this

Anons, do you consider PDEs applied or pure math?

>>16724688
I'd say there's roughly four types of PDE work: purely theoretical work focusing on existence/uniqueness/properties of solutions and estimates/inequalities, theoretical numerical work on the properties of algorithms to find solutions such as proving their rate of convergence, more computational numerical work on algorithms that wants to create the most efficient algorithms leveraging GPUs for example and lastly, applied work that is in a domain specific context that involves PDEs.
I would say 1 and 2 are pure, 3 is mixed and 4 is applied. But many papers fall in multiple categories, though there is not much overlap between 1 and the rest.

>>16724686
"it" referring to spaced repetition software, I presume? If so, then I recommend open source software like Anki or Mnemosyne (Anki is not open source for IOS fyi).

>>16724708
If it helps me actually retain stuff I’m happy to pay for a good product

>>16724688
PDEs are something which can be approached from a ton of different depending on your needs and interests. There's a lot of "pure" approaches to PDEs that look at the concept of a differential equation on more abstract spaces (rather than some Euclidean system with dynamics that evolve with a constantly forward proceeding time variable). Think a work which contains measure theoretic notions of derivative and integral on measure spaces (which are quite general in form, and not necessarily uniform Euclidean spaces).

There's also a big focus on the "practical" parts of PDEs, like deriving numerical solution mechanisms for classes of non-linear PDE systems for which closed form analytical solutions don't exist (or aren't known). This is much more of what I'd consider "applied."

There's also a decent amount of people focusing on PDEs on Riemannian smooth manifolds. As an example, you could look at PDEs which must vary such that a constant relationship is respected at all times (e.g., some function of the variables in your system of PDEs must be equal to 0 for all t). Studying PDEs in the context of smooth manifolds theory could give you a chance to flex some geometry and topology in the process (either pure or applied).

Tl;dr, it depends on what you want to do. There's a big wide world out there.

>>16724688
>>16724704
>>16724728
Is it just me, or is the field of PDEs less "respected" than other fields of pure mathematics, such as something completely abstract like algebraic geometry? I don't know if this comes from the applied side of PDEs or from their real-world applications.

>>16724709
Usually, you should at least try for free and open source software first, as often times it'll be "good enough" for the vast majority of people, for the vast majority of applications. Also, addons and customization are usually better supported in FOSS (though, I personally dislike Anki's ecosystem for third party addons). For Anki and Mnemosyne, they support LaTeX and/or MathJax by default, and so should be sufficient for those needing to type math symbols onto cards.

>>16724731
I'm an EE who studies pure mathematics as a hobby because I enjoy it (and, every once in a while I stumble upon something actually useful for my professional research). The last "pure math" course that was really relevant and helpful for me was my measure and introductory functional analysis course I took as an elective during grad school. Even then, it's mostly just useful for determining whether or not it's even worth pursuing a numerical solution to a research problem I'm working on.

I wouldn't worry about what others within the field "respect." For a really long time optimization theory and calculus of variations were looked upon as lowly peasant fields that only applied mathematicians, engineers and physicists concerned themselves with. Nowadays analysts trip over themselves to show that whatever abstract nonsense they are working on has some optimization theory application so they can keep being funded.

>>16724765
>Meausure theory in EE grad school
??? What applications are there? Are their probalistic interpretations of EE I don’t know about (I know nothing about EE)

>>16724781
Signal processing, most likely

>>16724781
With respect to the signal processing side of EE, you'd have a harder time finding a research discipline which doesn't benefit from measure and some introductory functional analysis.

Statistical signal processing and information theory have pretty much exclusively been handled by EEs historically. Estimation theory and signal detection are basically all about functions of random variables and conditional expectations (i.e., integrals conditioning on a sigma-algebra induced by a measurement). If you look into the research world for multi-target tracking, or signal detection theory, or information theoretic machine learning, it's all EEs, and measure theoretic mixture models are everywhere when you're looking at the cutting edge research.

There's even a big part of multi-target tracking research that uses random measure theory (specifically, random finite set trackers). Take a look at the probability hypothesis density filter if you want an example that uses random finite sets.

A lot of signal processing deals with Fourier analysis and Laplace transforms. It's pretty helpful to know how integrals work when you're doing applied Fourier analysis at a research level. Additionally, a lot of research level signal processing deals with finding optimal processes for determining the eigenvectors and eigenvaluesof random matrices, as well as determining error bounds for those estimates. This uses measure all the time.

Finally, if you're in the optimization and non-linear programming side of EE, L_p spaces and inner product spaces, and dual spaces (which you'll encounter in any decent measure course) are all super helpful.

>>16724591
I'm not saying what you should do, but I've found it helps to be capable of a little bit of programming in pure math as well.
My favourite trick when faced with a difficult recurrence relation is to compute many terms by simple memoization and then perform linear or log-linear or ... regression against what I guess the rates of growth are.
Of course computer algebra is basically indispensable today.
What do you think about topics like https://www2.math.upenn.edu/~wilf/AeqB.html ?
>>16724672
>were you always good at math?
no
>Do you think you can get better at math?
yes
>Nothing stuck.
It's normal to forget things, but they are supposed to "come back to you" pretty quickly upon studying them a second time.

>>16724688
K-theory is as pure as it gets and it’s essentially about PDEs.

Have you ever heard of or know any (preferably notable or successful) mathematicians that have struggled with 'serious' mental health issues?

>>16725327
André Bloch, Nash, Grothendieck

>>16722505
>alpha [≈] 13.60395°
Shouldn't alpha = arcsin(6/25)?
That's about 13.88654 degrees.

>>16712226 (OP)
Does the notation for derivative suck or am I slow?

A gambler has a 1% chance of jackpot, but only enough money for 20 pulls per day. What percentage of jackpot does he really have per day?

Further, approximately how many days until jackpot?

>>16725561
which notation are you referring to? there are like four

>>16725327
Watch the movie a beautiful mind, cry for a bit, then get back to studying. As a genuine schizo it was inspiring. Love that man

>>16725327
Cantor was put into a mental asylum towards the end of his life. Also Godel thought everybody was trying to poison him.

>>16725327
Gauss hated his children because he was afraid they’d eclipse him.

>>16725558

>>16725558
I don't get what you wanted to ask? Where does the 6/25 came from?

Let F be the field extension of the rationals obtained by adjoining all roots of unity. Does F contain a square root of 2? I think no, but how do we show this?

>>16727383
Sorry ignore that, I'm pretty sure adjoining an 8th root of unity gives a square root of 2. I can see this by looking at the complex numbers since [math] (\sqrt{2}/2)(1+i) + (\sqrt{2}/2)(1-i) = \sqrt{2} [/math]. However, I'm not sure how to prove this "directly" without using complex numbers, does anyone have an idea?

>>16722505

The large yellow circle passes through the centers of the thirteen small circles.

The large blue circle passes through the points of tangency.

(A regular thirteen-gon can be inscribed in the large yellow circle and circumscribed about the large blue circle.)

The angle between the yellow and blue lines is pi/13 radians = 180/13 degrees.
180/13 = 13 + 846153/999999
Which is close to your angle alpha.

The radius of each small circle is sin(pi/13) ≈ 0.2393.
Which is close to your radius of, what appears to be, 0.24.

https://www.wolframalpha.com/input?i=Flatten%5B%7By+%3D%3D+0%2C+x%5E2+%2B+y%5E2+%3D%3D+1%5E2%2C+Table%5B%28x+-+Cos%5B%282+k+%CF%80%29%2F13%5D%29%5E2+%2B+%28y+-+Sin%5B%282+k+%CF%80%29%2F13%5D%29%5E2+%3D%3D+Sin%5B%CF%80%2F13%5D%5E2%2C+%7Bk%2C+13%7D%5D%2C+x%5E2+%2B+y%5E2+%3D%3D+Cos%5B%CF%80%2F13%5D%5E2%2C+y+%3D%3D+Tan%5B%CF%80%2F13%5D+x%7D%5D

>>16726361
A = (0, 0)
B = (1, 0)
C' = Intersect(d, f)

u = length(AB) = 1
v = length(BC') = 6/25

sin(alpha) = v/u = 6/25

The image in post # 16722505 indirectly states, that alpha ≈ beta.
I thought, that they might be equal.
Or that rounding errors caused them to be unequal.
I was going to use the following two websites to see if that's the case. mathworld.wolfram.com/Circle-CircleTangents.html www.wolframalpha.com

But that image indirectly states, that alpha ≠ arcsin(6/25).
Thus I couldn't continue.

>>16727725
>I was going to use the following two websites to see if that's the case.
both
mathworld.wolfram.com/Circle-CircleTangents.html
and
www.wolframalpha.com

>>16727725
Yes. Alpha and beta are not in fact equal. and my hope to be a greek god shattered. The error is 1% at 90 degree. Less for the smaller angle. This is for practical purposes okay. I have some issues with pen and paper constructions because error accumulated by inner and outer tangents using smaller circles. I will try bigger circles next time.

>>16712666
Assuming you those blank spaces are ones, you can transform it into a lower triangle matrix and take the product of the main diagonal, if memory serves.

>>16712666
det(A) is trivially zero if you use the property in pic related.

>>16728364
retard

>>16728675
>>16712739
nigger

>>16728364
No idiot

https://www.wolframalpha.com/input?i=determinant+%5B%5B1%2C1%2C0%2C1%5D%2C%5B1%2C1%2C1%2C0%5D%2C%5B0%2C1%2C1%2C1%5D%2C%5B1%2C0%2C1%2C1%5D%5D

Does anyone know if there is an identity like
[math] d \star \alpha = \mathrm{tr}_{g^{-1}} ( d \alpha ) \,\mathrm{vol}_g [/math]
where [math] \alpha [/math] is a differential 1-form on an oriented Riemannian manifold [math] (M,g) [/math]?
Here [math] \mathrm{vol}_g [/math] is the volume form on M for g, and [math] \star [/math] is the Hodge star operator.
I'm trying to read a paper and it seems to use something like the above identity, but I'm not sure if that's what's being done.

How can I go from
[math]T(X,Y) = \nabla_XY-\nabla_YX - [X,Y][/math]
to
[math]T = (\Gamma_{jk}^i-\Gamma_{kj}^i)\mathrm{d}x^j\otimes\mathrm{d}x^k\otimes\partial_i[/math]
?

>>16728940

>>16728940
Compute the derivatives as usual (there is a table of covariant derivatives for tensor of various degrees and ignore the commutator since christoffel symbols implies affine connection and no torsion.

>>16728967
>christoffel symbols implies affine connection and no torsion
That is wrong: Christoffel symbols can be used for any affine connection, even one with nonzero torsion.

I tried looking a the imo problems (just was curious) and was glad to remember that as a grown man no matter how much I try and study I won't be as smart as talented people but that those people will be replaced by ai so it'll all be ok

>>16728796
>>16712739

>>16729122
If the matrix in >>16712666 was supposed to have all 1's then it wouldn't have been written the way it was. Use your brain

Hello math addicts, I want to learn high school to professor level math subjects to satisfy my curiosity and have fun. My problem is that I am aimless in where to go and learn these things and the fear that I am too old engage in or let alone use this knowledge for a future career. What shall I do big brained bros?

I have two math degrees, but doing math IRL is way too fucking hard.
I work in AI/ML and am always trying to find clever ways to do some kind of math-informed transformation but it's honestly such a hopeless pursuit.
Feels like 99% of math is just circlejerking because the moment you try to apply it to something IRL you end up with intractable problems.

>>16729182
If this just commonplace for others than I'll just try books, youtube, and lurk on this general. Godspeed matheads

>>16729084
When it comes down to it, it's funny how we praise being smart, but being smart boils down to have a better working memory and being able to bruteforce eliminate candidate solutions faster.
That's it. It's all pretty banal.
To me it seems inevitable that AI will eventually replace us, because if the thinking machine can do all of that faster, then what use are we to anyone.

>>16728959
What textbook is this?

>>16729323
I typed it up myself in LaTeX

>>16729148
Pick a general area of math you think is neat based off of something like numberphile videos or something like that. Linear algebra, number theory, analysis, whatever. Then, find the limits of your own knowledge of what you do know (i.e., maybe you remember some trig identities, but do you know calculus? Maybe you remember geometry, but how do you feel about taking inverses of real valued functions? Etc.) Then, try and bridge the gap slowly between yourself and whatever goal you have. If you want to learn linear algebra, look up a final exam from some class somewhere online, and study a textbook on your own until you find yourself able to answer the questions on said exam. You can repeat this process for more and more subjects as you go on. Also "professor level math" implies you want a PhD level knowledge. In which case, get a PhD.

>>16727620

What is the general way to make a function f(x,y) that consists of drawing one or more closed curves (not necessarily well-behaved, just as long as they're closed) in the xy plane and letting the function be 0 in the exterior and 1 in the interior or interiors?
Sure, it could be smooth curves like the image but it could also be voronoi cells, or polygons, or anything.

>>16715708
I'm probably going to be homeless. I figure I'll get a math textbook and a notebook and do math all day while I beg for food, but the voices I hear say "everyone hates people like that".

Not a trollpost :/

>>16712226 (OP)
I BTFO'd every single nerd in my class at maths while being the exact opposite of a nerd, bullying nerds, and smoking hash during the break, how does that make you feel?

>>16715918
It is not only natural in c, it is also natural in X
Do you want people to omit the X too??

>>16731978
Yes? It’s just the evaluation map of the product/internal object adjunction in Cat.

petition to rename "category theory" to "monoidoid theory"

>>16731964
It means you're both cool and good at math. It can happen

>>16732108
Lol, an actually good math joke

What's an example of a noncompact boundaryless 3-manifold that isn't homeomorphic to [math] X \times \mathbb{R} [/math] for some 2-manifold X?

>>16731816
For one simple (i.e. non-self-crossing) closed curve, just find a homeomorphism of the plane that changes it to the unit circle. I'll leave the rest for you to figure out